Activity and efficiency of the building sector in Morocco: A review of status and measures in Ifrane

Hamza El Hafdaoui; Ahmed Khallaayoun; Kamar Ouazzani; Hamza El Hafdaoui; Ahmed Khallaayoun; Kamar Ouazzani

doi:10.3934/energy.2023024

AIMS Energy

2023, Volume 11, Issue 3: 454-485. doi: 10.3934/energy.2023024

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Review Special Issues

Activity and efficiency of the building sector in Morocco: A review of status and measures in Ifrane

1.
School of Science & Engineering, Al Akhawayn University in Ifrane, Morocco
2.
National School of Applied Sciences, Sidi Mohamed Ben Abdellah University, Morocco
3.
Higher School of Technology, Sidi Mohamed Ben Abdellah University, Morocco

Received: 12 January 2023 Revised: 22 April 2023 Accepted: 25 April 2023 Published: 11 May 2023

One-third of all greenhouse gas emissions come from the world's building stock while accounting for 40% of global energy use. There is no way to combat global warming or attain energy independence without addressing the inefficiency of the building sector. This sector is the second consumer of electricity after the industrial sector in Morocco and is ranked third emitter after the energy sector and transportation sector. Using Ifrane as a case study, this paper examines and reviews the city's energy use and the initiatives taken to improve building efficiency. The findings showed that, during the analyzed period, i.e., from 2014 to 2022, Ifrane's annual electricity consumption climbed steadily from 35 to 43 GWh. The government of Morocco has implemented effective laws, guidelines and regulations, as well as publicized ways to reduce energy consumption and increase energy efficiency. However, gathered data and survey results revealed opportunities and challenges for enhancing Ifrane's efficient energy use.

The study also evaluates government programs, codes/standards and related actions for the improvement of household energy efficiency. As part of the review, the available literature was analyzed to assess the effectiveness of energy behavior and awareness, the impact of an economical and sustainable building envelope, the impact of building retrofitting programs, the importance of energy-performing devices and appliances, the adoption of smart home energy management systems, the integration of renewable energies for on-site clean energy generation and the role of policies and governance in the building sector in Ifrane. A benchmark evaluation and potential ideas are offered to guide energy policies and improve energy efficiency in Ifrane and other cities within the same climate zone.

Keywords:

Citation: Hamza El Hafdaoui, Ahmed Khallaayoun, Kamar Ouazzani. Activity and efficiency of the building sector in Morocco: A review of status and measures in Ifrane[J]. AIMS Energy, 2023, 11(3): 454-485. doi: 10.3934/energy.2023024

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Abstract

1. Introduction

The Schrödinger-Virasoro algebra is an infinite-dimensional Lie algebra that was introduced (see, e.g., [10]) in the context of non-equilibrium statistical physics. In [21], the author give a representation of the Schrödinger-Virasoro algebra by using vertex algebras, and introduced an extension of the Schrödinger-Virasoro algebra. To be precise, for $\varepsilon\in\{0,\frac{1}{2}\}$ , the Schrödinger-Virasoro algebra $\mathcal{SV}(\varepsilon)$ is a Lie algebra with the $\mathbb{C}$ basis

$\begin{equation*} \{L_{i},H_{j},I_{i}|i\in \mathbb{Z},j\in \varepsilon+\mathbb{Z}\} \end{equation*}$

and Lie brackets

$\begin{eqnarray*} &&[L_{m},L_{n}] = (m-n)L_{m+n}, \\ &&[L_{m},H_{n}] = (\frac{1}{2}m-n)H_{m+n}, \\ &&[L_{m}, I_{n}] = -nI_{m+n}, \\ &&[H_{m},H_{n}] = (m-n)I_{m+n}, \\ &&[H_{m},I_{n}] = [I_{m},I_{n}] = 0. \end{eqnarray*}$

The Lie algebra $\mathcal{SV}(\frac{1}{2})$ is called the original Schrödinger-Virasoro algebra, and $\mathcal{SV}(0)$ is called the twisted Schrödinger-Virasoro algebra. Recently, the theory of the structure and representations of both original and twisted Schrödinger-Virasoro algebra has been investigated in a series of studies. For instance, the Lie bialgebra structures, (bi)derivations, automorphisms, 2-cocycles, vertex algebra representations and Whittaker modules were investigated in [9,11,14,15,21].

Post-Lie algebras were introduced around 2007 by B. Vallette [22], who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Post-Lie algebras have arose the interest of a great many authors, see [4,5,12,13]. One of the most important problems in the study of post-Lie algebras is to find the post-Lie algebra structures on the (given) Lie algebras. In [13,18,20], the authors determined all post-Lie algebra structures on $sl(2, \mathbb{C})$ of special linear Lie algebra of order $2$ , the Witt algebra and the W-algebra $W(2,2)$ respectively.

In this paper, we shall study the graded post-Lie algebra structures on the Schrödinger-Virasoro algebra. We only study the twisted Schrödinger-Virasoro algebra $\mathcal{SV}(0)$ , the case for the original Schrödinger-Virasoro algebra $\mathcal{SV}(\frac{1}{2})$ is similar. For convenience we denote $\mathcal{S} = \mathcal{SV}(0)$ . It should be noted that the commutative post-Lie algebra structures on $\mathcal{S}$ already are given by [11], we will consider the general case.

Throughout this paper, we denote by $\mathbb{Z}$ the set of all integers. For a subset $S$ of $\mathbb{Z}$ and a fixed integer $k$ , denote $S^* = S\setminus \{0\}$ , $S_{>k} = \{t\in S|t>k\}$ , $S_{<k} = \{t\in S|t<k\}$ , $S_{\geqslant k} = \{t\in S|t\geqslant k\}$ and $S_{\leqslant k} = \{t\in S|t\leqslant k\}$ . We assume that the field in this paper always is the complex number field $\mathbb{C}$ .

The paper is organized as follows. In Section 2, we give general results on post-Lie algebras and some lemmas which will be used to our proof. In Section 3, we completely characterize the graded post-Lie algebra structures on Schrödinger-Virasoro algebra $\mathcal{S}$ . In Section 4, by using the post-Lie algebra structures we characterize the forms of the homogeneous Rota-Baxter operator on $\mathcal{S}$ .

2. Preliminaries

We will give the essential definitions and results as follows.

Definition 2.1. A post-Lie algebra $(V,\triangleright,[,])$ is a vector space V over a field $k$ equipped with two $k$ -bilinear products $x\triangleright y$ and $[x,y]$ satisfying that $(V,[,])$ is a Lie algebra and

$\begin{eqnarray} &&[x,y]\triangleright z = x\triangleright(y\triangleright z)-y\triangleright(x\triangleright z)-\langle x,y\rangle \triangleright z, \end{eqnarray}$

(1)

$\begin{eqnarray} &&x \triangleright[y,z] = [x \triangleright y,z]+[y,x\triangleright z] \end{eqnarray}$

(2)

for all $x,y\in V$ , where $\langle x,y\rangle = x\triangleright y-y\triangleright x$ . We also say that $(V,\triangleright, [,])$ is a post-Lie algebra structure on the Lie algebra $(V,[,])$ . If a post-Lie algebra $(V,\triangleright, [,])$ satisfies $x\triangleright y = y\triangleright x$ for all $x,y\in V$ , then it is called a commutative post-Lie algebra.

Suppose that $(L,[,])$ is a Lie algebra. Two post-Lie algebras $(L,[,],\triangleright_{1})$ and $(L,[,],\triangleright_{2})$ on the Lie algebra $L$ are called to the isomorphic if there is an automorphism $\tau$ of the Lie algebra $(L,[,])$ satisfies

$\tau(x\triangleright_{1}y) = \tau(x)\triangleright_{2}\tau(y), \forall x,y\in L.$

Remark 1. The left multiplications of the post-Lie algebra $(V,[,],\triangleright)$ are denoted by $\mathcal{L}$ , i.e., we have $\mathcal{L}(x)(y) = x\triangleright y$ for all $x,y\in V$ . By (2), we see that all operator $\mathcal{L}(x)$ are Lie algebra derivations of the Lie algebra (V, [, ]).

Lemma 2.2. [15] Denote by $Der(\mathcal{S})$ and by Inn $(\mathcal{S})$ the space of derivations and the space of inner derivations of $\mathcal{S}$ respectively. Then

$\begin{equation*} Der(\mathcal{S}) = Inn(\mathcal{S})\oplus\mathbb{C}D_{1}\oplus\mathbb{C}D_{2}\oplus\mathbb{C}D_{3} \end{equation*}$

where $D_1,D_2, D_3$ are outer derivations defined by

$\begin{eqnarray*} &&D_{1}(L_{n}) = 0,D_{1}(H_{n}) = H_{n},D_{1}(I_{n}) = 2I_{n}, \\ &&D_{2}(L_{n}) = nI_{n},D_{2}(H_{n}) = 0,D_{2}(I_{n}) = 0, \\ &&D_{3}(L_{n}) = I_{n},D_{3}(H_{n}) = 0,D_{3}(I_{n}) = 0. \end{eqnarray*}$

3. The graded post-Lie algebra structures on the Schrödinger-Virasoro algebra

Since the Schrödinger-Virasoro algebra $\mathcal{S}$ is graded, we suppose that the post-Lie algebra structure on the Schrödinger-Virasoro algebra $\mathcal{S}$ to be graded. Namely, we mainly consider the post-Lie algebra structure on Schrödinger-Virasoro algebra $\mathcal{S}$ which satisfies

$\begin{eqnarray} L_{m}\triangleright L_{n}& = &\phi(m,n)L_{m+n}, \end{eqnarray}$

(3)

$\begin{eqnarray} L_{m}\triangleright H_{n}& = &\varphi(m,n)H_{m+n}, \end{eqnarray}$

(4)

$\begin{eqnarray} L_{m}\triangleright I_{n}& = &\chi(m,n)I_{m+n}, \end{eqnarray}$

(5)

$\begin{eqnarray} H_{m}\triangleright L_{n}& = &\psi(m,n)H_{m+n}, \end{eqnarray}$

(6)

$\begin{eqnarray} H_{m}\triangleright H_{n}& = &\xi(m,n)I_{m+n}, \end{eqnarray}$

(7)

$\begin{eqnarray} I_{m}\triangleright L_{n}& = &\theta(m,n)I_{m+n}, \end{eqnarray}$

(8)

$\begin{eqnarray} H_{m}\triangleright I_{n}& = &I_{m}\triangleright H_{n} = I_{m}\triangleright I_{n} = 0, \end{eqnarray}$

(9)

for all $m,n\in\mathbb{Z}$ , where $\phi$ , $\varphi$ , $\chi$ , $\psi$ , $\xi$ , $\theta$ are complex-valued functions on $\mathbb{Z}\times\mathbb{Z}$ .

We start with the crucial lemma.

Lemma 3.1. There exists a graded post-Lie algebra structure on $\mathcal{S}$ satisfying (3)-(9) if and only if there are complex-valued functions $f, g, h$ on $\mathbb{Z}$ and complex numbers $a,\mu$ such that

$\begin{eqnarray} &&\phi(m,n) = (m-n)f(m), \end{eqnarray}$

(10)

$\begin{eqnarray} &&\varphi(m,n) = (\frac{m}{2}-n)f(m)+\delta_{m,0}\mu, \end{eqnarray}$

(11)

$\begin{eqnarray} &&\chi(m,n) = -nf(m)+2\delta_{m,0}\mu, \end{eqnarray}$

(12)

$\begin{eqnarray} &&\psi(m,n) = -(\frac{n}{2}-m)h(m), \end{eqnarray}$

(13)

$\begin{eqnarray} &&\xi(m,n) = (m-n)h(m), \end{eqnarray}$

(14)

$\begin{eqnarray} &&\theta(m,n) = mg(m)+\delta_{m,0}na, \end{eqnarray}$

(15)

$\begin{eqnarray} &&(m-n)(f(m+n)(1+f(m)+f(n))-f(n)f(m)) = 0, \end{eqnarray}$

(16)

$\begin{eqnarray} &&(m-n)\delta_{m+n,0}\mu(1+f(m)+f(n)) = 0, \end{eqnarray}$

(17)

$\begin{eqnarray} &&(\frac{m}{2}-n)(h(m+n)(1+f(m)+h(n))-f(m)h(n)) = 0, \end{eqnarray}$

(18)

$\begin{eqnarray} &&n\delta_{m+n,0}a(1+f(m)+g(n)) = 0, \end{eqnarray}$

(19)

$\begin{eqnarray} && n(m+n)(g(m+n)(1+f(m)+g(n))-f(m)g(n))\\ &&\ \ \ \ = \delta_{n,0}m^{2}a(f(m)-g(m)), \end{eqnarray}$

(20)

$\begin{eqnarray} &&(m-n)\delta_{m+n,0}a(1+h(m)+h(n)) = 0, \end{eqnarray}$

(21)

$\begin{eqnarray} &&(m-n)(g(m+n)(1+h(m)+h(n))-h(m)h(n)) = 0. \end{eqnarray}$

(22)

Proof. Suppose that there exists a graded post-Lie algebra structure satisfying (3)-(9) on $\mathcal{S}$ . By Remark 1, $\mathcal{L}(x)$ is a derivation of $\mathcal{S}$ . It follows by Lemma 2.2 that there are a linear map $\psi$ from $\mathcal{S}$ into itself and linear functions $\alpha, \beta, \gamma$ on $\mathcal{S}$ such that

$\begin{eqnarray*} x\triangleright y& = &(ad\psi(x)+\alpha(x)D_{1}+\beta(x)D_{2}+\gamma(x)D_{3})(y) \\ & = &[\psi(x),y]+\alpha(x)D_{1}(y)+\beta(x)D_{2}(y)+\gamma(x)D_{3}(y) \end{eqnarray*}$

where $D_i, i = 1,2,3$ are given by Lemma 2.2. This, together with (3)-(9), gives that

$\begin{eqnarray} L_{m}\triangleright L_{n}& = &[\psi(L_{m}),L_{n}]+\beta(L_{m})nI_{n}+\gamma(L_{m})I_{n} = \phi(m,n)L_{m+n}, \end{eqnarray}$

(23)

$\begin{eqnarray} L_{m}\triangleright H_{n}& = &[\psi(L_{m}),H_{n}]+\alpha(L_{m})H_{n} = \varphi(m,n)H_{m+n}, \end{eqnarray}$

(24)

$\begin{eqnarray} L_{m}\triangleright I_{n}& = &[\psi(L_{m}),I_{n}]+\alpha(L_{m})2I_{n} = \chi(m,n)I_{m+n}, \end{eqnarray}$

(25)

$\begin{eqnarray} H_{m}\triangleright L_{n}& = &[\psi(H_{m}),L_{n}]+\beta(H_{m})nI_{n}+\gamma(H_{m})I_{n} = \psi(m,n)H_{m+n}, \end{eqnarray}$

(26)

$\begin{eqnarray} H_{m}\triangleright H_{n}& = &[\psi(H_{m}),H_{n}]+\alpha(H_{m})H_{n} = \xi(m,n)I_{m+n}, \end{eqnarray}$

(27)

$\begin{eqnarray} H_{m}\triangleright I_{n}& = &[\psi(H_{m}),I_{n}]+\alpha(H_{m})2I_{n} = 0, \end{eqnarray}$

(28)

$\begin{eqnarray} I_{m}\triangleright L_{n}& = &[\psi(I_{m}),L_{n}]+\beta(I_{m})nI_{n}+\gamma(I_{m})I_{n} = \theta(m,n)I_{m+n}, \end{eqnarray}$

(29)

$\begin{eqnarray} I_{m}\triangleright H_{n}& = &[\psi(I_{m}),H_{n}]+\alpha(I_{m})H_{n} = 0, \end{eqnarray}$

(30)

$\begin{eqnarray} I_{m}\triangleright I_{n}& = &[\psi(I_{m}),I_{n}]+\alpha(I_{m})2I_{n} = 0. \end{eqnarray}$

(31)

Let

$\begin{eqnarray*} \psi(L_{m})& = \sum\nolimits_{i\in \mathbb{Z}}a^{(m)}_{i}L_{i}+\sum\nolimits_{i\in \mathbb{Z}}b^{(m)}_{i}H_{i}+\sum\nolimits_{i\in \mathbb{Z}}c^{(m)}_{i}I_{i}, \\ \psi(H_{m})& = \sum\nolimits_{i\in \mathbb{Z}}d^{(m)}_{i}L_{i}+\sum\nolimits_{i\in \mathbb{Z}}e^{(m)}_{i}H_{i}+\sum\nolimits_{i\in \mathbb{Z}}f^{(m)}_{i}I_{i}, \\ \psi(I_{m})& = \sum\nolimits_{i\in \mathbb{Z}}g^{(m)}_{i}L_{i}+\sum\nolimits_{i\in \mathbb{Z}}h^{(m)}_{i}H_{i}+\sum\nolimits_{i\in \mathbb{Z}}x^{(m)}_{i}I_{i} \end{eqnarray*}$

where $a^{(m)}_{i}, b^{(m)}_{i}, c^{(m)}_{i}, d^{(m)}_{i}, e^{(m)}_{i}, f^{(m)}_{i}, g^{(m)}_{i}, h^{(m)}_{i}, x^{(m)}_{i}\in\mathbb{C}$ for all $i\in\mathbb{Z}$ . Then by (23)-(31), similar to the proof of [18], we obtain that (10)-(22) hold.

The "if'' part is a direct checking. The proof is completed.

Lemma 3.2. Let $f,g,h$ be complex-valued functions on $\mathbb{Z}$ and $\mu,a\in \mathbb{C}$ satisfying (18) and (20). Then we have

$\begin{equation} g(n),h(n)\in \{0,-1\} \mathit{\text{for every}} n\neq 0. \end{equation}$

(32)

Proof. By letting $m = 0$ in (18) and (20), respectively, we have $nh(n)(1+h(n)) = 0$ and $n^{2}g(n)(1+g(n)) = 0$ . This implies (32).

Lemma 3.3. Let $f,g,h$ be complex-valued functions on $\mathbb{Z}$ and $\mu,a$ be complex numbers satisfying (17)-(22). If $f(\mathbb{Z}) = 0$ , then we have $\mu = a = 0$ and

$g(\mathbb{Z}) = h(\mathbb{Z}) = 0 \mathit{\text{or}}g(\mathbb{Z}) = h(\mathbb{Z}) = -1.$

Proof. Since $f(\mathbb{Z}) = 0$ , we take $m = -n = 1$ in (17) and (19) we have $\mu = 0$ and

$\begin{equation} a(1+g(-1)) = 0. \end{equation}$

(33)

By letting $n = 0$ and $m = -1$ in (20) we deduce that $ag(-1) = 0$ . This, together with (33), implies $a = 0$ . As $\mu = a = 0$ , so Equations (18), (20) and (22) become to

$\begin{eqnarray} &&(\frac{m}{2}-n)(h(m+n)(1+h(n)) = 0, \end{eqnarray}$

(34)

$\begin{eqnarray} &&n(m+n)(g(m+n)(1+g(n)) = 0, \end{eqnarray}$

(35)

$\begin{eqnarray} &&(m-n)(g(m+n)-h(m)h(n)+h(m)g(m+n)+h(n)g(m+n)) = 0. \end{eqnarray}$

(36)

We now prove the following four claims:

Claim 1. If $h(1) = 0$ , then $h(\mathbb{Z}) = 0$ .

By (34) with $n = 1$ we see that $h(m+1) = 0$ for all $m\neq 2$ . It follows that $h(\mathbb{Z}\backslash \{3\}) = 0$ . Since $h(2) = 0$ , by taking $n = 2, m = 1$ in (34) we have $-\frac{3}{2}h(3) = 0$ , which implies $h(3) = 0$ . We obtain $h(\mathbb{Z}) = 0$ .

Claim 2. If $h(1) = -1$ , then $h(\mathbb{Z}) = -1$ .

By (34) with $m+n = 1$ we see that $h(n) = -1$ for all $n\in \mathbb{Z}$ with $\frac{1-3n}{2}\neq 0$ . This means that $h(\mathbb{Z}) = -1$ .

Claim 3. If $g(1) = 0$ , then $g(\mathbb{Z}^*) = 0$ .

By (35) with $n = 1$ we see that $g(m+1) = 0$ for all $m\neq -1$ . It follows that $g(\mathbb{Z^*}) = 0$ .

Claim 4. If $g(1) = -1$ , then $g(\mathbb{Z}^*) = -1$ .

By (35) with $m+n = 1$ we see that $g(n) = -1$ for all $n\neq 0$ . This means that $g(\mathbb{Z^*}) = -1$ .

Now we consider the values of $h(1)$ and $g(1)$ according to (32).

Case i. If $h(1) = g(1) = 0$ , then by Claims 1 and 3 we have $h(\mathbb{Z}) = 0$ and $g(\mathbb{Z}^*) = 0$ . According to (36) with $n = -1$ and $m = 1$ we know $g(0) = 0$ . This means that $g(\mathbb{Z}) = 0$ .

Case ii. If $h(1) = g(1) = -1$ , then by Claims 2 and 4 we have $h(\mathbb{Z}) = -1$ and $g(\mathbb{Z}^*) = -1$ . According to (36) with $n = -1$ and $m = 1$ we see that $1+g(0) = 0$ and so that $g(0) = -1$ . This implies $g(\mathbb{Z}) = -1$ .

Case iii. If $h(1) = 0$ , $g(1) = -1$ , then we will get a contradiction. In fact, by Claims 1 and 4, we have $h(\mathbb{Z}) = 0$ and $g(\mathbb{Z}^*) = -1$ . From (36) with $m = 2, n = -1$ we see that $g(1) = 0$ which contradicts $g(1) = -1$ .

Case iv. If $h(1) = -1$ , $g(1) = 0$ , then we will also get a contradiction. In fact, by Claims 2 and 3, we have $h(\mathbb{Z}) = -1$ and $g(\mathbb{Z^*}) = 0$ . From (36) with with $m = 2$ , $n = -1$ we see that $g(1) = -1$ which contradicts $g(1) = 0$ . The proof is completed.

Lemma 3.4. Let $f,g,h$ be complex-valued functions on $\mathbb{Z}$ and $\mu,a$ be complex numbers satisfying (17)-(22). If $f{(\mathbb{Z}_{\geqslant2})} = -1, f{(\mathbb{Z}_{\leqslant 1})} = 0$ , then $\mu = a = 0$ and $g$ , $h$ must satisfy one of the following forms:

(i) $g{(\mathbb{Z})} = h{(\mathbb{Z})} = 0$ ;

(ii) $g{(\mathbb{Z})} = h{(\mathbb{Z})} = -1$ ;

(iii) $h{(\mathbb{Z}_{\leqslant 0})} = 0$ , $h{(\mathbb{Z}_{\geqslant 1})} = -1$ and

$g{(\mathbb{Z}_{\leqslant-1})} = 0$ , $g{(\mathbb{Z}_{\geqslant 1})} = -1$ , $g(0) = \hat{\lambda}$ for some $\hat{\lambda}\in \mathbb{C}$ .

Proof. By $f{(\mathbb{Z}_{\geqslant2})} = -1, f{(\mathbb{Z}_{\leqslant 1})} = 0$ , similar to the proof of Lemma 3.3, we know $\mu = a = 0$ . From this, we have by (18), (20) and (22) that

$\begin{eqnarray} &&h(m+n)(h(n)+1) = 0 \text{ if } m\leqslant 1, \frac{m}{2}-n\neq 0, \end{eqnarray}$

(37)

$\begin{eqnarray} &&g(m+n)(g(n)+1) = 0 \text{ if } m\leqslant 1, n\neq 0, m+n\neq 0, \end{eqnarray}$

(38)

$\begin{eqnarray} &&g(m+n)(1+h(m)+h(n)) = h(m)h(n) \text{ if } m\neq n. \end{eqnarray}$

(39)

We first prove the following six claims:

Claim 1. If $h(1) = 0$ , then $h(\mathbb{Z}) = 0$ .

By (37) with $n = 1$ we see that $h(m+1) = 0$ for all $\frac{m}{2}-1\neq 0$ with $m\leqslant 1$ . Hence, we deduce that $h(\mathbb{Z}_{\leqslant 2}) = 0$ . Note that $h(2) = 0$ , by (37) with $n = 2$ we see that $h(m+2) = 0$ for all $\frac{m}{2}-2\neq 0$ with $m\leqslant 1$ . We now have $h(\mathbb{Z}_{\leqslant 3}) = 0$ . If we repeat this process, we see that $h(\mathbb{Z}_{\leqslant k}) = 0$ for all $k = 1,2,3,\cdots$ . Note that $\bigcup_{k\geqslant 1}(\mathbb{Z}_{\leqslant k}) = \mathbb{Z}$ , so one has $h(\mathbb{Z}) = 0$ .

Claim 2. If $h(-1) = -1$ , then $h(\mathbb{Z}) = -1$ .

By (37) with $m+n = -1$ we see that $h(n) = h(-1-m) = -1$ for all $\frac{3m}{2}+1\neq 0$ with $m\leqslant 1$ . This deduces that $h(\mathbb{Z}_{\geqslant -2}) = -1$ . Note that $h(-2) = -1$ , by (37) with $m+n = -2$ we see that $h(-m-2) = -1$ for all $\frac{3m}{2}+2\neq 0$ with $m\leqslant 1$ . Thus, $h(\mathbb{Z}_{\geqslant -3}) = -1$ . If we repeat this process, we see that $h(\mathbb{Z}_{\geqslant k}) = -1$ for all $k = -1,-2,-3,\cdots$ . Note that $\bigcup_{k\leqslant -1}(\mathbb{Z}_{\geqslant k}) = \mathbb{Z}$ , so one has $h(\mathbb{Z}) = -1$ .

Claim 3. If $h(1) = -1$ , then $h(\mathbb{Z}_{\geqslant 1}) = -1$ .

By (37) with $m+n = 1$ we see that $h(n) = h(1-m) = -1$ for all $\frac{3m}{2}-1\neq0$ with $m\leqslant 1$ . This implies $h(\mathbb{Z}_{\geqslant 1}) = -1$ .

Claim 4. If $h(-1) = 0$ , then $h(\mathbb{Z}_{\leqslant 0}) = 0$ .

By (37) with $n = -1$ we see that $h(m-1) = 0$ for all $m\neq -2$ with $m\leqslant 1$ . It follows that $h(\mathbb{Z}_{\leqslant 0}\setminus \{-3\}) = 0$ . Let $m = -1, n = -2$ in (37), from $\frac{m}{2}\neq n$ we have $h(-3) = 0$ . Therefore, we get $h(\mathbb{Z}_{\leqslant 0}) = 0$ .

Next, similar to Claims 1 and 3, we from (38) obtain the following claims.

Claim 5. If $g(1) = 0$ , then $g(\mathbb{Z}^*) = 0$ .

Claim 6. If $g(1) = -1$ , then $g(\mathbb{Z}_{\geqslant 1}) = -1$ .

Now we discuss the values of $h(1)$ and $h(-1)$ . By (32), $h(1), h(-1)\in \{-1, 0\}$ .

Case i. When $h(1) = 0$ .

By Claim 1 we have $h(\mathbb{Z}) = 0$ . According to (39), one has $g(m+n) = 0$ for any $m,n\in \mathbb{Z}$ with $m\neq n$ . This implies $g(\mathbb{Z}) = 0$ .

Case ii. When $h(-1) = -1$ .

By Claim 2 we have $h(\mathbb{Z}) = -1$ . According to (39), one has $g(m+n) = -1$ for any $m,n\in \mathbb{Z}$ with $m\neq n$ . This implies $g(\mathbb{Z}) = -1$ .

Case iii. When $h(1) = -1$ and $h(-1) = 0$ .

By Claims 3 and 4 we have $h(\mathbb{Z}_{\leqslant 0}) = 0$ and $h(\mathbb{Z}_{\geqslant 1}) = -1$ . This, together with (39), yields $g(m+n) = 0$ for any $m,n\in \mathbb{Z}$ with $m,n\leqslant 0$ and $m\neq n$ , and $g(m+n) = -1$ for any $m,n\in \mathbb{Z}$ with $m,n\geqslant 1$ and $m\neq n$ . Consequently, we obtain $g{(\mathbb{Z}_{\leqslant-1})} = 0$ and $g{(\mathbb{Z}_{\geqslant 3})} = -1$ . By (32), $g(1)\in \{-1, 0\}$ . If $g(1) = 0$ , then Claim 5 tells us that $g(\mathbb{Z}^*) = 0$ which contracts $g{(\mathbb{Z}_{\geqslant 3})} = -1$ . Therefore, we have $g(1) = -1$ . From this with Claim 6 we have $g{(\mathbb{Z}_{\geqslant 1})} = -1$ . Let $g(0) = \hat{\lambda}$ for some $\hat{\lambda}\in \mathbb{C}$ .

It is easy to check that the values of $g$ given in Cases i-iii above are consistent with (38). They give the conclusions (i), (ii) and (iii) respectively. The proof is completed.

Lemma 3.5. Let $f,g,h$ be complex-valued functions on $\mathbb{Z}$ and $\mu,a$ be complex numbers satisfying (17)-(22). If $f{(\mathbb{Z}_{>0})} = -1,f{(\mathbb{Z}_{<0})} = 0 \,$ and $\, f{(0)} = c$ for some $c\in \mathbb{C}$ , then there are $\lambda, \hat{\tau}\in \mathbb{C}$ such that $\mu, a$ , $g$ , $h$ must be one of the following forms:

(i) $a = 0$ , $\mu\in \mathbb{C}$ and $g{(\mathbb{Z})} = h{(\mathbb{Z})} = 0$ ;

(ii) $a = 0$ , $\mu\in \mathbb{C}$ and $g{(\mathbb{Z})} = h{(\mathbb{Z})} = -1$ ;

(iii) $\mu \in \mathbb{C}$ , $h{(\mathbb{Z}_{> 0})} = -1$ , $h{(\mathbb{Z}_{<0})} = 0$ , $h(0) = \lambda$ and $g{(\mathbb{Z}^*_{\geqslant k})} = -1$ , $g{(\mathbb{Z}^*_{\leqslant k-1})} = 0$

for some $k\in \{-2,-1,1,2,3\}$ , $g(0) = \hat{\tau}$ and $a = 0$ when $k\neq 1$ ;

(iv) $a = 0$ , $\mu\in \mathbb{C}$ and $h{(\mathbb{Z}_{\geqslant t})} = -1$ , $h{(\mathbb{Z}_{\leqslant t-1})} = 0$ for some $t\in \mathbb{Z}\setminus\{0,1\}$ and

$g{(\mathbb{Z}_{\geqslant s})} = -1$ , $g{(\mathbb{Z}_{\leqslant s-1})} = 0$ for some $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ .

Proof. Take $m = -n\neq 0$ in (18)-(22), one has

$\begin{eqnarray} &&h(0)(1+f(-n)+h(n)) = f(-n)h(n), \text{ for all } n\neq 0, \end{eqnarray}$

(40)

$\begin{eqnarray} &&a(1+f(-n)+g(n)) = 0, \text{ for all } n\neq 0, \end{eqnarray}$

(41)

$\begin{eqnarray} &&a(1+h(-n)+h(n)) = 0, \text{ for all } n\neq 0, \end{eqnarray}$

(42)

$\begin{eqnarray} &&g(0)(1+h(-n)+h(n)) = h(-n)h(n), \text{ for all } n\neq 0. \end{eqnarray}$

(43)

Note that $f{(\mathbb{Z}_{>0})} = -1,f{(\mathbb{Z}_{<0})} = 0 \,$ and $\, f{(0)} = c$ for some $c\in \mathbb{C}$ . It is follows by (18), (20) and (22) that

$\begin{eqnarray} &&h(n)(h(m+n)+1) = 0 \text{ for all } m > 0, \frac{m}{2}-n\neq 0; \end{eqnarray}$

(44)

$\begin{eqnarray} &&h(m+n)(h(n)+1) = 0 \text{ for all } m < 0, \frac{m}{2}-n\neq 0; \end{eqnarray}$

(45)

$\begin{eqnarray} &&g(n)(g(m+n)+1) = 0 \text{ for all } m > 0, n\neq 0, m+n\neq 0; \end{eqnarray}$

(46)

$\begin{eqnarray} &&g(m+n)(g(n)+1) = 0 \text{ for all } m < 0, n\neq 0, m+n\neq 0; \end{eqnarray}$

(47)

$\begin{eqnarray} &&g(m+n)(1+h(m)+h(n)) = h(m)h(n) \text{ for all } m\neq n. \end{eqnarray}$

(48)

For any $t\in \mathbb{Z}^*$ , we first prove some claims as follows.

Claim 1. If $h(t) = 0$ , then $h(\mathbb{Z}_{\leqslant t}) = 0$ .

In fact, by (44) with $n = t-m$ we deduce $h(t-m) = 0$ for all $m>0$ with $m\neq \frac{2}{3}t$ . This implies $h(\mathbb{Z}_{\leqslant t}\setminus\{\frac{1}{3}t\}) = 0$ . On the other hand, by (45) with $n = t$ we see that $h(m+t) = 0$ for all $m<0$ with $m\neq 2t$ . This gives that $h(\mathbb{Z}_{\leqslant t}\setminus\{3t\}) = 0$ . Clearly, $3t\neq \frac{1}{3}t$ since $t\neq 0$ . Thereby, we obtain $h(\mathbb{Z}_{\leqslant t}) = 0$ .

Claim 2. If $h(t) = -1$ , then $h(\mathbb{Z}_{\geqslant t}) = -1$ .

This proof is similar to Claim 1 by using (44) and (45). Also, similar to Claims 1 and 2, by (46) and (47) we can obtain the following two claims:

Claim 3. If $g(t) = 0$ , then $g(\mathbb{Z}^*_{\leqslant t}) = 0$ .

Claim 4. If $g(t) = -1$ , then $g(\mathbb{Z}^*_{\geqslant t}) = -1$ .

According to (32), by Claims 1 and 2, $h$ must be one of the following forms:

(1) $h(\mathbb{Z}^{*}) = 0$ ;

(2) $h(\mathbb{Z}^{*}) = -1$ ;

(3) $h(\mathbb{Z}_{>0}) = -1$ , $h(\mathbb{Z}_{<0}) = 0$ and $h(0) = \lambda$ for some $\lambda\in \mathbb{C}$ ;

(4) $h(\mathbb{Z}_{\geqslant t}) = -1$ , $h(\mathbb{Z}_{\leqslant t-1}) = 0$ for some $t\in \mathbb{Z}\setminus\{0,1\}$ .

In view of the above result, the next proof will be divided into the following cases.

Case i. When $h(\mathbb{Z}^{*}) = 0$ .

By taking $n = 1$ in (40), one has $h(0) = 0$ . Hence we see that $h(\mathbb{Z}) = 0$ . This together with (48) yields $g(\mathbb{Z}) = 0$ . In addition, we have by (43) that $a = 0$ .

Case ii. When $h(\mathbb{Z}^{*}) = -1$ .

By taking $n = -1$ in (40), one has $h(0) = -1$ . Hence we see that $h(\mathbb{Z}) = -1$ . This together with (48) yields $g(\mathbb{Z}) = -1$ . In addition, by (43) we get $a = 0$ .

Case iii. When $h(\mathbb{Z}_{>0}) = -1$ , $h(\mathbb{Z}_{<0}) = 0$ and $h(0) = \lambda$ for some $\lambda\in \mathbb{C}$ .

By (48) we see that $g(m+n) = -1$ for any $m,n\in \mathbb{Z}$ with $m,n> 0$ and $m\neq n$ , and $g(m+n) = 0$ for any $m,n\in \mathbb{Z}$ with $m,n< 0$ and $m\neq n$ . Consequently, we obtain $g{(\mathbb{Z}_{\leqslant -3})} = 0$ and $g{(\mathbb{Z}_{\geqslant 3})} = -1$ . By (32), $g(i)\in \{-1, 0\}$ for $i\in \{-2,-1,1,2\}$ . In view of Claims 3 and 4, we can assume that $g(k) = -1$ and $g(k-1) = 0$ for some $k\in \{-2,-1,1,2,3\}$ . In all, by Claims 3 and 4 we get $g(\mathbb{Z}_{\geqslant k}^*) = -1$ and $g(\mathbb{Z}_{\leqslant k-1}^*) = 0$ . Next, if $k\in \{-1,-2\}$ then by taking $n = k$ in (41) we have $a = 0$ ; and if $k\in \{2,3\}$ then by taking $n = k-1$ in (41) we also have $a = 0$ . But $a$ can be arbitrary if $k = 1$ .

Case iv. When $h(\mathbb{Z}_{\geqslant t}) = -1$ , $h(\mathbb{Z}_{\leqslant t-1}) = 0$ for some $t\in \mathbb{Z}\setminus\{0,1\}$ .

Note that $t\geqslant 2$ or $t \leqslant -1$ , then by taking $n = 1$ in (42) we have $a = 0$ . Next, by(48) we see that $g(m+n) = -1$ for any $m,n\in \mathbb{Z}$ with $m,n\geqslant t$ and $m\neq n$ , and $g(m+n) = 0$ for any $m,n\in \mathbb{Z}$ with $m,n\leqslant t-1$ and $m\neq n$ . Consequently, we obtain $g{(\mathbb{Z}_{\leqslant 2t-3})} = 0$ and $g{(\mathbb{Z}_{\geqslant 2t+1})} = -1$ . By (32), $g(i)\in \{-1, 0\}$ for $i\in \{2t-2,2t-1,2t,2t+1\}$ . In view of Claims 3 and 4, we can assume that $g(s) = -1$ and $g(s-1) = 0$ for some $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ . Note that $0\notin \{2t-2,2t-1,2t,2t+1\}$ since $t\neq 0,1$ , by Claims 3 and 4 we get $g(\mathbb{Z}_{\geqslant s}) = -1$ and $g(\mathbb{Z}_{\leqslant s-1}) = 0$ . The proof is completed.

Lemma 3.6. Let $f,g,h$ be complex-valued functions on $\mathbb{Z}$ and $\mu,a$ be complex numbers. Then (17)-(22) hold if and only if $f,g,h,a,\mu$ meet one of the situations listed in Table 2.

Table 2. Values of

$f,g,h$ satisfying (16)-(22), where

$a,\mu\in\mathbb{C}$ ,

$k\in \{-2,-1,1,2,3\}$ ,

$t\in \mathbb{Z}\setminus\{0,1\}$ and

$s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ .

$Cases$	$f(n)$ from Table 1	$a$ , $\mu$	$h(n),g(n)$
$\mathcal{W}_{1}^{\mathcal{P}_{1}}$	$\mathcal{P}_{1}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{1}}$	$\mathcal{P}_{1}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{1}^{\mathcal{P}_{2}}$	$\mathcal{P}_{2}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{2}}$	$\mathcal{P}_{2}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{3}^c}$	$\mathcal{P}^{c}_{3}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{3}^c}$	$\mathcal{P}^{c}_{3}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{3},k}$	$\mathcal{P}^{c}_{3}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z}_{> 0})}=-1$ , $h{(\mathbb{Z}_{< 0})}=0$ and
			$g{(\mathbb{Z}^_{\geqslant k})}=-1$ , $g{(\mathbb{Z}^_{\leqslant k-1})}=0$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{3}^c,k=1}$	$\mathcal{P}^{c}_{3}$	$\forall a$ and $\forall \mu$	$h{(\mathbb{Z}_{> 0})}=-1$ , $h{(\mathbb{Z}_{< 0})}=0$ and
			$g{(\mathbb{Z}_{> 0})}=-1$ , $g{(\mathbb{Z}_{< 0})}=0$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{3}^c,s,t}$	$\mathcal{P}^{c}_{3}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z}_{\geqslant t})}=-1$ , $h{(\mathbb{Z}_{\leqslant t-1})}=0$ and
			$g{(\mathbb{Z}_{\geqslant s})}=-1$ , $g{(\mathbb{Z}_{\leqslant s-1})}=0$
$\mathcal{W}_{1,\mu}^{\mathcal{P}_{4}^c}$	$\mathcal{P}^{c}_{4}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2,\mu}^{\mathcal{P}_{4}^c}$	$\mathcal{P}^{c}_{4}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3,\mu}^{\mathcal{P}^{c}_{4},k}$	$\mathcal{P}^{c}_{4}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z}_{> 0})}=0$ , $h{(\mathbb{Z}_{< 0})}=-1$ and
			$g{(\mathbb{Z}^_{\geqslant k})}=0$ , $g{(\mathbb{Z}^_{\leqslant k-1})}=-1$
$\mathcal{W}_{4,a,\mu}^{\mathcal{P}^{c}_{4},k=1}$	$\mathcal{P}^{c}_{4}$	$\forall a$ and $\forall \mu$	$h{(\mathbb{Z}_{> 0})}=0$ , $h{(\mathbb{Z}_{< 0})}=-1$ and
			$g{(\mathbb{Z}_{> 0})}=0$ , $g{(\mathbb{Z}_{< 0})}=-1$
$\mathcal{W}_{5,\mu}^{\mathcal{P}_{4}^{c},s,t}$	$\mathcal{P}^{c}_{4}$	$a=0$ and $\forall \mu$	$h{(\mathbb{Z}_{\geqslant t})}=0$ , $h{(\mathbb{Z}_{\leqslant t-1})}=-1$ and
			$g{(\mathbb{Z}_{\geqslant s})}=0$ , $g{(\mathbb{Z}_{\leqslant s-1})}=-1$
$\mathcal{W}_{1}^{\mathcal{P}_{5}}$	$\mathcal{P}_{5}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{5}}$	$\mathcal{P}_{5}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3}^{\mathcal{P}_{5}}$	$\mathcal{P}_{5}$	$a=\mu=0$	$h{(\mathbb{Z}_{\leqslant 0})}=0$ , $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
			$g{(\mathbb{Z}_{\leqslant -1})}=0$ , $g{(\mathbb{Z}_{\geqslant 1})}=-1$
$\mathcal{W}_{1}^{\mathcal{P}_{6}}$	$\mathcal{P}_{6}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{6}}$	$\mathcal{P}_{6}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3}^{\mathcal{P}_{6}}$	$\mathcal{P}_{6}$	$a=\mu=0$	$h{(\mathbb{Z}_{\leqslant 0})}=-1$ , $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
			$g{(\mathbb{Z}_{\leqslant -1})}=-1$ , $g{(\mathbb{Z}_{\geqslant 1})}=0$
$\mathcal{W}_{1}^{\mathcal{P}_{7}}$	$\mathcal{P}_{7}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{7}}$	$\mathcal{P}_{7}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3}^{\mathcal{P}_{7}}$	$\mathcal{P}_{7}$	$a=\mu=0$	$h{(\mathbb{Z}_{\leqslant 0})}=-1$ , $h{(\mathbb{Z}_{\geqslant 1})}=0$ and
			$g{(\mathbb{Z}_{\leqslant -1})}=-1$ , $g{(\mathbb{Z}_{\geqslant 1})}=0$
$\mathcal{W}_{1}^{\mathcal{P}_{8}}$	$\mathcal{P}_{8}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=0$
$\mathcal{W}_{2}^{\mathcal{P}_{8}}$	$\mathcal{P}_{8}$	$a=\mu=0$	$h{(\mathbb{Z})}=g{(\mathbb{Z})}=-1$
$\mathcal{W}_{3}^{\mathcal{P}_{8}}$	$\mathcal{P}_{8}$	$a=\mu=0$	$h{(\mathbb{Z}_{\leqslant 0})}=0$ , $h{(\mathbb{Z}_{\geqslant 1})}=-1$ and
			$g{(\mathbb{Z}_{\leqslant -1})}=0$ , $g{(\mathbb{Z}_{\geqslant 1})}=-1$

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Proof. The proof of the "if" direction can be directly verified. We now prove the "only if" direction. In view of $f$ satisfying (16), by Theorem 2.4 of [10] we know that $f$ is determined by Table 1.

Table 1. Values of f satisfying (16), where c ∈

$\mathbb{C}$ .

$Cases$	$f(n)$
$\mathcal{P}_{1}$	$f{(\mathbb{Z})}=0$
$\mathcal{P}_{2}$	$f{(\mathbb{Z})}=-1$
$\mathcal{P}^{c}_{3}$	$f{(\mathbb{Z}_{>0})}=-1,f{(\mathbb{Z}_{<0})}=0 \,$ and $\, f{(0)}=c$
$\mathcal{P}^{c}_{4}$	$f{(\mathbb{Z}_{>0})}=0,f{(\mathbb{Z}_{<0})}=-1 \,$ and $\, f{(0)}=c$
$\mathcal{P}_{5}$	$f{(\mathbb{Z}_{\geqslant2})}=-1 \,$ and $\, f{(\mathbb{Z}_{\leqslant1})}=0$
$\mathcal{P}_{6}$	$f{(\mathbb{Z}_{\geqslant2})}=0 \,$ and $\, f{(\mathbb{Z}_{\leqslant1})}=-1$
$\mathcal{P}_{7}$	$f{(\mathbb{Z}_{\geqslant-1})}=0 \,$ and $\, f{(\mathbb{Z}_{\leqslant-2})}=-1$
$\mathcal{P}_{8}$	$f{(\mathbb{Z}_{\geqslant-1})}=-1 \,$ and $\, f{(\mathbb{Z}_{\leqslant-2})}=0$

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When $f$ takes the form of Case $\mathcal{P}_{1}$ in Table 1, by the results of Lemma 3.3, we see that $\mu,a,g,h$ must satisfy the condition of Cases $\mathcal{W}_{1}^{\mathcal{P}_{1}}$ and $\mathcal{W}_{2}^{\mathcal{P}_{1}}$ in Table 2. From Lemma 3.3, Cases $\mathcal{W}_{1}^{\mathcal{P}_{1}}, i = 1,2$ is easy to say. In the same way, when $f$ takes the form of Case $\mathcal{P}_{2}$ in Table 1, then we obtain the forms of Cases $\mathcal{W}_{1}^{\mathcal{P}_{2}}$ and $\mathcal{W}_{2}^{\mathcal{P}_{2}}$ in Table 2.

When $f$ takes the form of Case $\mathcal{P}_{3}^c$ in Table 1, by the results of Lemma 3.5, we see that $\mu,a,g,h$ must satisfy the one condition of Cases $\mathcal{W}_{i,\mu}^{\mathcal{P}_{3}^c}, i = 1,2$ , $\mathcal{W}_{3,\mu}^{\mathcal{P}_{3}^c,k}$ , $\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{3}^c,k = 1}$ and $\mathcal{W}_{5,\mu}^{\mathcal{P}_{3}^c,s,t}$ in Table 2. From Lemma 3.5, the results of Cases $\mathcal{W}_{i,\mu}^{\mathcal{P}_{3}^c}, i = 1,2$ are easily obtained; and Case $\mathcal{W}_{3,\mu}^{\mathcal{P}_{3}^c,k}$ satisfies $\mu \in \mathbb{C}$ , $h{(\mathbb{Z}_{> 0})} = -1$ , $h{(\mathbb{Z}_{<0})} = 0$ , $h(0) = \lambda$ and $g{(\mathbb{Z}^*_{\geqslant k})} = -1$ , $g{(\mathbb{Z}^*_{\leqslant k-1})} = 0$ , for some $k\in \{-2,-1,1,2,3\}$ , $g(0) = \hat{\tau}$ with $a = 0$ when $k\neq 1$ and $a$ is arbitrary if $k = 1$ ; Case $\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{3}^c,k = 1}$ satisfies $\mu \in \mathbb{C}$ , $h{(\mathbb{Z}_{> 0})} = -1$ , $h{(\mathbb{Z}_{<0})} = 0$ , $h(0) = \lambda$ and $g{(\mathbb{Z}_{> 0})} = -1$ , $g{(\mathbb{Z}_{< 0})} = 0$ for some $k = 1$ , $g(0) = \hat{\tau}$ ; Case $\mathcal{W}_{5,\mu}^{\mathcal{P}_{3}^c,s,t}$ satisfies $a = 0$ , $\mu\in \mathbb{C}$ and $h{(\mathbb{Z}_{\geqslant t})} = -1$ , $h{(\mathbb{Z}_{\leqslant t-1})} = 0$ for some $t\in \mathbb{Z}\setminus\{0,1\}$ and $g{(\mathbb{Z}_{\geqslant s})} = -1$ , $g{(\mathbb{Z}_{\leqslant s-1})} = 0$ for some $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ . In the same way, when $f$ takes the form of Case $\mathcal{P}_{4}^c$ in Table 1, then we obtain the results of Cases $\mathcal{W}_{i,\mu}^{\mathcal{P}_{4}^c}, i = 1,2$ , $\mathcal{W}_{3,\mu}^{\mathcal{P}_{4}^c,k}$ , $\mathcal{W}_{4,a,\mu}^{\mathcal{P}_{4}^c,k = 1}$ and $\mathcal{W}_{5,\mu}^{\mathcal{P}_{4}^c,s,t}$ in Table 2, respectively.

When $f$ takes the form of Case $\mathcal{P}_{5}$ in Table 1, by the results of Lemma 3.4, we see that $\mu,a,g,h$ must satisfy the condition of Cases $\mathcal{W}_{i}^{\mathcal{P}_{5}}, i = 1,2,3$ in Table 2. From Lemma 3.4, the results of Cases $\mathcal{W}_{i}^{\mathcal{P}_{5}}, i = 1,2,$ are easily obtained; and for Case $\mathcal{W}_{3}^{\mathcal{P}_{5}}$ , we get $h{(\mathbb{Z}_{\leqslant 0})} = 0$ , $h{(\mathbb{Z}_{\geqslant 1})} = -1$ and $g{(\mathbb{Z}_{\leqslant-1})} = 0$ , $g{(\mathbb{Z}_{\geqslant 1})} = -1$ , $g(0) = \hat{\lambda}$ for some $\hat{\lambda}\in \mathbb{C}$ . Similarly, when $f$ takes the form of Case $\mathcal{P}_{k}, k = 6,7,8$ in Table 1, then we obtain the forms of Cases $\mathcal{W}_{i}^{\mathcal{P}_{k}}$ , $i = 1,2,3$ , $k = 6,7,8$ in Table 2. The proof is completed.

Lemma 3.7. Let $(\mathcal{P}(\phi_{i}, \varphi_{i}, \chi_{i}, \psi_{i}, \xi_{i}, \theta_{i}), \triangleright_{i})$ , $i = 1, 2$ be two algebras with the same linear space as $\mathcal{S}$ and equipped with $\mathbb{C}$ -bilinear products $x\triangleright_{i}y$ such that

$\begin{eqnarray*} &&L_{m}\triangleright_{i}L_{n} = \phi_{i}(m,n)L_{m+n},\,\,\, L_{m}\triangleright_{i}H_{n} = \varphi_{i}(m,n)H_{m+n},\\ &&L_{m}\triangleright_{i}I_{n} = \chi_{i}(m,n)I_{m+n},\,\,\, H_{m}\triangleright_{i}L_{n} = \psi_{i}(m,n)H_{m+n},\\ &&H_{m}\triangleright_{i}H_{n} = \xi_{i}(m,n)I_{m+n},\,\,\, I_{m}\triangleright_{i}L_{n} = \theta_{i}(m,n)I_{m+n},\\ &&H_{m}\triangleright_{i}I_{n} = I_{m}\triangleright_{i}H_{n} = I_{m}\triangleright_{i}I_{n} = 0 \end{eqnarray*}$

for all $m, n\in\mathbb{Z}$ , where $\phi_{i}, \varphi_{i}, \chi_{i}, \psi_{i}, \xi_{i}, \theta_{i}$ , $i = 1, 2$ are complex-valued functions on $\mathbb{Z}\times\mathbb{Z}$ . Furthermore, let $\tau:\mathcal{P}(\phi_{1}, \varphi_{1}, \chi_{1}, \psi_{1}, \xi_{1}, \theta_{1})\rightarrow\mathcal{P}(\phi_{2}, \varphi_{2}, \chi_{2}, \psi_{2}, \xi_{2}, \theta_{2})$ be a linear map determined by

$\tau(L_{m}) = -L_{-m},\tau(H_{m}) = -H_{-m},\tau(I_{m}) = -I_{-m}$

for all $m\in\mathbb{Z}$ . In addition, suppose that $(\mathcal{P}(\phi_{1},\varphi_{1},\chi_{1},\psi_{1},\xi_{1},\theta_{1}),[,],\triangleright_{1})$ is a post-Lie algebra. Then $(\mathcal{P}(\phi_{2},\varphi_{2},\chi_{2},\psi_{2},\xi_{2},\theta_{2}),[,],,\triangleright_{2})$ is a post-Lie algebra and $\tau$ is an isomorphism on post-Lie algebras if and only if

$\begin{equation} \left\{ \begin{array}{c} \phi_{2}(m,n) = -\phi_{1}(-m,-n);\\ \varphi_{2}(m,n) = -\varphi_{1}(-m,-n);\\ \chi_{2}(m,n) = -\chi_{1}(-m,-n);\\ \psi_{2}(m,n) = -\psi_{1}(-m,-n);\\ \xi_{2}(m,n) = -\xi_{1}(-m,-n);\\ \theta_{2}(m,n) = -\theta_{1}(-m,-n).\\ \end{array} \right. \end{equation}$

(49)

Proof. Clearly, $\tau$ is a Lie automorphism of $\mathcal{S}$ . Suppose $(\mathcal{P}(\phi_{2},\varphi_{2},\chi_{2},\psi_{2},\xi_{2},\theta_{2}),[,], \triangleright_{2})$ is a post-Lie algebra and $\tau:\mathcal{P}(\phi_{1},\varphi_{1},\chi_{1},\psi_{1},\xi_{1},\theta_{1}) \rightarrow \mathcal{P}(\phi_{2},\varphi_{2},\chi_{2},\psi_{2},\xi_{2},\theta_{2})$ is a post-Lie isomorphism. Then we have

$\begin{eqnarray*} &\tau(L_{m}\triangleright_{i}L_{n}) = -\phi_{i}(m,n)L_{-(m+n)}, \\ &\tau(L_{m}\triangleright_{i}H_{n}) = -\varphi_{i}(m,n)H_{-(m+n)}, \\ &\tau(L_{m}\triangleright_{i}I_{n}) = -\chi_{i}(m,n)I_{-(m+n)}, \\ &\tau(H_{m}\triangleright_{i}L_{n}) = -\psi_{i}(m,n)H_{-(m+n)}, \\ &\tau(H_{m}\triangleright_{i}H_{n}) = -\xi_{i}(m,n)I_{-(m+n)}, \\ &\tau(I_{m}\triangleright_{i}L_{n}) = -\theta_{i}(m,n)I_{-(m+n)} \end{eqnarray*}$

for $i = 1,2$ . This tell us that that (49) holds. Conversely, we first suppose that (49) hold. Then, by using Lemma 3.1 and $(\phi_{1},\varphi_{1},\chi_{1},\psi_{1},\xi_{1},\theta_{1},[,],\triangleright_{1})$ is a post-Lie algebra, we know that there are complex-valued functions $f_{1},g_{1},h_{1}$ on $\mathbb{Z}$ and complex numbers $a_1,\mu_{1}$ satisfying (10)-(22) with replacing $(\phi,\varphi,\chi,\psi,\xi,\theta,f,g,h,\mu,a)$ by $(\phi_{1},\varphi_{1},\chi_{1},\psi_{1},\xi_{1},\theta_{1},f_1,g_1, h_1,\mu_1,a_1)$ . Next, let $f_{2}(m) = f_{1}(-m)$ , $g_{2}(m) = g_{1}(-m)$ , $h_{2}(m) = h_{1}(-m)$ , $\mu_{2} = -\mu_{1}$ and $a_{2} = a_{1}$ , then we see that (10)-(22) hold with replacing $(\phi,\varphi,\chi,\psi,\xi,\theta,f,g,h,\mu,a)$ by $(\phi_2,\varphi_2,\chi_2,\psi_2,\xi_2,\theta_2,f_1,g_1, h_1,\mu_1,a_1)$ . By Lemma 3.1, $\mathcal{P}(\phi_2,\varphi_2,\chi_2,\psi_2,\xi_2,\theta_2)$ is a post-Lie algebra.

The remainder is to prove that $\tau$ is an isomorphism between post-Lie algebra. But one has

$\begin{eqnarray*} &&\tau(L_{m}\triangleright_{1}L_{n}) = -\phi_{1}(m,n)L_{-(m+n)} = \phi_{2}(-m,-n)L_{-(m+n)} = \tau(L_{m})\triangleright_{2}\tau(L_{n}),\\ &&\tau(L_{m}\triangleright_{1}H_{n}) = -\varphi_{1}(m,n)H_{-(m+n)} = \varphi_{2}(-m,-n)H_{-(m+n)} = \tau(L_{m})\triangleright_{2}\tau(H_{n}),\\ &&\tau(L_{m}\triangleright_{1}I_{n}) = -\chi_{1}(m,n)I_{-(m+n)} = \chi_{2}(-m,-n)I_{-(m+n)} = \tau(L_{m})\triangleright_{2}\tau(I_{n}),\\ &&\tau(H_{m}\triangleright_{1}L_{n}) = -\psi_{1}(m,n)H_{-(m+n)} = \psi_{2}(-m,-n)H_{-(m+n)} = \tau(H_{m})\triangleright_{2}\tau(L_{n}),\\ &&\tau(H_{m}\triangleright_{1}H_{n}) = -\varphi_{1}(m,n)I_{-(m+n)} = \varphi_{2}(-m,-n)I_{-(m+n)} = \tau(H_{m})\triangleright_{2}\tau(H_{n}), \end{eqnarray*}$

$\begin{eqnarray*} &&\tau(I_{m}\triangleright_{1}L_{n}) = -\theta_{1}(m,n)I_{-(m+n)} = \phi_{2}(-m,-n)I_{-(m+n)} = \tau(I_{m})\triangleright_{2}\tau(L_{n}) \end{eqnarray*}$

and $\tau(H_{m}\triangleright_{1}I_{n}) = \tau(H_{m})\triangleright_{2}\tau(I_{n}) = 0$ , $\tau(I_{m}\triangleright_{1}H_{n}) = \tau(I_{m})\triangleright_{2}\tau(H_{n})$ = 0, $\tau(I_{m}\triangleright_{1}I_{n}) = \tau(I_{m})\triangleright_{2}\tau(I_{n})$ = 0. The proof is completed.

Theorem 3.8. A graded post-Lie algebra structure on $\mathcal{S}$ satisfying (3)-(9) must be one of the following types, for all $m, n\in \mathbb{Z}$ (in every case $I_{m}\triangleright H_{n} = H_{m}\triangleright I_{n} = I_{m}\triangleright I_{n} = 0$ ),

$(\mathcal{W}_1^{\mathcal{P}_1} ):$ $L_m \triangleright_1^{\mathcal{P}_1} L_n = 0$ , $L_m \triangleright_1^{\mathcal{P}_1} H_n = 0$ , $L_m \triangleright_1^{\mathcal{P}_1} I_n = 0$ , $H_m \triangleright_1^{\mathcal{P}_1} L_n = 0$ , $H_m \triangleright_1^{\mathcal{P}_1} H_n = 0$ , $I_m \triangleright_1^{\mathcal{P}_1} L_n = 0$ ;

$(\mathcal{W}_2^{\mathcal{P}_1} )$ : $L_m \triangleright_2^{\mathcal{P}_1} L_n = 0$ , $L_m \triangleright_2^{\mathcal{P}_1} H_n = 0$ , $L_m \triangleright_2^{\mathcal{P}_1} I_n = 0$ , $H_m \triangleright_2^{\mathcal{P}_1} L_n = (\frac{n}{2}-m)H_{m+n}$ , $H_m \triangleright_2^{\mathcal{P}_1} H_n = (n-m)I_{m+n}$ , $I_m \triangleright_2^{\mathcal{P}_1} L_n = -mI_{m+n}$ ;

$( \mathcal{W}_1^{\mathcal{P}_2} ):$ $L_m \triangleright_1^{\mathcal{P}_2} L_n = (n-m)L_{m+n}$ , $L_m \triangleright_1^{\mathcal{P}_2} H_n = (n-\frac{m}{2})H_{m+n}$ , $L_m \triangleright_1^{\mathcal{P}_2} I_n = nI_{m+n}$ , $H_m \triangleright_1^{\mathcal{P}_2} L_n = 0$ , $H_m \triangleright_1^{\mathcal{P}_2} H_n = 0$ , $I_m \triangleright_1^{\mathcal{P}_2} L_n = 0$ ;

$(\mathcal{W}_2^{\mathcal{P}_2}):$ $L_m \triangleright_2^{\mathcal{P}_2} L_n = (n-m)L_{m+n}$ , $L_m \triangleright_2^{\mathcal{P}_2} H_n = (n-\frac{m}{2})H_{m+n}$ , $L_m \triangleright_2^{\mathcal{P}_2} I_n = nI_{m+n}$ , $H_m \triangleright_2^{\mathcal{P}_2} L_n = (\frac{n}{2}-m)H_{m+n}$ , $H_m \triangleright_2^{\mathcal{P}_2} H_n = (n-m)I_{m+n}$ , $I_m \triangleright_2^{\mathcal{P}_2} L_n = -mI_{m+n}$ ;

$(\mathcal{W}_{i,a,\mu,\lambda}^{\mathcal{P}^{c}_3,s,k,t}): \ i = 1,2,3, 4,5$

$L_m \triangleright_{i}^{\mathcal{P}_3^c} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n}, & m> 0,\\ -ncL_n,& m = 0,\\ 0, & m<0; \end{array}} \right.$

$L_m \triangleright_{i}^{\mathcal{P}_3^c} H_n = \left\{ { \begin{array}{cc} (n-\frac{m}{2})H_{m+n}, & m> 0,\\ (-nc+\mu) H_n, & m = 0,\\ 0, & m<0; \end{array}} \right.$

$L_m \triangleright_{i}^{\mathcal{P}_3^c} I_n = \left\{ { \begin{array}{cc} nI_{m+n}, & m> 0,\\ (-nc+2\mu) I_n, & m = 0,\\ 0, & m<0; \end{array}} \right.$

$\ \ \quad H_m \triangleright_{i}^{\mathcal{P}_3^c} L_n = \delta_{i,2}(\frac{n}{2}-m)H_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +(\delta_{i,3}+\delta_{i,4}) \begin{cases} (\frac{n}{2}-m)H_{m+n}, & m> 0,\\ -\frac{n}{2}\lambda H_{n}, & m = 0,\\ 0, & m< 0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5} \begin{cases} (\frac{n}{2}-m)H_{m+n}, &m\geqslant t ,\\ 0, & m\leqslant t-1 ; \end{cases}$

$\ \ \quad H_m \triangleright_{i}^{\mathcal{P}_3^c} H_n = \delta_{i,2}(n-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +(\delta_{i,3}+\delta_{i,4}) \begin{cases} (n-m)I_{m+n}, & m> 0,\\ -n\lambda I_{n}, & m = 0,\\ 0, & m< 0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5} \begin{cases} (n-m)I_{m+n}, &m\geqslant t ,\\ 0, & m\leqslant t-1 ; \end{cases}$

$\ \ \quad I_m \triangleright_{i}^{\mathcal{P}_3^c} L_n = \delta_{i,2}(-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,3} \begin{cases} -mI_{m+n}, &m\geqslant k ,\\ 0, & m\leqslant k-1 ; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,4} \begin{cases} -mI_{m+n}, & m> 0,\\ naI_{n}, & m = 0,\\ 0, & m<0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5} \begin{cases} -mI_{m+n}, & m\geqslant s,\\ 0, & m\leqslant s-1; \end{cases}$

$(\mathcal{W}_{i,a,\mu,\lambda}^{\mathcal{P}_4^{c},s,k,t}): \ i = 1,2,3, 4,5$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_4^{c}} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n},& m< 0,\\ -ncL_n,& m = 0,\\ 0, & m>0; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_4^{c}} H_n = \left\{{ \begin{array}{cc} (n-\frac{m}{2})H_{m+n},& m< 0,\\ (-nc+\mu)H_n,& m = 0,\\ 0, & m>0; \end{array}} \right.$

$\ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_4^{c}} I_n = \left\{ { \begin{array}{cc} nI_{m+n}, &m< 0,\\ (-nc+2\mu) I_n, &m = 0,\\ 0, & m>0; \end{array}} \right.$

$H_m \triangleright_{i}^{\mathcal{P}_4^{c}} L_n = \delta_{i,2}(\frac{n}{2}-m)H_{n+m}$

$\ \qquad \qquad \quad \ \ \ \ +(\delta_{i,3}+\delta_{i,4})\begin{cases} 0, & m > 0,\\ -\frac{n}{2}\lambda H_{n}, & m = 0,\\ (\frac{n}{2}-m)H_{m+n}, & m< 0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5}\begin{cases} 0, & m\geqslant t ,\\ (\frac{n}{2}-m)H_{m+n}, & m \leqslant t-1 ; \end{cases}$

$H_m \triangleright_{i}^{\mathcal{P}_4^{c}} H_n = \delta_{i,2}(n-m)I_{n+m}$

$\ \qquad \qquad \quad \ \ \ \ +(\delta_{i,3}+\delta_{i,4})\begin{cases} 0, & m > 0,\\ -n\lambda I_{n}, & m = 0,\\ (n-m)I_{m+n}, & m< 0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5}\begin{cases} 0, & m\geqslant t ,\\ (n-m)I_{m+n}, & m \leqslant t-1 ; \end{cases}$

$I_m \triangleright_{i}^{\mathcal{P}_4^{c}} L_n = \delta_{i,2}(-m)I_{n+m}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,3}\begin{cases} 0, & m \geqslant k ,\\ -mI_{m+n}, & m \leqslant k-1 ; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,4}\begin{cases} 0, & m > 0,\\ naI_{n}, & m = 0,\\ -mI_{m+n}, & m< 0; \end{cases}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,5}\begin{cases} 0, & m \geqslant s,\\ -mI_{m+n}, & m\leqslant s-1; \end{cases}$

$(\mathcal{W}_{i}^{\mathcal{P}_5}): \ i = 1,2,3,$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_5} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n}, & m\geqslant 2,\\ 0, & m \leqslant 1; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_5} H_n = \left\{{ \begin{array}{cc} (n-\frac{m}{2})L_{m+n}, & m\geqslant 2,\\ 0, & m \leqslant 1; \end{array}} \right.$

$\ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_5} I_n = \left\{ { \begin{array}{cc} nI_{m+n}, & m \geqslant 2,\\ 0, & m\leqslant 1; \end{array}} \right.$

$H_m \triangleright_{i}^{\mathcal{P}_5} L_n = \delta_{i,2}(\frac{n}{2}-m)H_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,3}\begin{cases} 0, & m\leqslant 0,\\ (\frac{n}{2}-m)H_{m+n}, & m \geqslant 1; \end{cases}$

$H_m \triangleright_{i}^{\mathcal{P}_5} H_n = \delta_{i,2}(n-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,3}\begin{cases} 0, & m\leqslant 0,\\ (n-m)I_{m+n}, & m \geqslant 1; \end{cases}$

$I_m \triangleright_{i}^{\mathcal{P}_5} L_n = \delta_{i,2}(-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i,3}\begin{cases} 0, & m\leqslant 0,\\ -mI_{m+n}, & m \geqslant 1; \end{cases}$

$(\mathcal{W}_{i}^{\mathcal{P}_6}):\ i = 1,2,3,$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_6} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n}, & m\leqslant 1,\\ 0, & m \geqslant 2; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_6} H_n = \left\{{ \begin{array}{cc} (n-\frac{m}{2})H_{m+n}, & m\leqslant 1,\\ 0, & m \geqslant 2; \end{array}} \right.$

$\ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_6} I_n = \left\{ { \begin{array}{cc} nI_{m+n}, & m \leqslant 1,\\ 0, & m \geqslant 2; \end{array}} \right.$

$H_m \triangleright_{i}^{\mathcal{P}_6} L_n = \delta_{i, 2}(\frac{n}{2}-m)H_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3}\begin{cases} (\frac{n}{2}-m)H_{m+n}, & m \leqslant 0,\\ 0, & m \geqslant 1; \end{cases}$

$H_m \triangleright_{i}^{\mathcal{P}_6} H_n = \delta_{i, 2}(n-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3}\begin{cases} (n-m)I_{m+n}, & m \leqslant 0,\\ 0, & m \geqslant 1; \end{cases}$

$I_m \triangleright_{i}^{\mathcal{P}_6} L_n = \delta_{i, 2}(-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3}\begin{cases} -mI_{m+n}, & m \leqslant -1,\\ 0, & m \geqslant 0; \end{cases}$

$(\mathcal{W}_{i}^{\mathcal{P}_7}):\ i = 1,2,3,$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_7} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n}, & m\leqslant -2,\\ 0, & m \geqslant -1; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_7} H_n = \left\{{ \begin{array}{cc} (n-\frac{m}{2})H_{m+n}, & m\leqslant -2,\\ 0, & m \geqslant -1; \end{array}} \right.$

$\ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_7} I_n = \left\{ { \begin{array}{cc} nI_{m+n},& m \leqslant -2,\\ 0, & m \geqslant -1; \end{array}} \right.$

$H_m \triangleright_{i}^{\mathcal{P}_7} L_n = \delta_{i, 2}(\frac{n}{2}-m)H_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} (\frac{n}{2}-m)H_{m+n}, & m \leqslant 0,\\ 0, & m \geqslant 1; \end{cases}$

$H_m \triangleright_{i}^{\mathcal{P}_7} H_n = \delta_{i, 2}(n-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} (n-m)I_{m+n}, & m \leqslant 0,\\ 0, & m \geqslant 1; \end{cases}$

$I_m \triangleright_{i}^{\mathcal{P}_7} L_n = \delta_{i, 2}(-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} -mI_{m+n}, & m \leqslant -1,\\ 0, & m \geqslant 0; \end{cases}$

$(\mathcal{W}_{i}^{\mathcal{P}_8}):\ i = 1,2,3,$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_8} L_n = \left\{{ \begin{array}{cc} (n-m)L_{m+n}, & m\geqslant -1,\\ 0, & m \leqslant -2; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_8} H_n = \left\{{ \begin{array}{cc} (n-\frac{m}{2})H_{m+n}, & m\geqslant -1,\\ 0, & m \leqslant -2; \end{array}} \right.$

$\ \ \ \ \ \ \ L_m \triangleright_{i}^{\mathcal{P}_8} I_n = \left\{{ \begin{array}{cc} nI_{m+n}, & m\geqslant -1,\\ 0, & m \leqslant -2; \end{array}} \right.$

$H_m \triangleright_{i}^{\mathcal{P}_8} L_n = \delta_{i, 2}(\frac{n}{2}-m)H_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} 0, & m \leqslant 0,\\ (\frac{n}{2}-m)H_{m+n}, & m \geqslant 1 \end{cases}$

$H_m \triangleright_{i}^{\mathcal{P}_8} H_n = \delta_{i, 2}(n-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} 0, & m \leqslant 0,\\ (n-m)I_{m+n}, & m \geqslant 1 \end{cases}$

$I_m \triangleright_{i}^{\mathcal{P}_8} L_n = \delta_{i, 2}(-m)I_{m+n}$

$\ \qquad \qquad \quad \ \ \ \ +\delta_{i, 3} \begin{cases} 0, & m \leqslant 0,\\ -mI_{m+n}, & m \geqslant 1 \end{cases}$

where $c,a,\mu,\lambda\in\mathbb{C}$ , $k\in \{-2,-1,1,2,3\}$ , $t\in \mathbb{Z}\setminus\{0,1\}$ and $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ . Conversely, the above types are all the graded post-Lie algebra structures satisfying (3)-(9) on $\mathcal{S}$ . Furthermore, the post-Lie algebras $\mathcal{W}^{\mathcal{P}_{3}^c,s,k,t}_{i,a,\mu,\lambda}$ , $\mathcal{W}^{\mathcal{P}_{5}}_{j}$ and $\mathcal{W}^{\mathcal{P}_{6}}_{j}$ are isomorphic to the post-Lie algebras $\mathcal{W}^{\mathcal{P}_{4}^c,s,k,t}_{i,a,\mu,\lambda}$ , $\mathcal{W}^{\mathcal{P}_{7}}_{j}$ and $\mathcal{W}^{\mathcal{P}_{8}}_{j}$ , $i = 1,2,3,4,5$ and $j = 1,2,3$ respectively, and other post-Lie algebras are not mutually isomorphic.

Proof. Suppose that $(\mathcal{S},[,],\triangleright)$ is a class of post-Lie algebra structures satisfying (3)-(9) on the Schrödinger-Virasoro algebra $\mathcal{S}$ . By Lemma 3.3-3.5, there are complex-valued functions $f$ , $g$ , $h$ on $\mathbb{Z}$ and complex numbers $\mu,a$ such that one of $26$ cases in Table 2 holds. From this with Lemma 3.1, we obtain $26$ classes of graded post-Lie algebra structures on $\mathcal{S}$ . We claim that $h(0) = \lambda$ and $g(0) = \hat{\tau}$ in $\mathcal{W}_{i,a,\mu,\lambda}^{\mathcal{P}_{j}^c,s,k,t}, j = 3,4$ and $i = 1,2,3,4,5$ and $g(0) = \hat{\lambda}$ in $\mathcal{W}^{\mathcal{P}_{j}}_{i}$ , $j = 5,6,7,8$ and $i = 1,2,3$ . We claim that $g(0) = \hat{\lambda}$ and $g(0) = \hat{\tau}$ will not appear in every structures, when $m = 0$ , for example, in Case $\mathcal{W}^{\mathcal{P}_{5}}_{i}$ , $i = 1,2,3$ , then $I_{m}\triangleright_{3}^{P_{5}}L_{n} = 0 \hat{\lambda} I_{0+n} = 0$ , one has $I_{m}\triangleright_{3}^{P_{5}}L_{n} = 0$ for $m \leqslant 0$ , and in Case $\mathcal{W}^{\mathcal{P}_{3}^c,s,k,t}_{i,a,\mu,\lambda}$ , $i = 1,2,3,4,5$ , then $H_{m}\triangleright_{3,\lambda}^{P_{3}}L_{n} = -(\frac{n}{2}-0)\lambda H_{0+n} = 0$ , one has $H_{m}\triangleright_{3,\lambda}^{P_{3}}L_{n} = -\frac{n}{2}\lambda H_{n}$ for $m = 0$ . Hence we can obtain $26$ classes of graded post-Lie algebra structures on $\mathcal{S}$ listed in the theorem.

Conversely, every type of the $26$ cases means that there are complex-valued functions $f$ and $g$ , $h$ on $\mathbb{Z}$ and complex numbers $a,\mu$ such that (10)-(15) hold and, the Equations (16)-(22) are easily verified. Thus, by Lemma 3.1 we see that they are the all graded post-Lie algebra structures satisfying (3)-(9) on the Schrödinger-Virasoro algebra $\mathcal{S}$ .

Finally, by Lemma 3.7 with maps $L_{m}\rightarrow-L_{-m}$ , $H_{m}\rightarrow-H_{-m}$ , $I_{m}\rightarrow-I_{-m}$ we know that the post-Lie algebras $\mathcal{W}^{\mathcal{P}_{3}^c,s,k,t}_{i,a,\mu,\lambda}$ , $\mathcal{W}^{\mathcal{P}_{5}}_{j}$ and $\mathcal{W}^{\mathcal{P}_{6}}_{j}$ are isomorphic to the post-Lie algebras $\mathcal{W}^{\mathcal{P}_{4}^c,s,k,t}_{i,a,\mu,\lambda}$ , $\mathcal{W}^{\mathcal{P}_{7}}_{j}$ and $\mathcal{W}^{\mathcal{P}_{8}}_{j}$ , $i = 1,2,3,4,5$ and $j = 1,2,3$ respectively. Clearly, the other post-Lie algebras are not mutually isomorphic. The proof is completed.

4. Application to Rota-Baxter operators

The Rota-Baxter algebra was introduced by the mathematician Glen E. Baxter [2] in 1960 in his probability study, and was popularized mainly by the work of Rota [G. Rota1, G. Rota2] and his school. Recently, the Rota-Baxter algebra relation were introduced to solve certain analytic and combinatorial problem and then applied to many fields in mathematics and mathematical physics (see [6,7,19,23] and the references therein). Now let us recall the definition of Rota-Baxter operator.

Definition 4.1. Let $L$ be a complex Lie algebra. A Rota-Baxter operator of weight $\lambda\in\mathbb{C}$ is a liner map $R:L\rightarrow L$ satisfying

$\begin{equation} [R(x),R(y)] = R([R(x),y]+[x,R(y)])+\lambda R([x,y]),\,\,\forall\text{x,y}\in L. \end{equation}$

(50)

Note that if $R$ is a Rota-Baxter operator of weight $\lambda\neq0$ , then $\lambda^{-1}R$ is a Rota-Baxter operator of weight 1. Therefore, one only needs to consider Rota-Baxter operators of weight $0$ and $1$ .

4.1. Rota-Baxter operators of weight $1$

In this section, we mainly consider the homogeneous Rota-Baxter operator $R$ of weight 1 on the Schrödinger-Virasoro $\mathcal{S}$ given by

$\begin{equation} R(L_{m}) = f(m)L_{m},\ \ \ R(H_{m}) = h(m)H_{m},\ \ \ R(I_{m}) = g(m)I_{m} \end{equation}$

(51)

for all $m\in\mathbb{Z}$ , where $f,g,h$ are complex-valued functions on $\mathbb{Z}$ .

Lemma 4.2. (see [1]) Let $(L, [,])$ be a Lie algebra and $R:L\rightarrow L$ a Rota-Baxter operator of weight 1. Define a new operation $x\triangleright y = [R(x),y]$ on L. Then $(L,[,],\triangleright)$ is a post-Lie algebra.

Theorem 4.3. A homogeneous Rote-Baxrer operator $R$ of weight 1 satisfying (51) on the Schrödinger-Virasoro $\mathcal{S}$ must be one of the following types

$(\mathcal{R}_1^{\mathcal{P}_1} ):$ $R(L_m) = 0, R(H_n) = 0, R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_1}$ ): $R(L_m) = 0, R(H_n) = -H_n, R(I_n) = -I_n;$

$( \mathcal{R}_1^{\mathcal{P}_2} ):$ $R(L_m) = -L_m, R(H_n) = 0, R(I_n) = 0;$

$( \mathcal{R}_2^{\mathcal{P}_2}):$ $R(L_m) = -L_m, R(H_n) = -H_n, R(I_n) = -I_n;$

( $\mathcal{R}_1^{\mathcal{P}_3^c}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m>0, \\ cL_{0}, \ \ & m = 0,\\ 0, & m< 0; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_3^c}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m>0, \\ cL_{0}, \ \ & m = 0,\\ 0, & m< 0; \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n;$

( $\mathcal{R}_{3,\hat{\tau},\lambda}^{\mathcal{P}^{c}_{3},k}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m>0, \\ cL_{0}, \ \ & m = 0,\\ 0, & m< 0; \end{cases}$ $R(H_n) = \begin{cases} -H_{n}, \ \ & n> 0, \\ \lambda H_{0}, \ \ & n = 0,\\ 0, & n< 0; \end{cases}$

$\quad R(I_n) = \begin{cases} -I_{n}, \ \ & n\geqslant k, \\ \hat{\tau} I_{0}, \ \ & n = 0,\\ 0, & n\leqslant k-1; \end{cases}$

( $\mathcal{R}_{5}^{\mathcal{P}^{c}_{3},s,t}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m>0, \\ cL_{0}, \ \ & m = 0,\\ 0, & m< 0; \end{cases}$ $R(H_n) = \begin{cases} -H_{n}, \ \ & n\geqslant t, \\ 0 , \ \ & n\leqslant t-1; \end{cases}$

$R(I_n) = \begin{cases} -I_{n}, \ \ & n\geqslant s, \\ 0, \ \ & n\leqslant s-1; \end{cases}$

( $\mathcal{R}_1^{\mathcal{P}_4^{c}}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m< 0,\\ cL_{0}, \ \ & m = 0,\\ 0, & m>0; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_4^{c}}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m< 0,\\ cL_{0}, \ \ & m = 0,\\ 0, & m>0; \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n;$

( $\mathcal{R}_{3,\hat{\tau},\lambda}^{\mathcal{P}^{c}_{4},k}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m< 0,\\ cL_{0}, \ \ & m = 0,\\ 0, & m>0; \end{cases}$ $R(H_n) = \begin{cases} 0, \ \ & n> 0, \\ \lambda H_{0}, \ \ & n = 0,\\ -H_{n}, \ \ & n<0 ; \end{cases}$

$R(I_n) = \begin{cases} 0, \ \ & n\geqslant k, \\ \hat{\tau} I_{0}, \ \ & n = 0,\\ -I_{n}, \ \ & n\leqslant k-1 ; \end{cases}$

( $\mathcal{R}_{5}^{\mathcal{P}^{c}_{4},s,t}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m>0, \\ cL_{0}, \ \ & m = 0,\\ 0, & m< 0; \end{cases}$ $R(H_n) = \begin{cases} 0, \ \ & n\geqslant t, \\ -H_n, \ \ & n\leqslant t-1; \end{cases}$

$R(I_n) = \begin{cases} 0, \ \ & n\geqslant s, \\ -I_n, \ \ & n\leqslant s-1; \end{cases}$

( $\mathcal{R}_1^{\mathcal{P}_5}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m\geqslant 2,\\ 0, \ \ & m\leqslant 1; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_5}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m\geqslant 2,\\ 0, \ \ & m\leqslant 1; \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n;$

( $\mathcal{R}_{3,\hat{\lambda}}^{\mathcal{P}_5}$ ): $R(L_m) = \begin{cases} -L_{m}, \ \ & m\geqslant 2,\\ 0, \ \ & m\leqslant 1; \end{cases}$ $R(H_n) = \begin{cases} 0, \ \ & n\leqslant 0,\\ -H_{n}, \ \ & n\geqslant 1; \end{cases}$

$\qquad\quad R(I_n) = \begin{cases} 0, \ \ & n\leqslant -1,\\ \hat{\lambda} I_{0}, \ \ & n = 0,\\ -I_{n}, \ \ & n \geqslant 1; \end{cases}$

( $\mathcal{R}_1^{\mathcal{P}_6}$ ): $R(L_m) = \begin{cases} -L_{m}, & m \leqslant 1,\\ 0, \ \ & m\geqslant 2; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_6}$ ): $R(L_m) = \begin{cases} -L_{m}, & m \leqslant 1,\\ 0, \ \ & m\geqslant 2; \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n;$

( $\mathcal{R}_{3,\hat{\lambda}}^{\mathcal{P}_6}$ ): $R(L_m) = \begin{cases} -L_{m}, & m \leqslant 1,\\ 0, \ \ & m\geqslant 2; \end{cases}$ $R(H_n) = \begin{cases} -H_{n}, & n \leqslant 0,\\ 0, \ \ & n\geqslant 1 ; \end{cases}$

$\qquad\quad R(I_n) = \begin{cases} -I_{n}, & n \leqslant -1,\\ \hat{\lambda} I_{0}, \ \ & n = 0,\\ 0, \ \ & n\geqslant 1 ; \end{cases}$

( $\mathcal{R}_1^{\mathcal{P}_7}$ ): $R(L_m) = \begin{cases} -L_{m}, & m\leqslant -2 ,\\ 0, \ \ & m\geqslant -1; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0;$

( $\mathcal{R}_2^{\mathcal{P}_7}$ ): $R(L_m) = \begin{cases} -L_{m}, & m\leqslant -2,\\ 0, \ \ & m\geqslant -1; \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n$ ;

( $\mathcal{R}_{3,\hat{\lambda}}^{\mathcal{P}_7}$ ): $R(L_m) = \begin{cases} -L_{m}, & m\leqslant -2,\\ 0, \ \ & m\geqslant -1; \end{cases}$ $R(H_n) = \begin{cases} 0, \ \ & n\geqslant 1,\\ -H_{n}, \ \ & n\leqslant 0; \end{cases}$

$\qquad\quad R(I_n) = \begin{cases} 0, & n\geqslant 1,\\ \hat{\lambda} I_{0}, \ \ & n = 0,\\ -I_{n}, \ \ & n\leqslant -1; \end{cases}$

( $\mathcal{R}_1^{\mathcal{P}_8}$ ): $R(L_m) = \begin{cases} -L_{m}, & m\geqslant -1,\\ 0, \ \ & m\leqslant -2; \end{cases}$ $R(H_n) = 0,$ $R(I_n) = 0$ ;

$(\mathcal{R}_2^{\mathcal{P}_8} ):$ $R(L_m) = \begin{cases} -L_{m}, & m\geqslant -1,\\ 0, \ \ & m\leqslant -2, \end{cases}$ $R(H_n) = -H_n,$ $R(I_n) = -I_n$ ;

$( \mathcal{R}_{3,\hat{\lambda}}^{\mathcal{P}_8} ):$ $R(L_m) = \begin{cases} -L_{m}, &m\geqslant -1,\\ 0, \ \ & m\leqslant -2, \end{cases}$ $R(H_n) = \begin{cases} -H_{n}, & n\geqslant 1,\\ 0, \ \ &n\leqslant 0; \end{cases}$

$R(I_n) = \begin{cases} -I_{n}, & n\geqslant 1,\\ \hat{\lambda} I_{0}, \ \ & n = 0,\\ 0, \ \ &n\leqslant -1 \end{cases}$

for all $m,n\in\mathbb{Z}$ , where $c,\lambda,\hat{\lambda},\hat{\tau}\in\mathbb{C}$ , $k\in \{-2,-1,1,2,3\}$ with $k\neq1$ , $t\in \mathbb{Z}\setminus\{0,1\}$ and $s\in \{2t-2,2t-1,2t,2t+1,2t+2\}$ .

Proof. In view of Lemma 4.2, if we define a new operation $x\triangleright y = [R(x),y]$ on $\mathcal{S}$ , then ( $\mathcal{S},[,],\triangleright$ ) is a post-Lie algebra. By (51), we have

$\begin{eqnarray} &&L_{m}\triangleright L_{n} = [R(L_{m}),L_{n}] = (m-n)f(m)L_{m+n}, \end{eqnarray}$

(52)

$\begin{eqnarray} &&L_{m}\triangleright H_{n} = [R(L_{m}),H_{n}] = (\frac{m}{2}-n)f(m)H_{m+n}, \end{eqnarray}$

(53)

$\begin{eqnarray} &&L_{m}\triangleright I_{n} = [R(L_{m}),I_{n}] = -nf(m)I_{m+n}, \end{eqnarray}$

(54)

$\begin{eqnarray} &&H_{m}\triangleright L_{n} = [R(H_{m}),L_{n}] = -(\frac{n}{2}-m)h(m)H_{m+n}, \end{eqnarray}$

(55)

$\begin{eqnarray} &&H_{m}\triangleright H_{n} = [R(H_{m}),H_{n}] = (m-n)h(m)I_{m+n}, \end{eqnarray}$

(56)

$\begin{eqnarray} &&I_{m}\triangleright L_{n} = [R(I_{m}),L_{n}] = mg(m)I_{m+n} \end{eqnarray}$

(57)

and $I_{m}\triangleright H_{n} = [R(I_{m}),H_{n}] = H_{m}\triangleright I_{n} = [R(H_{m}),I_{n}] = I_{m}\triangleright I_{n} = [R(I_{m}),I_{n}] = 0$ for all $m,n\in\mathbb{Z}$ . This means that $(\mathcal{S},[,],\triangleright)$ is a graded post-Lie algebras structure satisfying (3)-(9) with $\phi(m,n) = (m-n)f(m)$ , $\varphi(m,n) = (\frac{m}{2}-n)f(m)$ , $\chi(m,n) = -nf(m)$ , $\psi(m,n) = -(\frac{n}{2}-m)h(m)$ , $\xi(m,n) = (m-n)h(m)$ and $\theta(m,n) = mg(m)$ .

A similar discussion to Lemma 3.1 gives

$\begin{eqnarray*} &&(m-n)(f(m+n)-f(n)f(m)+f(m)f(m+n)+f(n)f(m+n)) = 0, \\ &&(\frac{m}{2}-n)(h(m+n)-f(m)h(n)+f(m)h(m+n)+h(n)h(m+n)) = 0, \\ && n(m+n)(g(m+n)(1+f(m)+g(n))-f(m)g(n)) = 0, \\ &&(m-n)(g(m+n)-h(m)h(n)+h(m)g(m+n)+h(n)g(m+n)) = 0. \end{eqnarray*}$

From this we conclude that Equations (10)-(22) hold with $a = \mu = 0$ . In the same way of Lemma 3.6, we see that $f,g,h$ must satisfy Table 2 with $a = \mu = 0$ . This excludes Cases $\mathcal{W}_{4,a,\mu}^{\mathcal{P}^{c}_{3},k = 1}$ and $\mathcal{W}_{4,a,\mu}^{\mathcal{P}^{c}_{4},k = 1}$ . Thus, $f$ , $g$ , $h$ must be of the $24$ cases listed in Table 2 with $a = \mu = 0$ , which can yield the $24$ forms of $R$ one by one. It is easy to verify that every form of $R$ listed in the above is a Rota-Baxter operator of weight $1$ satisfying (51). The proof is completed.

4.2. Remark on Rota-Baxter operators of weight zero and pre-Lie algebras

The natural question is: how we can characterize the Rota-Baxter operators of weight zero on the Schrödinger-Virasoro $\mathcal{S}$ ? This is related to the so called pre-Lie algebra which is a class of Lie-admissible algebras whose commutators are Lie algebras. Pre-Lie algebras appeared in many fields in mathematics and physics under different names like left-symmetric algebras, Vinberg algebras and quasi-associative algebras (see the survey article [3] and the references therein). Now we recall the definition of pre-Lie algebra as follows.

Definition 4.4. A pre-Lie algebra $A$ is a vector space $A$ with a bilinear product $\triangleright$ satisfying

$\begin{equation} (x\triangleright y)\triangleright z - x\triangleright(y\triangleright z) = (y\triangleright x)\triangleright z - y\triangleright(x\triangleright z),\;\;\forall x,y,z\in A. \end{equation}$

(58)

As a parallel result of Lemma 4.2, one has the following conclusion.

Proposition 1. (see [8]) Let $(L,[\cdot, \cdot])$ be a Lie algebra with a Rota-Baxter operator $R$ of weight 0 on it. Define a new operation $x\triangleright y = [R(x),y]$ for any $x,y\in L$ . Then $(L,\triangleright)$ is a pre-Lie algebra.

Using a similar method on classification of Rota-Baxter operators of weight $1$ in the above subsection, by Proposition 1 we can get the forms of Rota-Baxter operators of weight zero when the corresponding structure of pre-Lie algebra are known. For example, consider the homogeneous Rota-Baxter operator $R$ of weight zero on the Schrödinger-Virasoro algebra $\mathcal{S}$ satisfying (51). According to Proposition 1, if we define a new operation $x\triangleright y = [R(x),y]$ on $\mathcal{S}$ , then ( $\mathcal{S},\triangleright$ ) is a pre-Lie algebra. By (51), we have Equations (52)-(57) hold. At this point we can apply the relevant results on pre-Lie algebra satisfying (52)-(57). But the classification of graded pre-Lie algebra structures on $\mathcal{S}$ is also an unsolved problem, as far as we know. In fact, we can direct characterize the Rota-Baxter operators of weight zero on the Schrödinger-Virasoro $\mathcal{S}$ satisfying (51) following the approach of [6]. Due to limited space, it will not be discussed here.

Acknowledgments

We would like to express our sincere thanks to the anonymous referees for their careful reading and valuable comments towards the improvement of this article.

References

[1]	UN-Habitat, The Value of Sustainable Urbanization. World Cities Report, Nairobi, 2020. Available from: https://unhabitat.org/sites/default/files/2020/10/wcr_2020_report.pdf.
[2]	Agence Française de Développement, Sustainable Cities. Focus, Paris, 2019. Available from: https://upfi-med.eib.org/wp-content/uploads/2020/04/AFD_CIS_FOCUS-VILLES_DURABLES_ENG_WEB-VF-BAT-1.pdf.
[3]	Dixon T, Connaughton J, Green S (2018) Sustainable Futures in the Built Environment to 2050: A Foresight Approach to Construction and Development, John Wiley & Sons. https://doi.org/10.1002/9781119063834
[4]	IEA (International Energy Agency), Energy technology perspectives pathways to a clean energy system, 2012. Available from: https://iea.blob.core.windows.net/assets/7136f3eb-4394-47fd-9106-c478283fcf7f/ETP2012_free.pdf.
[5]	UNEP (United Nation for Environment Programme), Transport. Investing in energy and resource efficiency, UNEP, 2011. Available from: https://wedocs.unep.org/bitstream/handle/20.500.11822/22013/10.0_transport.pdf?sequence=1&%3BisAllowed.
[6]	Bulkeley H, Broto VC, Maassen A (2013) Low-carbon transitions and the reconfiguration of urban infrastructure. Urban Studies 51: 1471–1486. https://doi.org/10.1177/0042098013500089 doi: 10.1177/0042098013500089
[7]	Dowling R, McGuirk P, Bulkeley H (2014) Retrofitting cities: Local governance in Sydney, Australia. Cities 38: 18–24. https://doi.org/10.1016/j.cities.2013.12.004 doi: 10.1016/j.cities.2013.12.004
[8]	Rutherford J, Jaglin S (2015) Introduction to the special issue—Urban energy governance: Local actions, capacities and politics. Energy Policy 78: 173–178. https://doi.org/10.1016/j.enpol.2014.11.033 doi: 10.1016/j.enpol.2014.11.033
[9]	UN-Habitat, Sustainable Urban Energy Planning—A handbook for cities and towns in developing countries. UNEP, Nairobi, 2009. Available from: https://seors.unfccc.int/applications/seors/attachments/get_attachment?code=LUZ4E1JJHTISK0JLBY55WLV36ICQR6WT.
[10]	Webb J, Hawkey D, Tingey M (2016) Governing cities for sustainable energy: The UK case. Cities 54: 28–35. https://doi.org/10.1016/j.cities.2015.10.014 doi: 10.1016/j.cities.2015.10.014
[11]	Ji C, Choi M, Hong T, et al. (2021) Evaluation of the effect of a building energy efficiency certificate in reducing energy consumption in Korean apartments. Energy Build 248: 111168. https://doi.org/10.1016/j.enbuild.2021.111168 doi: 10.1016/j.enbuild.2021.111168
[12]	El Hafdaoui H, Jelti F, Khallaayoun A, et al. (2023) Energy and environmental national assessment of alternative fuel buses in Morocco. World Electr Veh J 14: 105. https://doi.org/10.3390/wevj14040105 doi: 10.3390/wevj14040105
[13]	Prafitasiwi AG, Rohman MA, Ongkowijoyo CS (2022) The occupant's awareness to achieve energy efficiency in campus building. Results Eng 14: 10039. https://doi.org/10.1016/j.rineng.2022.100397 doi: 10.1016/j.rineng.2022.100397
[14]	Cô té-Roy L, Moser S (2022) A kingdom of new cities: Morocco's national Villes Nouvelles strategy. Geoforum 131: 27–38. https://doi.org/10.1016/j.geoforum.2022.02.005 doi: 10.1016/j.geoforum.2022.02.005
[15]	Delmastro, Chiara; De Bienassis, Tanguy; Goodson, Timothy; Lane, Kevin; Le Marois, Jean-Baptiste; Martinez-Gordon, Rafael; Husek, Martin, "Buildings, " IEA, 2021. Available from: https://www.iea.org/reports/buildings.
[16]	Ministère de la Transition Energétique et du Développement Durable, Stratégie Bas Carbone à Long Terme—Maroc 2050, Rabat, 2021. Available from: https://unfccc.int/sites/default/files/resource/MAR_LTS_Dec2021.pdf.
[17]	IEA (International Energy Agency), Transport—Improving the sustainability of passenger and freight transport, 2021. Available from: https://www.iea.org/topics/transport.
[18]	Ministère de la Transition Energétique et du Développement Durable (MTEDD), Consommation Energetique par l'Administration—Fès et Meknès, SIREDD, Rabat, Morocco, 2012. Available from: https://siredd.environnement.gov.ma/fes-meknes/indicateur/DetailIndicateurPartial?idIndicateur=2988.
[19]	Chegari B, Tabaa M, Moutaouakkil F, et al. (2020) Local energy self-sufficiency for passive buildings: Case study of a typical Moroccan building. J Build Eng 29: 101164. https://doi.org/10.1016/j.jobe.2019.101164 doi: 10.1016/j.jobe.2019.101164
[20]	Oubourhim A, El-Hami K (2020) Efficiency energy standards and labelling for residential appliances in Morocco. In Advanced Intelligent Systems for Sustainable Development, Marrakesh, Morocco, Springer, 97–109. https://doi.org/10.1007/978-3-030-36475-5_10
[21]	El Majaty S, Touzani A, Kasseh Y (2023) Results and perspectives of the application of an energy management system based on ISO 50001 in administrative buildings—case of Morocco. Mater Today: Proc 72: 3233–323. https://doi.org/10.1016/j.matpr.2022.07.094 doi: 10.1016/j.matpr.2022.07.094
[22]	Merini I, Molina-García A, García-Cascales MS, et al. (2020) Analysis and comparison of energy efficiency code requirements for buildings: A Morocco—Spain case study. Energies 13: 5979. https://doi.org/10.3390/en13225979 doi: 10.3390/en13225979
[23]	Sghiouri H, Mezrhab A, Karkri M, et al. (2018) Shading devices optimization to enhance thermal comfort and energy performance of a residential building in Morocco. J Build Eng 18: 292–302. https://doi.org/10.1016/j.jobe.2018.03.018 doi: 10.1016/j.jobe.2018.03.018
[24]	Jihad AS, Tahiri M (2018) Forecasting the heating and cooling load of residential buildings by using a learning algorithm "gradient descent", Morocco. Case Studies Therm Eng 12: 85–93. https://doi.org/10.1016/j.csite.2018.03.006 doi: 10.1016/j.csite.2018.03.006
[25]	Romani Z, Draoui A, Allard F (2015) Metamodeling the heating and cooling energy needs and simultaneous building envelope optimization for low energy building design in Morocco. Energy Build 102: 139–148. https://doi.org/10.1016/j.enbuild.2015.04.014 doi: 10.1016/j.enbuild.2015.04.014
[26]	Sghiouri H, Charai M, Mezrhab A, et al. (2020) Comparison of passive cooling techniques in reducing overheating of clay-straw building in semi-arid climate. Build Simul 13: 65–88. https://doi.org/10.1007/s12273-019-0562-0 doi: 10.1007/s12273-019-0562-0
[27]	Bendara S, Bekkouche MA, Benouaz T, et al. (2019) Energy efficiency and insulation thickness according to the compactness index case of a studio apartment under saharan weather conditions. J Sol Energy Eng 141: 04101. https://doi.org/10.1115/1.4042455 doi: 10.1115/1.4042455
[28]	Rochd A, Benazzouz A, Ait Abdelmoula I, et al. (2021) Design and implementation of an AI-based & IoT-enabled home energy management system: A case study in Benguerir—Morocco. Energy Rep 7: 699–719. https://doi.org/10.1016/j.egyr.2021.07.084 doi: 10.1016/j.egyr.2021.07.084
[29]	Lebied M, Sick F, Choulli Z, et al. (2018) Improving the passive building energy efficiency through numerical simulation—A case study for Tetouan climate in northern of Morocco. Case Studies Therm Eng 11: 125–134. https://doi.org/10.1016/j.csite.2018.01.007 doi: 10.1016/j.csite.2018.01.007
[30]	Bouhal T, Fertahi S e.-D, Agrouaz Y, et al. (2018) Technical assessment, economic viability and investment risk analysis of solar heating/cooling systems in residential buildings in Morocco. Sol Energy 170: 1043–1062. https://doi.org/10.1016/j.solener.2018.06.032 doi: 10.1016/j.solener.2018.06.032
[31]	Swan LG, Ugursal VI (2009) Modeling of end-use energy consumption in the residential sector: A review of modeling techniques. Renewable Sustainable Energy Rev 13: 1819–1835. https://doi.org/10.1016/j.rser.2008.09.033 doi: 10.1016/j.rser.2008.09.033
[32]	Martos A, Pacheco-Torres R, Ordóñ ez J, et al. (2016) Towards successful environmental performance of sustainable cities: Intervening sectors—A review. Renewable Sustainable Energy Rev 57: 479–495. https://doi.org/10.1016/j.rser.2015.12.095 doi: 10.1016/j.rser.2015.12.095
[33]	Howard B, Parshall L, Thompson J, et al. (2012) Spatial distribution of urban building energy consumption by end use. Energy Build 45: 141–151. https://doi.org/10.1016/j.enbuild.2011.10.061 doi: 10.1016/j.enbuild.2011.10.061
[34]	Pereira IM, Sad de Assis E (2013) Urban energy consumption mapping for energy management. Energy Policy 59: 257–269. https://doi.org/10.1016/j.enpol.2013.03.024 doi: 10.1016/j.enpol.2013.03.024
[35]	Mutani G, Todeschi V (2021) GIS-based urban energy modelling and energy efficiency scenarios using the energy performance certificate database. Energy Efficiency 14: 1–28. https://doi.org/10.1007/s12053-021-09962-z doi: 10.1007/s12053-021-09962-z
[36]	Todeschi V, Boghetti R, Kämpf JH, et al. (2021) Evaluation of urban-scale building energy-use models and tools—Application for the city of Fribourg, Switzerland. Sustainability 13: 1595. https://doi.org/10.3390/su13041595 doi: 10.3390/su13041595
[37]	Population of Ifrane 2023, AZNations, 2023. Available from: https://www.aznations.com/population/ma/cities/ifrane-1.[Accessed 27 March 2023].
[38]	Ministère de l'Intérieur, Monographie Générale. La Région de Fès-Meknès, 2015. Available from: https://knowledge-uclga.org/IMG/pdf/regiondefesmeknes-2.pdf.
[39]	Sick F, Schade S, Mourtada A, et al. (2014) Dynamic building simulations for the establishment of a Moroccan thermal regulation for buildings. J Green Build 9: 145–165. https://doi.org/10.3992/1943-4618-9.1.145 doi: 10.3992/1943-4618-9.1.145
[40]	Boujnah M, Jraida K, Farchi A, et al. (2016) Comparison of the calculation methods of heating and cooling. Int J Current Trends Eng Technol 2. Available from: http://ijctet.org/assets/upload/7371IJCTET2016120301.pdf.
[41]	Kharbouch Y, Ameur M (2021) Prediction of the impact of climate change on the thermal performance of walls and roof in Morocco. Int Rev Appl Sci Eng 13: 174–184. https://doi.org/10.1556/1848.2021.00330 doi: 10.1556/1848.2021.00330
[42]	Morocco sets regulations for energy efficiency. Oxford Business Group, 2015.[Online]. Available: https://oxfordbusinessgroup.com/analysis/morocco-sets-regulations-energy-efficiency.[Accessed 11 November 2022].
[43]	AMEE (Agence Marocaine pour l'Efficacité Energétique), Règlement Thermique de Construction au Maroc. Rabat, 2018. Available from: https://www.amee.ma/sites/default/files/inline-files/Lereglementthermique.pdf.
[44]	Bouroubat K, La construction durable: étude juridique comparative. HAL Open Science, Paris, 2017. Available from: https://theses.hal.science/tel-01617586/document.
[45]	M'Gbra N, Touzani A (2013) Energy efficiency codes in residential buildings and energy efficiency improvement in commercial and hospital buildings in Morocco. Mid-Term Evaluation Report on the UNDP/GEP Project, 5–34. Available from: https://procurement-notices.undp.org/view_file.cfm?doc_id=35481.
[46]	El Wardi FZ, Khabbazi A, Bencheikh C, et al. (2017) Insulation material for a model house in Zaouiat Sidi Abdessalam. In International Renewable and Sustainable Energy Conference (IRSEC), Tangier. https://doi.org/10.1109/IRSEC.2017.8477582
[47]	Gounni A, Ouhaibi S, Belouaggadia N, et al. (2022) Impact of COVID-19 restrictions on building energy consumption using Phase Change Materials (PCM) and insulation: A case study in six climatic zones of Morocco. J Energy Storage 55: 105374. https://doi.org/10.1016/j.est.2022.105374 doi: 10.1016/j.est.2022.105374
[48]	MHPV (Ministère de l'Habitat et de la Politique de la Ville), Guide des Bonnes Pratiques pour la Maitrise de l'Energie à l'Echelle de la Ville et de l'Habitat. Rabat, 2014. Available from: www.mhpv.gov.ma/wp-content/uploads/2021/11/Guide-de-bonnes-pratiques-pour-la-maitrise-de-l-energie.pdf.
[49]	PEEB (Programme for Energy Efficiency in Buildings), Building Sector Brief: Morocco. Agence Française de Développement, Paris, 2019. Available from: https://www.peeb.build/imglib/downloads/PEEB_Morocco_Country Brief_Mar 2019.pdf.
[50]	HCP (Haut-Commissariat au Plan), Les Indicateurs Sociaux du Maroc. Rabat, 2022. Available from: https://www.hcp.ma/Les-Indicateurs-sociaux-du-Maroc-Edition-2022_a3192.html#: ~: text=L'objectif%20de%20cette%20publication, une%20%C3%A9valuation%20des%20politiques%20publiques.
[51]	Lahlimi Alami A, Prospective Maroc—Energie 2030. HCP (Haut-Commissariat au Plan), Rabat, 2022. Available from: https://www.hcp.ma/downloads/?tag=Prospective+Maroc+2030.
[52]	Energy Efficiency in Buildings. AMEE, 2016. Available from: https://www.amee.ma/en/node/118.[Accessed 8 September 2022].
[53]	MTEDD (Ministère de la Transition Energétique et du Développement Durable), Campagne de Sensibilisation sur l'Economie d'Energie. 29 June 2022. Available from: https://www.mem.gov.ma/Pages/actualite.aspx?act=333.[Accessed 11 November 2022].
[54]	Ferreira D, Dey AK, Kostakos V (2011) Understanding human-smartphone concerns: A study of battery life. In International Conference of Pervasive Computing. https://doi.org/10.1007/978-3-642-21726-5_2
[55]	Karunarathna WKS, Jayaratne W, Dasanayaka S, et al. (2023) Factors affecting household's use of energy-saving appliances in Sri Lanka: An empirical study using a conceptualized technology acceptance model. Energy Effic, 16. https://doi.org/10.1007/s12053-023-10096-7
[56]	Waris I, Hameed I (2020) Promoting environmentally sustainable consumption behavior: an empirical evaluation of purchase intention of energy-efficient appliances. Energy Effic 13: 1653–1664. https://doi.org/10.1007/s12053-020-09901-4 doi: 10.1007/s12053-020-09901-4
[57]	HCP (Haut-Commissariat au Plan), Le secteur de l'emploi au Maroc. World Bank, Washington DC, 2021. Available from: https://www.hcp.ma/region-oriental/docs/Paysage%20de%20l%27%27emploi%20au%20Maroc%20_%20Recenser%20les%20obstacles%20a%20un%20marche%20du%20travail%20inclusif.pdf.
[58]	Gustafson S, Hartman W, Sellers B, et al. (2015) Energy sustainability in Morocco. Worcester Polytechnic Institute, Worcester. Available from: https://web.wpi.edu/Pubs/E-project/Available/E-project-101615-143625/unrestricted/energy-iqp_report-final2.pdf.
[59]	Hu Q, Qian X, Shen X, et al. (2022) Investigations on vapor cloud explosion hazards and critical safe reserves of LPG tanks. J Loss Prev Process Ind 80: 104904. https://doi.org/10.1016/j.jlp.2022.104904 doi: 10.1016/j.jlp.2022.104904
[60]	Zinecker A, Gagnon-Lebrun F, Touchette Y, et al. (2018) Swap: Reforming support for butane gas to invest in solar in Morocco. Int Inst Sustainable Dev. Available from: https://www.iisd.org/system/files/publications/swap-morocco-fr.pdf.
[61]	MEME (Ministère de l'Energie, des Mines et de l'Environnement), Feuille de Route Nationale pour la Valorisation Energétique de la Biomasse. Rabat, 2021. Available from: https://www.mem.gov.ma/Lists/Lst_rapports/Attachments/32/Feuille de Route Nationale pour la Valorisation Energétique de la Biomasse à l'horizon 2030.pdf.
[62]	Loutia M (2016) The applicability of geothermal energy for heating purposes in the region of Ifrane. Al Akhawayn University, Ifrane, 2016. Available from: www.aui.ma/sse-capstone-repository/pdf/spring2016/The Applicability Of Geothermal Energy For Heating Purposes In The Region of Ifrane.pdf.
[63]	Krarouch M, Lamghari S, Hamdi H, et al. (2020) Simulation and experimental investigation of a combined solar thermal and biomass heating system in Morocco. Energy Rep 6: 188–194. https://doi.org/10.1016/j.egyr.2020.11.270 doi: 10.1016/j.egyr.2020.11.270
[64]	HCP (Haut-Commissariat au Plan), Caractéristiques Démographiques et Socio-Economiques—Province Ifrane. Rabat, 2022. Available from: https://www.hcp.ma/region-meknes/attachment/1605477/.
[65]	HCP (Haut-Commissariat au Plan), Recensement Général de la Population et de l'Habitat 2014. HCP, Rabat, 2015. Available from: www.mhpv.gov.ma/wp-content/uploads/2019/12/RGPH-HABITAT.pdf.
[66]	Driouchi A, Zouag N (2006) Eléments pour le Renforcement de l'Insertion du Maroc dans l'Economie de Croissance. Haut-Commissariat au Plan, Ifrane, 2006. Available from: https://www.hcp.ma/downloads/?tag=Prospective+Maroc+2030.
[67]	MTEDD (Ministère de la Transition Energétique et du Développement Durable), Consommation énergétique par l'administration, 2019. Available from: https://siredd.environnement.gov.ma/fes-meknes/indicateur/DetailIndicateurPartial?idIndicateur=2988.[Accessed 3 September 2022].
[68]	Bami R (2022) Ifrane: L'énergie solaire remplace le bois. Yabiladi, 2022. Available from: https://www.yabiladi.com/article-societe-1636.html.[Accessed 18 November 2022].
[69]	MEMEE (Ministère de l'Energie, des Mines, de l'Eau et de l'Environnement), Stratégie Energétique Nationale—Horizon 2030. Rabat, 2021. Available from: https://www.mem.gov.ma/Lists/Lst_rapports/Attachments/33/Strat%C3%A9gue%20Nationale%20de%20l'Efficacit%C3%A9%20%C3%A9nerg%C3%A9tique%20%C3%A0%20l'horizon%202030.pdf.
[70]	Laroussi I (2017) Cost Study and Analysis of PV Installation per Watt Capacity in Ifrane. Al Akhawayn University, Ifrane, 2017. Available from: http://www.aui.ma/sse-capstone-repository/pdf/fall2017/PV%20INSTALLATION%20COST%20IN%20MOROCCO.%20ILIAS%20LAROUSSI.pdf.
[71]	Arechkik A, Sekkat A, Loudiyi K, et al. (2019) Performance evaluation of different photovoltaic technologies in the region of Ifrane, Morocco. Energy Sustainable Dev 52: 96–103. https://doi.org/10.1016/j.esd.2019.07.007 doi: 10.1016/j.esd.2019.07.007
[72]	Biodiesel Produced at AUI. Al Akhawayn University, Ifrane, 28 April 2016. Available from: http://www.aui.ma/en/media-room/news/al-akhawayn-news/3201-biodiesel-produced-at-aui.html.[Accessed 9 November 2022].
[73]	Derj A, Clean Energies Based Refurbishment of the Heating System of Al Akhawayn University Swimming Pool. Al Akhawayn University, Ifrane, 2015. Available from: www.aui.ma/sse-capstone-repository/pdf/Clean Energies Based Refurbishment of the Heating System of Al Akhawayn University Swimming Pool.pdf.
[74]	Farissi A, Driouach L, Zarbane K, et al. (2021) Covid-19 impact on moroccan small and medium-sized enterprises: Can lean practices be an effective solution for getting out of crisis? Manage Syst Prod Eng 29: 83–90. https://doi.org/10.2478/mspe-2021-0011
[75]	Yoo S-H (2005) Electricity consumption and economic growth: evidence from Korea. Energy Policy 33: 1627–1632. https://doi.org/10.2478/mspe-2021-0011 doi: 10.2478/mspe-2021-0011
[76]	Fatmi A (2022) Student Handbook & Planner. Al Akhawayn University, Ifrane. Available from: www.aui.ma/Student-handbook_2021-2022.pdf.
[77]	World Bank, The Social and Economic Impact of the Covid-19 Crisis in Morocco. Haut-Commissariat au Plan, Rabat, 2021. Available from: https://thedocs.worldbank.org/en/doc/852971598449488981-0280022020/original/ENGTheSocialandEconomicImpactoftheCovid19CrisisinMorocco.pdf.
[78]	Kharbouch Y, Mimet A, El Ganaoui M, et al. (2018) Thermal energy and economic analysis of a PCM-enhanced household envelope considering different climate zones in Morocco. Int J Sustainable Energy 37: 515–532. https://doi.org/10.1080/14786451.2017.1365076 doi: 10.1080/14786451.2017.1365076
[79]	Lachheb A, Allouhi A, Saadani R, et al. (2021) Thermal and economic analyses of different glazing systems for a commercial building in various Moroccan climates. Int J Energy Clean Environ 22: 15–41. https://doi.org/10.1615/InterJEnerCleanEnv.2020034790 doi: 10.1615/InterJEnerCleanEnv.2020034790
[80]	Nacer H, Radoine H, Mastouri H, et al. (2021) Sustainability assessment of an existing school building in Ifrane Morocco using LEED and WELL certification and environmental approach. In 9th International Renewable and Sustainable Energy Conference (IRSEC). https://doi.org/10.1109/IRSEC53969.2021.9741142
[81]	Houzir M, Plan Sectoriel—Eco Construction et Bâ timent Durable. UNEP, Rabat, 2016. Available from: https://switchmed.eu/wp-content/uploads/2020/04/02.-Sectoral-plan-construction-Morocco-in-french.pdf.
[82]	Beccali M, Finocchiaro P, Gentile V et al. (2017) Monitoring and energy performance assessment of an advanced DEC HVAC system in Morocco. In ISES Solar World Conference. https://doi.org/10.18086/swc.2017.28.01
[83]	Taimouri O, Souissi A (2019) Validation of a cooling loads calculation of an office building in Rabat Morocco based on manuel heat balance (Carrier Method). Int J Sci Technol Res 8: 2478–2484. Available from: https://www.ijstr.org/final-print/dec2019/Validation-Of-A-Cooling-Loads-Calculation-Of-An-Office-Building-In-Rabat-Morocco-Based-On-Manuel-Heat-Balance-carrier-Method.pdf.
[84]	IEA (International Energy Agency), Decree n. 2-17-746 on Mandatory energy audits and energy audit organizations, 2019. Available from: https://www.iea.org/policies/8571-decree-n-2-17-746-on-mandatory-energy-audits-and-energy-audit-organisations.[Accessed 26 October 2022].
[85]	Chramate I, Assadiki R, Zerrouq F, et al. (2018) Energy audit in Moroccan industries. Asial Life Sciences. Available from: https://www.researchgate.net/publication/330553987_Energy_audit_in_Moroccan_industries.
[86]	Lillemo SC (2014) Measuring the effect of procrastination and environmental awareness on households' energy-saving behaviours: An empirical approach. Energy Policy 66: 249–256. https://doi.org/10.1016/j.enpol.2013.10.077 doi: 10.1016/j.enpol.2013.10.077
[87]	Kang NN, Cho SH, Kim JT (2012) The energy-saving effects of apartment residents' awareness and behavior. Energy Build 46: 112–122. https://doi.org/10.1016/j.enbuild.2011.10.039 doi: 10.1016/j.enbuild.2011.10.039
[88]	Biresselioglu ME, Nilsen M, Demir MH, et al. (2018) Examining the barriers and motivators affecting European decision-makers in the development of smart and green energy technologies. J Cleaner Prod 198: 417–429. https://doi.org/10.1016/j.jclepro.2018.06.308 doi: 10.1016/j.jclepro.2018.06.308
[89]	Hartwig J, Kockat J (2016) Macroeconomic effects of energetic building retrofit: input-output sensitivity analyses. Constr Manage Econ 34: 79–97. https://doi.org/10.1080/01446193.2016.1144928 doi: 10.1080/01446193.2016.1144928
[90]	Pikas E, Kurnitski J, Liias R, et al. (2015) Quantification of economic benefits of renovation of apartment buildings as a basis for cost optimal 2030 energy efficiency strategies. Energy Build 86: 151–160. https://doi.org/10.1016/j.enbuild.2014.10.004 doi: 10.1016/j.enbuild.2014.10.004
[91]	Ferreira M, Almeida M (2015) Benefits from energy related building renovation beyond costs, energy and emissions. Energy Procedia 78: 2397–2402. https://doi.org/10.1016/j.egypro.2015.11.199 doi: 10.1016/j.egypro.2015.11.199
[92]	Song X, Ye C, Li H, et al. (2016) Field study on energy economic assessment of office buildings envelope retrofitting in southern China. Sustainable Cities Soc 28: 154–161. https://doi.org/10.1016/j.scs.2016.08.029 doi: 10.1016/j.scs.2016.08.029
[93]	Kaynakli O (2012) A review of the economical and optimum thermal insulation thickness for building applications. Renewable Sustainable Energy Rev 16,415–425. https://doi.org/10.1016/j.rser.2011.08.006
[94]	Bambara J, Athienitis AK (2018) Energy and economic analysis for greenhouse envelope design. Trans ASABE 61: 1795–1810. https://doi.org/10.13031/trans.13025 doi: 10.13031/trans.13025
[95]	Struhala K, Ostrý M (2022) Life-Cycle Assessment of phase-change materials in buildings: A review. J Cleaner Prod 336: 130359. https://doi.org/10.1016/j.jclepro.2022.130359 doi: 10.1016/j.jclepro.2022.130359
[96]	Arumugam P, Ramalingam V, Vellaichamy P (2022) Effective PCM, insulation, natural and/or night ventilation techniques to enhance the thermal performance of buildings located in various climates—A review. Energy Build 258: 111840. https://doi.org/10.1016/j.enbuild.2022.111840 doi: 10.1016/j.enbuild.2022.111840
[97]	Jaffe AB, Stavins RN (1994) The energy-efficiency gap—What does it mean? Energy Policy 22: 804–810. https://doi.org/10.1016/0301-4215(94)90138-4 doi: 10.1016/0301-4215(94)90138-4
[98]	Backlund S, Thollander P, Palm J, et al. (2012) Extending the energy efficiency gap. Energy Policy 51: 392–396. https://doi.org/10.1016/j.enpol.2012.08.042 doi: 10.1016/j.enpol.2012.08.042
[99]	Gerarden TD, Newell RG, Stavins RN (2017) Assessing the energy-efficiency gap. J Econ Lit 55: 1486–1525. https://doi.org/10.1257/jel.20161360 doi: 10.1257/jel.20161360
[100]	Chai K-H, Yeo C (2012) Overcoming energy efficiency barriers through systems approach—A conceptual framework. Energy Policy 46: 460–472. https://doi.org/10.1016/j.enpol.2012.04.012 doi: 10.1016/j.enpol.2012.04.012
[101]	Allcott H (2011) Consumers' perceptions and misperceptions of energy costs. Am Econ Rev 101: 98–104. https://doi.org/10.1257/aer.101.3.98 doi: 10.1257/aer.101.3.98
[102]	Davis LW, Metcalf GE (2016) Does better information lead to better choices? Evidence from energy-efficiency labels. J Assoc Environ Resour Econ 3: 589–625. https://doi.org/10.1086/686252 doi: 10.1086/686252
[103]	Shen J (2012) Understanding the Determinants of Consumers' Willingness to Pay for Eco-Labeled Products: An Empirical Analysis of the China Environmental Label. J Serv Sci Manage 5: 87–94. https://doi.org/10.4236/jssm.2012.51011 doi: 10.4236/jssm.2012.51011
[104]	Poortinga W, Steg L, Vlek C, et al. (2003) Household preferences for energy-saving measures: A conjoint analysis. J Econ Psychol 24: 49–64. https://doi.org/10.1016/S0167-4870(02)00154-X doi: 10.1016/S0167-4870(02)00154-X
[105]	Banerjee A, Solomon BD (2003) Eco-labeling for energy efficiency and sustainability: a meta-evaluation of US programs. Energy Policy 31: 109–123. https://doi.org/10.1016/S0301-4215(02)00012-5 doi: 10.1016/S0301-4215(02)00012-5
[106]	Sammer K, Wüstenhagen R (2006) The influence of eco-labelling on consumer behaviour—results of a discrete choice analysis for washing machines. Bus Strategy Environ 15: 185–199. https://doi.org/10.1002/bse.522 doi: 10.1002/bse.522
[107]	Shen L, Sun Y (2016) Performance comparisons of two system sizing approaches for net zero energy building clusters under uncertainties. Energy Build 127: 10–21. https://doi.org/10.1016/j.enbuild.2016.05.072 doi: 10.1016/j.enbuild.2016.05.072
[108]	Good C, Andresen I, Hestnes AG (2015) Solar energy for net zero energy buildings—A comparison between solar thermal, PV and photovoltaic–thermal (PV/T) systems. Sol Energy 123: 986–996. https://doi.org/10.1016/j.solener.2015.10.013 doi: 10.1016/j.solener.2015.10.013
[109]	Harkouss F, Fardoun F, Biwole PH (2018) Passive design optimization of low energy buildings in different climates. Energy 165: 591–613, 2018. https://doi.org/10.1016/j.energy.2018.09.019 doi: 10.1016/j.energy.2018.09.019
[110]	Penna P, Prada A, Cappelletti F, et al. (2015) Multi-objectives optimization of energy efficiency measures in existing buildings. Energy Build 95: 57–69. https://doi.org/10.1016/j.enbuild.2014.11.003 doi: 10.1016/j.enbuild.2014.11.003
[111]	Serbouti A, Rattal M, Boulal A, et al. (2018) Multi-Objective optimization of a family house performance and forecast of its energy needs by 2100. Int J Eng Technol 7: 7–10. Available from: https://www.sciencepubco.com/index.php/ijet/article/view/23235.
[112]	ONEEP (Office National de l'Electricité et de l'Eau Potable), Tarif Général (MT). ONEE, 1 January 2017. Available from: http://www.one.org.ma/FR/pages/interne.asp?esp=1&id1=2&id2=35&id3=10&t2=1&t3=1.[Accessed 24 November 2022].
[113]	ONEEP (Office National de l'Electricité et de l'Eau Potable), Nos tarifs. ONEEP, 1 January 2017. Available from: http://www.one.org.ma/FR/pages/interne.asp?esp=1&id1=3&id2=113&t2=1.[Accessed 24 November 2022].
[114]	Abdou N, EL Mghouchi Y, Hamdaoui S, et al. (2021) Multi-objective optimization of passive energy efficiency measures for net-zero energy building in Morocco. Build Environ 204: 108141. https://doi.org/10.1016/j.buildenv.2021.108141 doi: 10.1016/j.buildenv.2021.108141
[115]	Srinivas M (2011) Domestic solar hot water systems: Developments, evaluations and essentials for viability with a special reference to India. Renewable Sustainable Energy Rev 15: 3850–3861. https://doi.org/10.1016/j.rser.2011.07.006 doi: 10.1016/j.rser.2011.07.006
[116]	Hudon K (2014) Chapter 20—Solar Energy—Water Heating. In Future Energy: Improved, Sustainable and Clean Options for our Planet. Elsevier Science, 433–451. https://doi.org/10.1016/B978-0-08-099424-6.00020-X
[117]	Bertoldi P (2022) Policies for energy conservation and sufficiency: Review of existing. Energy Build 26: 112075. https://doi.org/10.1016/j.enbuild.2022.112075 doi: 10.1016/j.enbuild.2022.112075
[118]	Bertoldi P (2020) Chapter 4.3—Overview of the European Union policies to promote more sustainable behaviours in energy end-users, Energy and Behaviour: Towards a Low Carbon Future. Academic Press: 451–477. https://doi.org/10.1016/B978-0-12-818567-4.00018-1
[119]	Herring H (2006) Energy efficiency—A critical view. Energy 31: 10–20. https://doi.org/10.1016/j.energy.2004.04.055 doi: 10.1016/j.energy.2004.04.055
[120]	Sorrell S, Gatersleben B, Druckman A (2020) The limits of energy sufficiency: A review of the evidence for rebound effects and negative spillovers from behavioural change. Energy Res Soc Sci 64: 101439. https://doi.org/10.1016/j.erss.2020.101439 doi: 10.1016/j.erss.2020.101439
[121]	Sachs W (1999) The Power of Limits: An Inquiry into New Models of Wealth, in Planet Dialectics. Explorations in Environment and Development, London, ZED-BOOKS. Available from: https://www.researchgate.net/publication/310580761_The_power_of_limits.
[122]	Brischke LA, Lehmann F, Leuser L, et al. (2015) Energy sufficiency in private households enabled by adequate appliances. In ECEEE Summer Study proceedings. Available from: https://epub.wupperinst.org/frontdoor/deliver/index/docId/5932/file/5932_Brischke.pdf.
[123]	Spangenberg JH, Lorek S (2019) Sufficiency and consumer behaviour: From theory to policy. Energy Policy 129: 1070–1079. https://doi.org/10.1016/j.enpol.2019.03.013 doi: 10.1016/j.enpol.2019.03.013
[124]	Heindl P, Kanschik P (2016) Ecological sufficiency, individual liberties, and distributive justice: Implications for policy making. Ecol Econ 126: 42–50. https://doi.org/10.1016/j.ecolecon.2016.03.019 doi: 10.1016/j.ecolecon.2016.03.019
[125]	IEA (International Energy Agency), Energy Policies beyond IEA Countries: Morocco 2019. IEA, 2019. Available from: https://www.iea.org/reports/energy-policies-beyond-iea-countries-morocco-2019.
[126]	Palermo V, Bertoldi P, Apostolou M, et al. (2020) Assessment of climate change mitigation policies in 315 cities in the Covenant of Mayors initiative. Sustainable Cities Soc 60: 102258. https://doi.org/10.1016/j.scs.2020.102258 doi: 10.1016/j.scs.2020.102258
[127]	Kona A, Bertoldi P, Kilkis S (2019) Covenant of mayors: Local energy generation, methodology, policies and good practice examples. Energies 12: 985. https://doi.org/10.3390/en12060985 doi: 10.3390/en12060985
[128]	Tsemekidi Tzeiranaki S, Bertoldi P, Diluiso F, et al. (2019) Analysis of the EU residential energy consumption: Trends and determinants. Energies 12: 1065. https://doi.org/10.3390/en12061065 doi: 10.3390/en12061065
[129]	Köppen W (1900) Klassification der Klimate nach Temperatur, Niederschlag and Jahreslauf. Petermanns Geographische Mitteilungen 6: 593–611. Available from: koeppen-geiger.vu-wien.ac.at/pdf/Koppen_1918.pdf.
[130]	Chen D, Chen HW (2013) Using the Köppen classification to quantify climate variation and change: An example for 1901–2010. Environ Dev 6: 69–79. https://doi.org/10.1016/j.envdev.2013.03.007 doi: 10.1016/j.envdev.2013.03.007
[131]	Perez-Garcia A, Guardiola AP, Gómez-Martínez F, et al. (2018) Energy-saving potential of large housing stocks of listed buildings, case study: l'Eixample of Valencia. Sustainable Cities Soc 42: 59–81. https://doi.org/10.1016/j.scs.2018.06.018 doi: 10.1016/j.scs.2018.06.018
[132]	Wang X, Ding C, Zhou M, et al. (2023) Assessment of space heating consumption efficiency based on a household survey in the hot summer and cold winter climate zone in China. Energy 274: 127381. https://doi.org/10.1016/j.energy.2023.127381 doi: 10.1016/j.energy.2023.127381
[133]	Cao X, Yao R, Ding C, et al. (2021) Energy-quota-based integrated solutions for heating and cooling of residential buildings in the Hot Summer and Cold Winter zone in China. Energy Build 236: 110767. https://doi.org/10.1016/j.enbuild.2021.110767 doi: 10.1016/j.enbuild.2021.110767
[134]	Deng Y, Gou Z, Gui X, et al. (2021) Energy consumption characteristics and influential use behaviors in university dormitory buildings in China's hot summer-cold winter climate region. J Build Eng 33: 101870. https://doi.org/10.1016/j.jobe.2020.101870 doi: 10.1016/j.jobe.2020.101870
[135]	Liu H, Kojima S (2017) Evaluation on the energy consumption and thermal performance in different residential building types during mid-season in hot-summer and cold-winter zone in China. Proc Eng 180: 282–291. https://doi.org/10.1016/j.proeng.2017.04.187 doi: 10.1016/j.proeng.2017.04.187
[136]	Geraldi MS, Melo AP, Lamberts R, et al. (2022) Assessment of the energy consumption in non-residential building sector in Brazil. Energy Build 273: 112371. https://doi.org/10.1016/j.enbuild.2022.112371 doi: 10.1016/j.enbuild.2022.112371
[137]	El Hafdaoui H, El Alaoui H, Mahidat S, et al. (2023) Impact of hot arid climate on optimal placement of electric vehicle charging stations. Energies 16: 753. https://doi.org/10.3390/en16020753 doi: 10.3390/en16020753

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