To reduce carbon dioxide emissions, electrified aircraft propulsion has been extensively studied worldwide. Among electrified aircraft, urban air mobility features short-range flying times. Both fully electric and hybrid types have been developed, undergoing field tests and awaiting commercial service induction. This paper describes a virtual electrical aircraft propulsion system designed as a 600 kW-rated aviation system for five passengers and one pilot, with some luggage, simulating a real flight. The propulsion system is powered by batteries and motors, comprising 12 distributed electric propulsion (DEP) systems. Each DEP inverter circuit is a two-level, three-phase inverter. A silicon carbide (SiC) metal–oxide–semiconductor field-effect transistor (MOSFET) module and an isolation gate driver Integrated Circuit (IC) are used to evaluate this inverter system. Through a virtual power profile, power redundancy is analyzed under normal conditions and unplanned worst-case scenarios. For a planned trajectory, the system's power profile is assessed during the takeoff, cruise, and landing phases. The DC-link voltage discharged is also reviewed for these flight stages. In some cases, due to external conditions, a quick escape motion during the cruise phase requires the propulsion system to operate at maximum power for a specific duration. In such unplanned events, flight time extends, leading to greater DC-link voltage discharge compared with a planned trajectory. To calculate power loss, switching losses are measured using a demo power board. These measurement results are used in loss simulation instead of datasheet values. Each DEP's output power value is derived from the system's virtual profile, and each DEP inverter's power loss is calculated through measurement-based simulation. Maximum power and current are evaluated as critical conditions in each virtual profile. Through loss and thermal simulations, the power redundancy of the proposed DEP inverter system is validated for use in electric propulsion systems.
Citation: Simon Kim, Donghyun Lee, Byunggil Kwak, Myeonghyo Kim. For electrified aircraft propulsion, SiC MOSFET inverter design with power redundancy in both escape event and failure[J]. Metascience in Aerospace, 2025, 2(1): 1-14. doi: 10.3934/mina.2025001
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To reduce carbon dioxide emissions, electrified aircraft propulsion has been extensively studied worldwide. Among electrified aircraft, urban air mobility features short-range flying times. Both fully electric and hybrid types have been developed, undergoing field tests and awaiting commercial service induction. This paper describes a virtual electrical aircraft propulsion system designed as a 600 kW-rated aviation system for five passengers and one pilot, with some luggage, simulating a real flight. The propulsion system is powered by batteries and motors, comprising 12 distributed electric propulsion (DEP) systems. Each DEP inverter circuit is a two-level, three-phase inverter. A silicon carbide (SiC) metal–oxide–semiconductor field-effect transistor (MOSFET) module and an isolation gate driver Integrated Circuit (IC) are used to evaluate this inverter system. Through a virtual power profile, power redundancy is analyzed under normal conditions and unplanned worst-case scenarios. For a planned trajectory, the system's power profile is assessed during the takeoff, cruise, and landing phases. The DC-link voltage discharged is also reviewed for these flight stages. In some cases, due to external conditions, a quick escape motion during the cruise phase requires the propulsion system to operate at maximum power for a specific duration. In such unplanned events, flight time extends, leading to greater DC-link voltage discharge compared with a planned trajectory. To calculate power loss, switching losses are measured using a demo power board. These measurement results are used in loss simulation instead of datasheet values. Each DEP's output power value is derived from the system's virtual profile, and each DEP inverter's power loss is calculated through measurement-based simulation. Maximum power and current are evaluated as critical conditions in each virtual profile. Through loss and thermal simulations, the power redundancy of the proposed DEP inverter system is validated for use in electric propulsion systems.
In the early 1930s, in order to generalize the formula of quantum mechanics, Jordan et al. introduced an important commutative non-associative algebra [1], which was initially called "r-order digital system". In 1947, Albert renamed this kind of algebra Jordan algebra and studied their structural theory [2]. Since then, Jordan algebras have attracted extensive attention. Particularly, Jacobson developed the representation theory of Jordan algebras [3,4]. Jordan superalgebras were first studied by Kac, who classified simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero [5]. Jordan superalgebras also have significant applications in quantum mechanics [6,7]. More results on Jordan superalgebras are available in [8,9].
Hom-type algebras were first introduced to study the q-deformation of Witt and Virasoro algebras [10,11], which played an important role in physics, mainly in conformal field theory. Bihom-type algebras are generalizations of Hom-type algebras, which were presented by Graziani et al. from the categorical point and applied to study certain deformations of quantum groups [12]. Up to now, the (Bi)hom-structures of various algebras have been intensively investigated. The construction relationship between Hom-type algebras and the module structure on them can be found in the literature [13,14,15,16,17]. Naturally, the construction between Bihom-type algebras is studied in the literature [18,19], and the results of representation and deformation can be found in [20,21,22]. In this paper, we first generalize bimodules and representations of Bihom-Jordan algebras [23,24] to Bihom-Jordan superalgebras and then develop the theory of representations and O-operators on Bihom-Jordan superalgebras.
The outline of the paper is presented as follows: In Section 2, we review some basics about Bihom-superalgebras, Bihom-Jordan superalgebras; we study Bihom-super modules and give some easy constructions of Bihom-Jordan superalgebras. In Section 3, we mainly study super-bimodules on Bihom-Jordan superalgebras and obtain some new constructions under the view of module. In Section 4, we study the representation of Bihom-Jordan superalgebra and give the definitions of O-operator and Rota–Baxter operator. At the same time, we also give the definition of Bihom-pre-Jordan superalgebra. Finally, the relationship between O-operator and Bihom-pre-Jordan superalgebra is studied. Actually, on the basis of this section, we can also continue to study cohomology theory.
Throughout the paper, all algebraic systems are supposed to be over a field of characteristic 0. Let A be a linear superspace over K that is a Z2-graded linear space with a direct sum A=A¯0⊕A¯1. The elements of Aj,j=0,1, are said to be homogenous and of parity j. The parity of a homogeneous element x is denoted by |x|. In the sequel, we will denote by H(A) the set of all homogeneous elements of A. In this paper, we need to use the elements, all of which are not specified, are homogeneous.
In this section, we recall some basic definitions about Bihom-Jordan superalgebras, provide some construction results. A Bihom-superalgebra is a quadruple (J,μ,α,β), where μ:J⊗J→J is an even bilinear map and α,β:J→J are even linear maps such that α∘μ=μ∘α and β∘μ=μ∘β (multiplicativity).
Definition 2.1. [25] Let (J,μ,α,β) be a Bihom-superalgebra.
The Bihom-associator of J is an even trilinear map asα,β:J⊗3⟶J defined by
asα,β=μ∘(μ⊗β−α⊗μ). | (2.1) |
For any ε,γ,δ∈H(J), asα,β(ε,γ,δ)=μ(μ(ε,γ),β(δ))−μ(α(ε),μ(γ,δ)).
In particular, when α=β=Id, Bihom-superalgebra is to degenerate to the superalgebra, so is Bihom-associator degenerates to the original associator. If α=β, Bihom-associator degenerates to the Hom-associator.
Definition 2.2. Let (J,μ,α,β) be a Bihom-superalgebra. Then
● A Bihom-sub-superalgebra of J is a Z2-graded linear subspace B⊆J, which satisfies μ(ε,γ)∈B,α(ε)∈B and β(ε)∈B, for all ε,γ∈H(J). Furthermore, if μ(ε,γ),μ(γ,ε)∈B, for all (ε,γ)∈J×B, then B is called a two-sided Bihom-ideal of J.
● J is regular if α and β are algebra automorphisms.
● J is involutive if α and β are two involutions, that is α2=β2=Id.
Definition 2.3. Let (J,μ,α,β) and (J′,μ′,α′β′) be two Bihom-superalgebras. If a homomorphism f:J→J′ satisfies the following conditions:
f∘μ=μ′∘(f⊗f),f∘α=α′∘fandf∘β=β′∘f. |
Then f is called Bihom-superalgebra morphism. And we call the set Γf={ε+f(ε)|ε∈H(J)}⊂J⊕J′ the graph of f.
Proposition 2.1. Let (J,μJ,αJ,βJ) and (B,μB,αB,βB) be two Bihom-Jordan superalgebras. Then an even linear map f:J→B is a morphism if and only if its graph Γf is a Bihom-subalgebra of (J⊕B,μ=μJ+μB,α=αJ+αB,β=βJ+βB).
Proof. Suppose that f is a morphism of Bihom-Jordan superalgebras. Clearly, Γf is a subspace of J⊕B, we only need to prove the Γf is closed under the μ,α,β. For all ε,γ∈H(J),
μ(ε+f(ε),γ+f(γ))=μJ(ε,γ)+μB(f(ε),f(γ))=μJ(ε,γ)+fμJ(ε,γ). |
Moreover, by fαJ=αBf and fβJ=βBf,
α(ε+f(ε))=αJ(ε)+αB(f(ε))=αJ(ε)+fαJ(ε),β(ε+f(ε))=βJ(ε)+βB(f(ε))=βJ(ε)+fβJ(ε). |
It follows that Γf is a Bihom-subalgebra of J⊕B.
Conversely, Γf is a Bihom-subalgebra of J⊕B, so
μ(ε+f(ε),γ+f(γ))=μJ(ε,γ)+μB(f(ε),f(γ))∈Γf, |
which implies that μB(f(ε),f(γ))=fμJ(ε,γ). Similarly, we also obtain αBf=fαJ and βBf=fβJ from α(Γf)⊆Γf and β(Γf)⊆Γf, respectively. Thus, f is a morphism of Bihom-Jordan superalgebras.
Definition 2.4. [25] A Bihom-associative superalgebra is a quadruple (J,μ,α,β), where α,β:J→J are even linear maps and μ:J×J→J is an even bilinear map such that αβ=βα,αμ=μα⊗2, βμ=μβ⊗2 and satisfying Bihom-associator is zero:
asα,β(ε,γ,δ)=0,forallε,γ,δ∈H(J). (Bihom-associativity condition) |
Clearly, when α=β, we obtain a Hom-associative superalgebra.
Definition 2.5. [19] A BiHom superalgebra (J,μ,α,β) is called a Bihom-Jordan superalgebra if for all ε,γ,δ,t∈H(J):
(i) αβ=βα,(ii) μ(β(ε),α(γ))=(−1)|ε||γ|μ(β(γ),α(ε)),(Bihom-super commutativity condition)(iii) ↺ε,γ,t(−1)|t|(|ε|+|δ|)~asα,β(μ(β2(ε),αβ(γ)),α2β(δ),α3(t))=0.(Bihom-Jordan super-identity) |
In particular, it is reduced to a Jordan superalgebra when α=β=Id.
Next, we give some common construction methods. Let (J,μ,α,β) be a Bihom-superalgebra. Define its plus Bihom-superalgebra as the Bihom-superalgebra J+=(J,∗,α,β), where
ε∗γ=12(μ(ε,γ)+(−1)|ε||γ|μ(α−1β(γ),αβ−1(ε))). |
Note that product ∗ is Bihom-supercommutative. In fact, for all ε,γ∈H(J),
β(ε)∗α(γ)=12(β(ε)α(γ)+(−1)|ε||γ|β(γ)α(ε))=(−1)|ε||γ|12(β(γ)α(ε)+β(ε)α(γ))=(−1)|ε||γ|β(γ)∗α(ε). |
Moreover, the plus Bihom-superalgebra J+=(J,∗,α,β) is a Bihom-Jordan superalgebra. Naturally, we define
ε⋄γ=μ(ε,γ)+(−1)|ε||γ|μ(α−1β(γ),αβ−1(ε)), |
the ⋄ is also Bihom-supercommutative. Then J⋄=(J,⋄,α,β) is also a Bihom-Jordan superlagebra.
Besides that, Bihom-Jordan superalgebra (J,μα,β=μ(α⊗β),α,β) can be obtained from Jordan superalgebra (J,μ). We also consider the quotient algebra obtained by modulo Bihom-ideal, given a Bihom-Jordan superalgebra (J,μ,α,β) and I is a Bihom-ideal. Define ˉμ,ˉα,ˉβ on J/I as follows:
ˉμ(ˉε,ˉγ)=¯μ(ε,γ),ˉα(ˉε)=¯α(ε),ˉβ(ˉε)=¯β(ε). |
Then (J/I,ˉμ,ˉα,ˉβ) is also a Bihom-Jordan superalgebra.
Example 2.1. Given a 3-dimensional Jordan superalgebra (J=Jˉ0⊕Jˉ1,μ) in [5], the bases of Jˉ0 and Jˉ1 are {ε} and {u,v}, respectively. The nontrivial multiplication is defined as follows:
μ(ε,ε)=ε,μ(ε,u)=12u,μ(ε,v)=12v,μ(u,v)=ε. |
We consider two even endomorphisms α and β, which satisfy α(ε)=ε,α(u)=−u,α(v)=−v, and β(ε)=−ε,β(u)=−u,β(v)=v. Then we obtain a Bihom-Jordan superalgebra (J,μ′=μ(α⊗β),α,β).
Example 2.2. In [5], let (J=Jˉ0⊕Jˉ1,μ) be a Jordan superalgebra with the nontrivial multiplication as follows:
μ(ε,ε)=2ε,μ(ε,u)=u,μ(ε,v)=v,μ(u,v)=1+kx, |
k∈K and k≠12, where {1,ε} and {u,v} are bases of Jˉ0 and Jˉ1, respectively. We define two even endomorphisms α and β satisfies α(1)=1,α(ε)=ε,α(u)=−u,α(v)=−v and β(1)=1,β(ε)=−ε,β(u)=u,β(v)=v. Then we obtain a Bihom-Jordan superalgebra (J,μ′=μ(α⊗β),α,β).
Definition 2.6. Let (J,μ,α,β) be a Bihom-superalgebra.
1) A Bihom-super-module (V,ϕ,ψ) is called an J-super-bimodule if it is equipped with an even left structure ρl and an even right structure map ρr on Z2-graded vector space V, ρl and ρr are given by
● ρl:(J⊗V,α⊗ϕ,β⊗ψ)→(V,ϕ,ψ),ρl(a,v)=a⋅v,
● ρr:(V⊗J,ϕ⊗α,ψ⊗β)→(V,ϕ,ψ),ρr(v,a)=v⋅a.
2) An even linear map f:(V,ϕ,ψ,ρl,ρr)→(V′,ϕ′,ψ′,ρ′l,ρ′r) is a morphism of the Bihom-super-modules such that the following commutative diagrams
![]() |
3) Let (V,ϕ,ψ,ρl,ρr) be an J-super-bimodule. Then the module Bihom-associator asVϕ,ψ of V is defined as:
asVϕ,ψ∘IdV⊗J⊗J=ρr∘(ρr⊗β)−ρr∘(ϕ⊗μ), | (2.2) |
asVϕ,ψ∘IdJ⊗V⊗J=ρr∘(ρl⊗β)−ρl∘(α⊗ρr), | (2.3) |
asVϕ,ψ∘IdJ⊗J⊗V=ρl∘(μ⊗ψ)−ρl∘(α⊗ρl). | (2.4) |
Definition 2.7. Let (J,μ,α,β) be a Bihom-associative superalgebra and (V,ϕ,ψ) be a Bihom-super-module. Then
1) A left Bihom-associative J-super-module structure consists of an even morphism ρl:J⊗V→V satisfies asVϕ,ψ=0 in (2.4).
2) A right Bihom-associative J-super-module structure consists of an even morphism ρr:V⊗J→V satisfies asVϕ,ψ=0 in (2.2).
3) A Bihom-associative J-super-bimodule structure consists of an even morphism ρl:J⊗V→V and an even morphism ρr:V⊗J→V such that (V,ϕ,ψ,ρl) is a left Bihom-associative J-super-module, (V,ϕ,ψ,ρr) is a right Bihom-associative J-super-module, and satisfies asVϕ,ψ=0 in (2.3).
In this section, we introduce super-bimodules of Bihom-Jordan superalgebras and give some of their constructions. Finally, we define an abelian extension in order to give an application in the next section. For convenience, the sign will subsequently be omitted from the product operation of elements in J.
Definition 3.1. Let (J,μ,α,β) be a Bihom-Jordan superalgebra. For all ε,γ,δ∈H(J),v∈H(V),
● A left Bihom-Jordan J-super-module is a Bihom-super-module (V,ϕ,ψ) that is equipped with an even left structure map ρl:J⊗V→V,ρl(a⊗v)=a⋅v such that ψ is invertible and the following conditions hold:
↺ε,γ,δ(−1)|γ||δ|β2α2(δ)⋅(αβ(ε)α2(γ)⋅ϕ3(v))=↺ε,γ,δ(−1)|ε||δ|αβ2(ε)α2β(γ)⋅(βα2(δ)⋅ϕ3(v)), | (3.1) |
β2α2(δ)⋅(βα2(γ)⋅(α2(ε)⋅ψ−1ϕ3(v)))+(−1)|ε||γ|+|ε||δ|+|γ||δ|β2α2(ε)⋅(βα2(γ)⋅(α2(δ)⋅ψ−1ϕ3(v)))+(−1)|ε||δ|+|ε||γ|((β2(ε)βα(δ))βα2(γ))⋅ϕ3ψ(v)=(−1)|γ||δ|β2α(γ)βα2(δ)⋅(βα2(ε)⋅ϕ3(v))+(−1)|γ||δ|+|ε||δ|β2α(γ)βα2(ε)⋅(βα2(δ)⋅ϕ3(v))+(−1)|ε||δ|+|ε||γ|β2α(ε)βα2(δ)⋅(βα2(γ)⋅ϕ3(v)). | (3.2) |
● A right Bihom-Jordan J-super-module is a Bihom-super-module (V,ϕ,ψ) that is equipped with an even right structure map ρr:V⊗J→V,ρr(v⊗a)=v⋅a such that the following conditions hold:
↺ε,γ,δ(−1)|ε||δ|(ϕψ2(v)⋅αβ(ε)α2(γ))⋅βα3(δ)=↺ε,γ,δ(−1)|γ||δ|(ϕψ2(v)⋅βα2(δ))⋅α2β(ε)α3(γ). | (3.3) |
((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|ε||γ|+|ε||δ|+|γ||δ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)+(−1)|γ||δ|ϕ2ψ2(v)⋅(βα(ε)α2(δ))α3(γ)=(ϕψ2(v)⋅βα2(ε))⋅α2β(γ)α3(δ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|(ϕψ2(v)⋅βα2(δ))⋅α2β(γ)α3(ε)+(−1)|ε||γ|(ϕψ2(v)⋅βα2(γ))⋅α2β(ε)α3(δ). | (3.4) |
Theorem 3.1. Let (J,μ,α,β) be a Bihom-Jordan superalgebra, (V,ϕ,ψ) be a Bihom-super-module and ρr:V⊗J→V,ρr(v⊗a)=v⋅a be an even linear map, which satisfies the following conditions: for all ε,γ∈H(J),v∈H(V),
ϕρr=ρr(ϕ⊗α),ψρr=ρr(ψ⊗β), | (3.5) |
ϕ(v)⋅β(ε)α(γ)=(v⋅β(ε))⋅βα(γ)+(−1)|ε||γ|(v⋅β(γ))⋅αβ(ε). | (3.6) |
Then (V,ϕ,ψ,ρr) is a right Bihom-Jordan J-super-module, called a right special Bihom-Jordan J-super-module.
Proof. For any ε,γ,δ∈H(J),v∈H(V),
↺ε,γ,δ(−1)|δ||γ|(ϕψ2(v)⋅βα2(δ))⋅α2β(ε)α3(γ)=↺ε,γ,δ(−1)|δ||γ|ϕ(ψ2(v)⋅βα(δ))⋅α2β(ε)α3(γ)(by(3.5))=↺ε,γ,δ(−1)|δ||γ|((ψ2(v)⋅βα(δ))⋅βα2(ε))⋅βα3(γ)+↺ε,γ,δ(−1)|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)(by(3.6))=↺ε,γ,δ(−1)|δ||γ|(ϕψ2(v)⋅βα(δ)α2(ε))⋅βα2(γ)−↺ε,γ,δ(−1)|δ||γ|+|δ||ε|((ψ2(v)⋅βα(ε))⋅βα2(δ))⋅βα3(γ)+↺ε,γ,δ(−1)|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)(by(3.6))=↺ε,γ,δ(−1)|ε||δ|(ϕψ2(v)⋅βα(ε)α2(γ))⋅βα2(δ). |
So Eq (3.3) holds. On the other hand,
(ϕψ2(v)⋅βα2(ε))⋅βα2(γ)α3(δ)+(−1)|ε||γ|(ϕψ2(v)⋅βα2(γ))⋅βα2(ε)α3(δ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|(ϕψ2(v)⋅βα2(δ))⋅βα2(γ)α3(ε)=ϕ(ψ2(v)⋅βα(ε))⋅βα2(γ)α3(δ)+(−1)|ε||γ|ϕ(ψ2(v)⋅βα(γ))⋅βα2(ε)α3(δ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|ϕ(ψ2(v)⋅βα(δ))⋅βα2(γ)α3(ε)(by(3.5))=((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|γ||δ|((ψ2(v)⋅βα(ε))⋅βα2(δ))⋅βα3(γ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)+(−1)|δ||ε|+|δ||γ|((ψ2(v)⋅βα(δ))⋅βα2(ε))⋅βα3(γ)+(−1)|ε||γ|((ψ2(v)⋅βα(γ))⋅βα2(ε))⋅βα3(δ)+(−1)|ε||γ|+|ε||δ|((ψ2(v)⋅βα(γ))⋅βα2(δ))⋅βα3(ε)(by(3.6))=((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|γ||δ|((ϕψ2(v)⋅βα(ε)α2(δ))⋅α3β(γ)−(−1)|γ||δ|+|ε||δ|((ψ2(v)⋅βα(δ))⋅βα2(ε))⋅βα3(γ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)+(−1)|δ||ε|+|δ||γ|((ψ2(v)⋅βα(δ))⋅βα2(ε))⋅βα3(γ)+(−1)|ε||γ|((ψ2(v)⋅βα(γ))⋅βα2(ε))⋅βα3(δ)+(−1)|ε||γ|+|ε||δ|((ψ2(v)⋅βα(γ))⋅βα2(δ))⋅βα3(ε)(by(3.6))=((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|γ||δ|((ϕψ2(v)⋅βα(ε)α2(δ))⋅α3β(γ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)+(−1)|ε||γ|((ψ2(v)⋅βα(γ))⋅βα2(ε))⋅βα3(δ)+(−1)|ε||γ|+|ε||δ|((ψ2(v)⋅βα(γ))⋅βα2(δ))⋅βα3(ε)=((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|γ||δ|ϕ2ψ2(v)⋅(βα(ε)α2(δ))α3(δ)−(−1)|ε||γ|(ϕψ2(v)⋅βα2(γ))⋅βα2(ε)α3(δ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)+(−1)|ε||γ|((ψ2(v)⋅βα(γ))⋅βα2(ε))⋅βα3(δ)+(−1)|ε||γ|+|ε||δ|((ψ2(v)⋅βα(γ))⋅βα2(δ))⋅βα3(ε)(by(3.6))=((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅βα3(δ)+(−1)|γ||δ|ϕ2ψ2(v)⋅(βα(ε)α2(δ))α3(δ)+(−1)|δ||ε|+|δ||γ|+|ε||γ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅βα3(ε)(by(3.6)). |
It follows Eq (3.4).
Similarly, we have the following result.
Theorem 3.2. Let (J,μ,α,β) be a Bihom-Jordan superalgebra, (V,ϕ,ψ) be a Bihom-super-module such that ψ is invertible, and ρl:J⊗V→V be an even linear map given by ρl(a⊗v)=a⋅v such that the following conditions hold:
ϕρl=ρl(α⊗ϕ),ψρl=ρl(β⊗ψ), | (3.7) |
β(ε)α(γ)⋅ψ(v)=βα(ε)⋅(α(γ)⋅v)+(−1)|ε||γ|βα(γ)⋅(α(ε)⋅v). | (3.8) |
Then (V,ϕ,,ψ,ρl) is a left Bihom-Jordan J-super-module called a left special super-module.
Proof. Similar to the proof of Theorem 3.1, the conclusion can be proved by repeatedly using Eqs (3.7) and (3.8).
Now, we give the definition super-bimodule of a BiHom-Jordan superalgebra.
Definition 3.2. Let (J,μ,α,β) be a Bihom-Jordan superalgebra. A Bihom-Jordan J-super-bimodule is a Bihom-super-module (V,ϕ,ψ) with an even left structure map ρl:J⊗V→V,ρl(a⊗v)=a⋅v and an even right structure map ρr:V⊗J→V,ρr(v⊗a)=v⋅a satisfying three conditions:
ρl(β⊗ϕ)=ρr(ψ⊗α)τ1, | (3.9) |
↺ε,γ,δ(−1)|δ|(|ε|+|v|)asVϕ,ψ(μ(β2(ε),αβ(γ)),ϕ2ψ(v),α3(δ))=0, | (3.10) |
(−1)|γ||δ|asVϕ,ψ(ψ2(v)⋅αβ(ε),βα2(γ),α3(δ))+(−1)|ε||γ|+|ε||δ|asVϕ,ψ(ψ2(v)⋅αβ(δ),βα2(γ),α3(ε))+(−1)|v||ε|+|v||γ|+|v||δ|asVϕ,ψ(μ(β2(ε),αβ(δ)),βα2(γ),ϕ3(v))=0. | (3.11) |
Remark 3.1. 1) If α=β=IdJ and ϕ=ψ=IdV then V is reduced to the so-called Jordan supermodule of the Jordan superalgebra (J,μ).
2) Clearly, a Bihom-Jordan A-super-bimodule is a right Bihom-Jordan super-module. Furthermore, it is a left Bihom-Jordan super-module if ψ is invertible.
Example 3.1. Here are some examples of Bihom-Jordan super-bimodules.
1) Let (J,μ,α,β) be a Bihom-Jordan superalgebra. Then (J,α,β) is a Bihom-Jordan J-super-bimodule where the structure maps are ρl=ρr=μ. More generally, if B is a Bihom-ideal of (J,μ,α,β), then (B,α,β) is a Bihom-Jordan J-super-bimodule where the structure maps are ρl(a,ε)=μ(a,ε)=μ(ε,a)=ρr(ε,a), for all (a,ε)∈H(J)×H(B).
2) If (J,μ) is a Jordan superalgebra and M is a Jordan J-super-bimodule in the usual sense, then (M,IdM,IdM) is a BiHom-Jordan J-super-bimodule where (J,μ,IdJ,IdJ) is a Bihom-Jordan superalgebra.
Theorem 3.3. Let (J,μ,α,β) be a Bihom-Jordan superalgebra and (V,ϕ,ψ,ρl,ρr) be a Bihom-Jordan J-super-bimodule. Define even linear maps ˜μ,˜α and ˜β on J⊕V,
● ˜μ:(J⊕V)⊗2→J⊕V,˜μ(ε+u,γ+v):=μ(ε,γ)+ε⋅v+u⋅γ,
● ˜α,˜β:(J⊕V)→J⊕V,
˜α(ε+u):=α(ε)+ϕ(v) and ˜β(ε+u):=β(ε)+ψ(v).
Then (J⊕V,˜μ,˜α,˜β) is a Bihom-Jordan superalgebra.
Proof. We omitted the calculation process; it is straightforward to see Bihom-super commutativity condition and Bihom-Jordan super-identity by Definition 3.2.
The next result shows that a special left and right Bihom-Jordan super-module has a Bihom-Jordan super-bimodule structure under a specific condition.
Theorem 3.4. Let (J,μ,α,β) be a regular Bihom-Jordan superalgebra, (V,ϕ,ψ) be both a left and a right special BiHom-Jordan J-module with the structure maps ρ1 and ρ2 respectively, such that ϕ is invertible, and the Bihom-associativity condition holds
ρ2∘(ρ1⊗β)=ρ1∘(α⊗ρ2). | (3.12) |
Define two even bilinear maps ρl:J⊗V→V and ρr:V⊗J→V by
ρl=ρ1+ρ2(ψϕ−1⊗αβ−1)∘τ1andρr=ρ1(βα−1⊗ϕψ−1)∘τ2+ρ2. | (3.13) |
Then (V,ϕ,ψ,ρl,ρr) is a Bihom-Jordan J-super-bimodule.
Proof. ρl and ρr are even structure maps from ρ1 and ρ2. We need to check out (3.9)–(3.11). First, for any (ε,v)∈H(J)×H(V),
ρl(β(ε),ϕ(v))=β(ε)⋅ϕ(v)+(−1)|a||v|ψϕ−1(ϕ(v))⋅αβ−1(β(ε))=β(ε)⋅ϕ(v)+(−1)|a||v|ψ(v)⋅α(ε), |
ρr(ψ⊗α)τ1(ε⊗v)=(−1)|a||v|ρr(ψ(v),α(ε))=(−1)|a||v|ψ(v)⋅α(ε)+βα−1(α(ε))⋅ϕψ−1(ϕ(v))=β(ε)⋅ϕ(v)+(−1)|a||v|ψ(v)⋅α(ε). |
So ρl(β⊗ϕ)=ρr(ψ⊗α)τ1. Next, for any ε,γ,δ∈H(J),v∈H(V)
asVϕ,ψ(μ(β2(ε),αβ(γ)),ϕ2ψ(v),α3(δ))=ρr(ρl(μ(β2(ε),αβ(γ)),ϕ2ψ(v)),βα3(δ))−ρl(αβ2(ε)α2β(γ),ρr(ϕ2ψ(v),α3(δ)))=ρr(β2(ε)αβ(γ)⋅ϕ2ψ(v),βα3(δ))+(−1)|ε||v|+|γ||v|ρr(ϕψ2(v)⋅αβ(ε)α2(γ),βα3(δ))−(−1)|v||δ|ρl(αβ2(ε)α2β(γ),βα2(δ)⋅ϕ3(v))−ρl(αβ2(ε)α2β(γ),ϕ2ψ(v)⋅α3(δ))(by(3.13))=(β2(ε)αβ(γ)⋅ϕ2ψ(v))⋅βα3(δ)+(−1)|δ||ε|+|δ||γ|+|δ||v|α2β2(δ)⋅(αβ(ε)α2(γ)⋅ϕ3(v))+(−1)|ε||v|+|γ||v|(ϕψ2(v)⋅αβ(ε)α2(γ))⋅βα3(δ)+(−1)|ε||v|+|γ||v|+|δ||v|+|δ||ε|+|δ||γ|α2β2(δ)⋅(ϕ2ψ(v)⋅α2(ε)α3β−1(γ))−(−1)|v||δ|αβ2(ε)α2β(γ)⋅(βα2(δ)⋅ϕ3(v))−(−1)|v||δ|+|δ||ε|+|δ||γ|+|v||ε|+|v||γ|(β2α(δ)⋅ψϕ2(v))⋅α2β(ε)α3(γ)−αβ2(ε)α2β(γ)⋅(ϕ2ψ(v)⋅α3(δ))−(−1)|v||ε|+|v||γ|+|δ||ε|+|δ||γ|(ϕψ2(v)⋅α2β(δ))⋅α2β(ε)α3(γ)(by(3.13))=(−1)|ε||v|+|γ||v|(ϕψ2(v)⋅αβ(ε)α2(γ))⋅βα3(δ)−(−1)|v||ε|+|v||γ|+|δ||ε|+|δ||γ|(ϕψ2(v)⋅α2β(δ))⋅α2β(ε)α3(γ)+(−1)|δ||ε|+|δ||γ|+|δ||v|α2β2(δ)⋅(αβ(ε)α2(γ)⋅ϕ3(v))−(−1)|v||δ|αβ2(ε)α2β(γ)⋅(βα2(δ)⋅ϕ3(v))(by(3.12)). |
So
↺ε,γ,δ(−1)|δ|(|ε|+|v|)asVϕ,ψ(β2(ε)αβ(γ),ϕ2ψ(v),α3(δ))=(−1)|v|(|ε|+|γ|+|δ|){↺ε,γ,δ(−1)|ε||δ|(ϕψ2(v)⋅αβ(ε)α2(γ))⋅βα3(δ)−↺ε,γ,δ(−1)|δ||γ|(ϕψ2(v)⋅βα2(δ))⋅α2β(ε)α3(γ)}+↺ε,γ,δ(−1)|δ||γ|α2β2(δ)⋅(αβ(ε)α2(γ)⋅ϕ3(v))−↺ε,γ,δ(−1)|δ||ε|αβ2(ε)α2β(γ)⋅(βα2(δ)⋅ϕ3(v))=(−1)|v|(|ε|+|γ|+|δ|)0+0=0. |
Finally, to prove (3.11), let us compute each of its three terms.
(−1)|γ||δ|asVϕ,ψ(ρr(ψ2(v),βα(ε)),βα2(γ),α3(δ))=(−1)|γ||δ|asVϕ,ψ(ψ2(v)⋅βα(ε),βα2(γ),α3(δ))+(−1)|γ||δ|+|ε||v|asVϕ,ψ(β2(ε)⋅ϕψ(v),βα2(γ),α3(δ))(by(3.13))=(−1)|γ||δ|ρr(ρr(ψ2(v)⋅βα(ε),βα2(γ)),α3β(δ))−(−1)|γ||δ|ρr(ϕψ2(v)⋅βα2(ε),βα2(γ)α3(δ))+(−1)|γ||δ|+|ε||v|ρr(ρr(β2(ε)⋅ϕψ(v),βα2(γ)),α3β(δ))−(−1)|γ||δ|+|ε||v|ρr(αβ2(ε)⋅ϕ2ψ(v),βα2(γ)α3(δ))=(−1)|γ||δ|ρr((ψ2(v)⋅βα(ε))⋅βα2(γ),α3β(δ))+(−1)|γ||δ|+|γ||v|+|ε||γ|ρr(β2α(γ)⋅(ϕψ(v)⋅α2(ε)),α3β(δ)−(−1)|γ||δ|(ϕψ2(v)⋅βα2(ε))⋅βα2(γ)α3(δ)−(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|β2α(γ)βα2(δ)⋅(ϕ2ψ(v)⋅α3(ε))+(−1)|γ||δ|+|ε||v|ρr((β2(ε)⋅ϕψ(v))⋅βα2(γ),α3β(δ))+(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|ρr(β2α(γ)⋅(αβ(ε)⋅ϕ2(v)),α3β(δ))−(−1)|γ||δ|+|ε||v|(αβ2(ε)⋅ϕ2ψ(v))⋅βα2(γ)α3(δ)−(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|β2α(γ)βα2(δ)⋅(α2β(γ)⋅ϕ3(v))(by(3.13))=(−1)|γ||δ|((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅α3β(δ)+(−1)|δ||ε|+|δ||v|α2β2(δ)⋅((ϕψ(v)⋅α2(ε))⋅α3(γ))+(−1)|γ||δ|+|γ||v|+|ε||γ|(β2α(γ)⋅(ϕψ(v)⋅α2(ε)))⋅α3β(δ)+(−1)|γ||v|+|ε||γ|+|δ||v|+|δ||ε|α2β2(δ)⋅(βα2(γ)⋅(ϕ2(v)⋅α3β−1(ε)))+(−1)|γ||δ|+|ε||v|((β2(ε)⋅ϕψ(v))⋅βα2(γ))⋅α3β(δ)+(−1)|ε||v|+|δ||ε|+|δ||v|α2β2(δ)⋅((αβ(ε)⋅ϕ2(v))α3(γ))+(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|(β2α(γ)⋅(αβ(ε)⋅ϕ2(v)))⋅α3β(δ)+(−1)|ε||v|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|α2β2(δ)⋅(α2β(γ)⋅(α2(ε)⋅ϕ3ψ−1(v)))−(−1)|γ||δ|(ϕψ2(v)⋅βα2(ε))⋅βα2(γ)α3(δ)−(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|β2α(γ)βα2(δ)⋅(ϕ2ψ(v)⋅α3(ε))⏟B−(−1)|γ||δ|+|ε||v|(αβ2(ε)⋅ϕ2ψ(v))⋅βα2(γ)α3(δ)⏟J−(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|β2α(γ)βα2(δ)⋅(α2β(γ)⋅ϕ3(v)).(by(3.13)andrearranging) |
Observe that
J=(−1)|γ||δ|+|ε||v|ϕ(β2(ε)⋅ϕψ(v))⋅β(α2(γ))α(α2(δ))=(−1)|γ||δ|+|ε||v|((β2(ε)⋅ϕψ(v))⋅βα2(γ))⋅βα3(δ)+(−1)|ε||v|((β2(ε)⋅ϕψ(v))⋅βα2(δ))⋅βα3(γ)(by(3.6))=(−1)|γ||δ|+|ε||v|((β2(ε)⋅ϕψ(v))⋅βα2(γ))⋅βα3(δ)+(−1)|ε||v|(αβ2(ε)⋅(ϕψ(v)⋅α2(δ)))⋅βα3(γ)(by(3.12))=(−1)|γ||δ|+|ε||v|((β2(ε)⋅ϕψ(v))⋅βα2(γ))⋅βα3(δ)⏟J1+(−1)|ε||v|α2β2(ε)⋅((ϕψ(v)⋅α2(δ))⋅α3(γ))⏟J2.(by(3.12)) |
B=(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|β(αβ(γ))α(αβ(δ))⋅ψ(ϕ2(v)⋅α3β−1(ε))=(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|α2β2(γ)⋅(α2β(δ)⋅(ϕ2(v)⋅α3β−1(ε)))+(−1)|γ||v|+|γ||ε|+|δ||v|+|δ||ε|α2β2(δ)⋅(α2β(γ)⋅(ϕ2(v)⋅α3β−1(ε)))(by(3.8))=(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|α2β2(γ)⋅((αβ(δ)ϕ2(v))⋅α3(ε))+(−1)|γ||v|+|γ||ε|+|δ||v|+|δ||ε|α2β2(δ)⋅(α2β(γ)⋅(ϕ2(v)⋅α3β−1(ε)))(by(3.12))=(−1)|γ||δ|+|γ||v|+|γ||ε|+|δ||v|+|δ||ε|(αβ2(γ)⋅(αβ(δ)⋅ϕ2(v)))⋅α3β(ε)⏟B1+(−1)|γ||v|+|γ||ε|+|δ||v|+|δ||ε|α2β2(δ)⋅(α2β(γ)⋅(ϕ2(v)⋅α3β−1(ε)))⏟B2.(by(3.12)) |
We substitute J1+J2 and B1+B2 for J and B to obtain
(−1)|γ||δ|asVϕ,ψ(ρr(ψ2(v),βα(ε)),βα2(γ),α3(δ))=(−1)|γ||δ|((ψ2(v)⋅βα(ε))⋅βα2(γ))⋅α3β(δ)+(−1)|δ||ε|+|δ||v|α2β2(δ)⋅((ϕψ(v)⋅α2(ε))⋅α3(γ))+(−1)|γ||δ|+|γ||v|+|ε||γ|(β2α(γ)⋅(ϕψ(v)⋅α2(ε)))⋅α3β(δ)+(−1)|ε||v|+|δ||ε|+|δ||v|α2β2(δ)⋅((αβ(ε)⋅ϕ2(v))α3(γ))+(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|(β2α(γ)⋅(αβ(ε)⋅ϕ2(v)))⋅α3β(δ)+(−1)|ε||v|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|α2β2(δ)⋅(α2β(γ)⋅(α2(ε)⋅ϕ3ψ−1(v)))−(−1)|γ||δ|(ϕψ2(v)⋅βα2(ε))⋅βα2(γ)α3(δ)−B1−J2−(−1)|γ||δ|+|ε||v|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|β2α(γ)βα2(δ)⋅(α2β(γ)⋅ϕ3(v)). |
Similarly, we have
(−1)|ε||γ|+|ε||δ|asVϕ,ψ(ρr(ψ2(v)⋅βα(δ)),βα2(γ),α3(ε))=(−1)|ε||γ|+|ε||δ|((ψ2(v)⋅βα(δ))⋅βα2(γ))⋅α3β(ε)+(−1)|ε||v|α2β2(ε)⋅((ϕψ(v)⋅α2(δ))⋅α3(γ))+(−1)|γ||v|+|δ||γ|+|ε||γ|+|ε||δ|(β2α(γ)⋅(ϕψ(v)⋅α2(δ)))⋅α3β(ε)+(−1)|δ||v|+|ε||v|α2β2(ε)⋅((αβ(δ)ϕ2(v))⋅α3(γ))+(−1)|δ||v|+|γ||δ|+|γ||v|+|ε||δ|+|ε||γ|(β2α(γ)⋅(αβ(δ)⋅ϕ2(v)))⋅α3β(ε)+(−1)|δ||v|+|γ||δ|+|γ||v|+|ε||v|α2β2(ε)⋅(α2β(γ)⋅(α2(δ)⋅ϕ3ψ−1(v)))−(−1)|ε||γ|+|ε||δ|(ϕψ2(v)⋅βα2(δ))⋅βα2(γ)α3(ε)−(−1)|ε||v|+|δ||v|+|ε||γ|+|ε||δ|α2β2(δ)⋅((ϕψ(v)⋅α2(ε))⋅α3(γ))−(−1)|γ||v|+|γ||δ|+|ε||v|+|ε||γ|(αβ2(γ)⋅(αβ(ε)⋅ϕ2(v)))⋅α3β(δ)−(−1)|ε||v|+|δ||v|+|γ||δ|+|γ||v|+|ε||γ|β2α(γ)α2β(ε)⋅(α2β(δ)⋅ϕ3(v)). |
In addition,
(−1)|v||ε|+|v||γ|+|v||δ|asVϕ,ψ(β2(ε)βα(δ),βα2(γ),ϕ3(v))=(−1)|v||ε|+|v||γ|+|v||δ|ρl((β2(ε)βα(δ))βα2(γ),ϕ3ψ(v))−(−1)|v||ε|+|v||γ|+|v||δ|ρl(β2α(ε)βα2(δ),ρl(βα2(γ),ϕ3(v))=(−1)|v||ε|+|v||γ|+|v||δ|((β2(ε)βα(δ))βα2(γ))⋅ϕ3ψ(v)+ϕ2ψ2(v)⋅((αβ(ε)α2(δ))α3(γ))−(−1)|v||ε|+|v||γ|+|v||δ|ρl(β2α(ε)βα2(δ),βα2(γ)⋅ϕ3(v))−(−1)|v||ε|+|v||δ|ρl(β2α(ε)βα2(δ),ψϕ2(v)⋅α3(γ))(by(3.13))=(−1)|v||ε|+|v||γ|+|v||δ|((β2(ε)βα(δ))βα2(γ))⋅ϕ3ψ(v)+ϕ2ψ2(v)⋅((αβ(ε)α2(δ))α3(γ))−(−1)|v||ε|+|v||γ|+|v||δ|(β2α(ε)βα2(δ))⋅(βα2(γ)⋅ϕ3(v))−(−1)|v||γ|+|γ||ε|+|γ||δ|(β2α(γ)ϕ2ψ(v))⋅βα2(ε)α3(δ)⏟D−(−1)|v||ε|+|v||δ|β2α(ε)βα2(δ)⋅(ψϕ2(v)⋅α3(γ))⏟C−(−1)|γ||ε|+|γ||δ|(ψ2ϕ(v)⋅α2β(γ))⋅α2β(ε)α3(δ).(by(3.13)) |
The same way, we replace C and D as follows
C=(−1)|v||ε|+|v||δ|β(αβ(ε))α(αβ(δ))⋅ψ(ϕ2(v)⋅α3β−1(γ))=(−1)|v||ε|+|v||δ|β2α2(ε)⋅(α2β(δ)⋅(ϕ2(v)⋅α3β−1(γ)))+(−1)|v||ε|+|v||δ|+|ε||δ|β2α2(δ)⋅(α2β(ε)⋅(ϕ2(v)⋅α3β−1(γ)))(by3.8)=(−1)|v||ε|+|v||δ|β2α2(ε)⋅((αβ(δ)⋅ϕ2(v))⋅α3(γ))⏟C1+(−1)|v||ε|+|v||δ|+|ε||δ|β2α2(δ)⋅((αβ(ε)⋅ϕ2(v))⋅α3(γ))⏟C2,(by3.12) |
D=(−1)|v||γ|+|γ||ε|+|γ||δ|ϕ(β2(γ)⋅ϕψ(v))⋅β(α2(ε))α(α2(δ))=(−1)|v||γ|+|γ||ε|+|γ||δ|((β2(γ)⋅ϕψ(v))⋅α2β(ε))⋅βα3(δ)+(−1)|v||γ|+|γ||ε|+|γ||δ|+|ε||δ|((β2(γ)⋅ϕψ(v))⋅α2β(δ))⋅βα3(ε)(by3.6)=(−1)|v||γ|+|γ||ε|+|γ||δ|(αβ2(γ)⋅(ϕψ(v)⋅α2(ε)))⋅βα3(δ)⏟D1+(−1)|v||γ|+|γ||ε|+|γ||δ|+|ε||δ|(αβ2(γ)⋅(ϕψ(v)⋅α2(δ)))⋅βα3(ε)⏟D2.(by3.12) |
Finally, we have
(−1)|γ||δ|asVϕ,ψ(ψ2(v)⋅αβ(ε),βα2(γ),α3(δ))+(−1)|ε||γ|+|ε||δ|asVϕ,ψ(ψ2(v)⋅αβ(δ),βα2(γ),α3(ε))+(−1)|v||ε|+|v||γ|+|v||δ|asVϕ,ψ(μ(β2(ε),αβ(δ)),βα2(γ),ϕ3(v))=(−1)|γ||δ|(3.2)+(−1)|v||ε|+|γ||ε|+|γ||v|+|δ||ε|+|δ||v|(3.4)=0. |
Hence, we prove that (V,ϕ,ψ,ρl,ρr) is a Bihom-Jordan J-super-bimodule.
Lemma 3.1. Let (J,μ,α,β) be a Bihom-associative superalgebra and (V,ϕ,ψ) be a Bihom-super-module.
1) If (V,ϕ,ψ) is a right Bihom-associative J-super-module with the structure map ρr, then (V,ϕ,ψ) is a right special Bihom-Jordan J⋄-super-module with the same structure map ρr.
2) If (V,ϕ,ψ) is a left Bihom-associative J-super-module with the structure map ρl such that ψ is invertible, then (V,ϕ,ψ) is a left special Bihom-Jordan J⋄-super-module with the same structure map ρl.
Proof. It also suffices to prove Eqs (3.6) and (3.8).
1) If (V,ϕ,ψ) is a right Bihom-associative J-super-module with the structure map ρr then for all (ε,γ,v)∈H(J)×H(J)×H(V). ϕ(v)⋅(β(ε)⋄α(γ))=ϕ(v)⋅(β(ε)α(γ)+(−1)|ε||γ|β(γ)α(ε))=(v⋅β(ε))⋅αβ(γ)+(−1)|ε||γ|(v⋅β(γ))⋅αβ(ε). Then (V,ϕ,ψ) is a right special Bihom-Jordan J⋄-super-module by Theorem 3.1.
2) Similarly, it is easy to obtain by Theorem 3.2.
End of lemma proof.
By Lemma 3.1 and Theorem 3.4, we obtain the following conclusion.
Proposition 3.1. Let (J,μ,α,β) be a Bihom-associative superalgebra and (V,ϕ,ψ,ρ1,ρ2) be a Bihom-associative J-super-bimodule such that ϕ and ψ are inversible. Then (V,ϕ,ψ,ρl,ρr) is a Bihom-Jordan J⋄-super-bimodule where ρl and ρr are defined as in Eq (3.13).
That is, a Bihom-associative J-super-bimodule gives rise to a Bihom-Jordan super-bimodule for J⋄.
Proposition 3.2. Let (J,μ,α,β) be a Bihom-Jordan superalgebra and Vϕ,ψ=(V,ϕ,ψ,ρl,ρr) be a Bihom-Jordan J-super-bimodule. Then for each n∈N such that ϕn=ψn=IdV, the maps
ρ(n)l=ρl∘(αn⊗ψn), | (3.14) |
and
ρ(n)r=ρr∘(ϕn⊗βn). | (3.15) |
as structure maps, (V,ϕ,ψ,ρ(n)l,ρ(n)r) is given to be a Bihom-Jordan J-super-bimodule. Denoted it by V(n)ϕ,ψ.
Proof. ρl and ρn are easy to prove special left and right super-modules, respectively, which are also left and right super-modules, and Eq (3.9) holds in V(n)ϕ,ψ. By direct calculation, we can convert asV(n)ϕ,ψ in V(n)ϕ,ψ to asVϕ,ψ in Vϕ,ψ, that is
asV(n)ϕ,ψ(β2(ε)αβ(γ),ϕ2ψ(v),α3(δ))=asVϕ,ψ(β2(αn(ε))αβ(αn(γ)),ϕ2ψ(v),α3(βn(δ))) |
, furthermore, we have
↺ε,γ,δ(−1)|δ|(|ε|+|v|)asV(n)ϕ,ψ(β2(ε)αβ(γ),ϕ2ψ(v),α3(δ))=↺αn(ε),αn(γ),βn(δ)(−1)|δ|(|ε|+|v|)asVϕ,ψ(β2(αn(ε))αβ(αn(γ)),ϕ2ψ(v),α3(βn(δ)))=0. |
Then we obtain Eq (3.10) in V(n)ϕ,ψ. Similarly, Eq (3.11) also holds in V(n)ϕ,ψ, which implies that V(n)ϕ,ψ is a Bihom-Jordan J-super-bimodule.
In the sequel, we present some results of Bihom-Jordan super-bimodules constructed by Jordan super-bimodules via endomorphisms.
Theorem 3.5. Let (J,μ) be a Jordan superalgebra, (V,ρl,ρr) be a Jordan J-super-bimodule, α,β be endomorphisms of J, which satisfies αβ=βα and ϕ,ψ be even linear self-maps of V such that ϕ∘ρl=ρl∘(α⊗ϕ), ϕ∘ρr=ρr∘(ϕ⊗α), ψ∘ρl=ρl∘(β⊗ψ) and ψ∘ρr=ρr∘(ψ⊗β). Denote Jα,β for the Bihom-Jordan superalgebra (J,μα,β=μ(α⊗β),α,β) and Vϕ,ψ for the Bihom-super-module (V,ϕ,ψ). Define two structure maps as follows:
~ρl=ρl(α⊗ψ)and~ρr=ρr(ϕ⊗β). | (3.16) |
Then Vϕ,ψ=(V,ϕ,ψ,~ρl,~ρr) is a Bihom-Jordan Jα,β-super-bimodule.
Proof. By direct calculation, it is easy to get asVϕ,ψ(μα,β(β2(ε),αβ(γ)),ϕ2ψ(v),α3(δ))=asV(α3β2(ε)α3β2(γ),ϕ3ψ2(v),α3β2(δ)), So it is clear Eqs (3.10) and (3.11) hold in Vϕ,ψ. Thus, (V,ϕ,ψ,~ρl,~ρr) is a Bihom-Jordan Jα,β-super-bimodule.
From Proposition 3.2 and Theorem 3.5, we have the following
Corollary 3.1. Let (J,μ) be a Jordan superalgebra, (V,ρl,ρr) be a Jordan J-super-bimodule, α,β be endomorphisms of J, which satify αβ=βα and ϕ,ψ be even linear self-maps of V such that ϕ∘ρl=ρl∘(α⊗ϕ), ϕ∘ρr=ρr∘(ϕ⊗α), ψ∘ρl=ρl∘(β⊗ψ) and ψ∘ρr=ρr∘(ψ⊗β). Denote Jα,β for the Bihom-Jordan superalgebra (J,μα,β=μ(α⊗β),α,β) and Vϕ,ψ for the Bihom-super-module (V,ϕ,ψ). Define two structure maps as follows:
~ρl(n)=ρl(αn+1⊗ψ)and~ρr(n)=ρr(ϕ⊗βn+1). | (3.17) |
Then Vϕ,ψ=(V,ϕ,ψ,~ρl(n),~ρr(n)) is a Bihom-Jordan Jα,β-super-bimodule for each n∈N.
Definition 3.3. An abelian extension of Bihom-Jordan superalgebra is a short exact sequence of Bihom-Jordan superalgebra:
0⟶(V,ϕ,ψ)i⟶(J,μJ,αJ,βJ)π⟶(B,μB,αB,βB)⟶0. |
where (V,ϕ,ψ) is a trivial Bihom-Jordan superalgebra, i and π are even morphisms of Bihom-superalgebras. If there exists an even morphism s:(B,μB,αB,βB)→(J,μJ,αJ,βJ) satisfies π∘s=IdB. Then the abelian extension is said to be split and s is called a section of π.
In this section, we study the representation and O-operator. Meanwhile, we characterize Bihom-pre-Jordan superalgebras by using O-operator.
Definition 4.1. Let (J,μ,α,β) be a Bihom-Jordan superalgebra, V be a Z2-graded vector spaces, ρ:J→End(V), ϕ,ψ∈Aug(V). Then (V,ρ,ϕ,ψ) is a representation of (J,μ,α,β), if the following conditions hold:
ϕψ=ψϕ, | (4.1) |
ρ(μ(μ(β2(ε),αβ(γ)),α2β(δ)))ϕ3ψ+(−1)|δ||γ|ρ(α2β2(ε))ϕψ−1ρ(αβ2(δ))ϕψ−1ρ(β2(γ))ϕψ+(−1)|γ||ε|+|δ||ε|ρ(α2β2(γ))ϕψ−1ρ(αβ2(δ))ϕψ−1ρ(β2(ε))ϕψ−ρ(μ(αβ2(ε),α2β(γ)))ρ(α2β(δ))ϕ3−(−1)|ε||δ|+|δ||γ|ρ(μ(αβ2(δ),α2β(ε)))ϕ2ψ−1ρ(β2(γ))ϕψ−(−1)|ε||δ|+|δ||γ|+|ε||γ|ρ(μ(αβ2(δ),α2β(γ)))ϕ2ψ−1ρ(β2(ε))ϕψ=0. | (4.2) |
↺ε,γ,δ(−1)|ε||δ|ρ(α2β2(ε))ϕψ−1ρ(μ(β2(γ),αβ(δ)))ϕ2ψ=↺ε,γ,δ(−1)|ε||δ|ρ(μ(αβ2(ε),α2β(γ)))ρ(α2β(δ))ϕ3. | (4.3) |
Example 4.1. Let (J,μ,α,β) be a regular Bihom-Jordan superalgebra. Define ad:J→End(J), for any ε,γ∈H(J), ad(ε)γ=μ(ε,γ). Then (J,ad,α,β) is a representation of (J,μ,α,β), which is called adjoint representation.
Proposition 4.1. Let (J,μ,α,β) be a Bihom-Jordan superalgebra. (V,ρ,ϕ,ψ) be a representation, define an even bilinear map μ and two even linear maps α and β on J⊕V as follows: for any ε,γ∈H(J),a,b∈H(V),
μ′(ε+a,γ+b)=μ(ε,γ)+ρ(ε)b+ρ(α−1β(γ))ϕψ−1(a), |
(α+ϕ)(ε+a)=α(ε)+ϕ(a),(β+ψ)(ε+a)=β(a)+ψ(a). |
Then (J⊕V,μ′,α+ϕ,β+ψ) is a Bihom-Jordan superalgebra, denoted by J⋉V and called semidirect product.
Proof. It can be verified directly by Definition 4.1.
We also consider the split null extension on J⋉V in Proposition 4.1.
Remark 4.1. Write elements a+v of J⊕V as (a,v). There is an injective homomorphism and a surjective homomorphism of Bihom-modules, respectively, as follows:
● i:V→J⊕V, i(v)=(0,v),
● π:J⊕V→, π(a,v)=a.
Moreover, i(V) is a Bihom-ideal of J⊕V such that J⊕V/i(V)≅J. On the other hand, there is an even morphisms σ:J→J⊕V given by σ(a)=(a,0), which is clearly a section of π. Therefore, we obtain the abelian split exact sequence:
![]() |
Definition 4.2. A BiHom superalgebra (J,⋅,α,β) is called a Bihom-pre-Jordan superalgebra if for all ε,γ,δ,t∈H(J):
1) αβ=βα, both α and β are reversible,
1)
((β2(ε)⋅αβ(γ))⋅α2β(δ))⋅α3β(w)+(−1)|ε||γ|((β2(γ)⋅αβ(ε))⋅α2β(δ))⋅α3β(w)+(−1)|δ|(|ε|+|γ|)(αβ2(δ)⋅(αβ(ε)α2(γ)))⋅α3β(w)+(−1)|δ|(|ε|+|γ|)+|ε||γ|(αβ2(δ)⋅(αβ(γ)α2(ε)))⋅α3β(w)+(−1)|δ||γ|α2β2(ε)⋅(α2β(δ)⋅(α2(γ)⋅α3β−1(w)))+(−1)|δ||ε|+|γ||ε|α2β2(γ)⋅(α2β(δ)⋅(α2(ε)⋅α3β−1(w)))−(αβ2(ε)⋅α2β(γ))⋅(α2β(δ)⋅α3(w))−(−1)|ε||γ|(αβ2(γ)⋅α2β(ε))⋅(α2β(δ)⋅α3(w))−(−1)|ε||δ|+|δ||γ|(αβ2(δ)⋅α2β(ε))⋅(α2β(γ)⋅α3(w))−(−1)|δ||γ|(αβ2(ε)⋅α2β(δ))⋅(α2β(γ)⋅α3(w))−(−1)|γ||δ|+|δ||ε|+|γ||ε|(αβ2(δ)⋅α2β(γ))⋅(α2β(ε)⋅α3(w))−(−1)|δ||ε|+|γ||ε|(αβ2(γ)⋅α2β(δ))⋅(α2β(ε)⋅α3(w))=0, | (4.4) |
3)
(−1)|w||γ|α2β2(w)⋅((αβ(ε)⋅α2(γ))⋅α3(δ))+(−1)|ε||γ|+|w||γ|α2β2(w)⋅((αβ(γ)⋅α2(ε))⋅α3(δ))+(−1)|ε||w|α2β2(ε)⋅((αβ(γ)⋅α2(w))⋅α3(δ))+(−1)|ε||w|+|γ||w|α2β2(ε)⋅((αβ(w)⋅α2(γ))⋅α3(δ))+(−1)|γ||ε|α2β2(γ)⋅((αβ(w)⋅α2(ε))⋅α3(δ))+(−1)|γ||ε|+|w||ε|α2β2(γ)⋅((αβ(ε)⋅α2(w))⋅α3(δ))−(−1)|w||ε|(αβ2(ε)⋅α2β(γ))⋅(α2β(w)⋅α3(δ))−(−1)|ε||γ|+|w||ε|(αβ2(γ)⋅α2β(ε))⋅(α2β(w)⋅α3(δ))−(−1)|ε||γ|(αβ2(γ)⋅α2β(w))⋅(α2β(ε)⋅α3(δ))−(−1)|γ||w|+|ε||γ|(αβ2(w)⋅α2β(γ))⋅(α2β(ε)⋅α3(δ))−(−1)|γ||w|(αβ2(w)⋅α2β(ε))⋅(α2β(γ)⋅α3(δ))−(−1)|w||ε|+|γ||w|(αβ2(ε)⋅α2β(w))⋅(α2β(γ)⋅α3(δ)). | (4.5) |
Actually, condition 3 is equivalent to
↺ε,γ,w{(−1)|ε||w|α2β2(ε)⋅((αβ(γ)⋅α2(w))⋅α3(δ))+(−1)|ε||w|+|γ||w|α2β2(ε)⋅((αβ(w)⋅α2(γ))⋅α3(δ))}=↺ε,γ,w{(−1)|ε||w|(αβ2(ε)⋅α2β(γ))⋅(α2β(w)⋅(δ))+(−1)|ε||w|+|ε||γ|(αβ2(γ)⋅α2β(ε))⋅(α2β(w)⋅(δ))}. |
Theorem 4.1. Let (J,⋅,α,β) be a Bihom-pre-Jordan superalgebra, define an even bilinear operator μ: for all ε,γ∈H(J)
μ(ε,γ)=ε⋅γ+(−1)|ε||γ|α−1β(γ)⋅αβ−1(ε), | (4.6) |
then (J,∙,α,β) is a Bihom-Jordan superalgebra.
Proof. By Eq (4.6), we get
μ(β(ε),α(γ))=β(ε)⋅α(γ)+(−1)|ε||γ|β(γ)⋅α(ε)=(−1)|ε||γ|μ(β(γ),α(ε)). |
That is to say the Bihom-super commutativity condition holds. Next, by direct calculation,
(−1)|w|(|ε|+|δ|)~asα,β(μ(β2(ε),αβ(γ)),α2β(δ),α3(w))=(−1)|w|(|ε|+|δ|)((β2(ε)⋅αβ(γ))⋅α2β(δ))⋅α3β(w)_①+(−1)|w||γ|α2β2(w)⋅((αβ(ε)⋅α2(γ))⋅α3(δ))⏟1′+(−1)|w|(|ε|+|δ|)+|δ|(|ε|+|γ|)(αβ2(δ)⋅(αβ(ε)⋅α2(γ)))⋅α3β(w)_②+(−1)|w||γ|+|δ|(|ε|+|γ|)α2β2(w)⋅(α2β(δ)⋅(α2(ε)⋅α3β−1(γ)))−(−1)|w|(|ε|+|δ|)(αβ2(ε)⋅α2β(γ))⋅(α2β(δ)⋅α3(w))_③−(−1)|w|(|δ|+|γ|)+|δ|(|ε|+|γ|)(αβ2(δ)⋅α2β(w))⋅(α2β(ε)⋅α3(γ))−(−1)|w||ε|(αβ2(ε)⋅α2β(γ))⋅(α2β(w)⋅α3(δ))⏟2′−(−1)|w||γ|+|δ|(|ε|+|γ|)(αβ2(w)⋅α2β(δ))⋅(α2β(ε)⋅α3(γ))+(−1)|w|(|ε|+|δ|)+|ε||γ|((β2(γ)⋅αβ(ε))⋅α2β(δ))⋅α3β(w)_④+(−1)|w||γ|+|ε||γ|α2β2(w)⋅((αβ(γ)⋅α2(ε))⋅α3(δ))⏟3′+(−1)|w|(|ε|+|δ|)+|δ|(|ε|+|γ|)+|ε||γ|(αβ2(δ)⋅(αβ(γ)⋅α2(ε)))⋅α3β(w)_⑤+(−1)|w||γ|+|δ|(|ε|+|γ|)+|ε||γ|α2β2(w)⋅(α2β(δ)⋅(α2(γ)⋅α3β−1(ε)))−(−1)|w|(|ε|+|δ|)+|ε||γ|(αβ2(γ)⋅α2β(ε))⋅(α2β(δ)⋅α3(w))_⑥−(−1)|w|(|δ|+|γ|)+|δ|(|ε|+|γ|)+|ε||γ|(αβ2(δ)⋅α2β(w))⋅(α2β(γ)⋅α3(ε))−(−1)|w||ε|+|ε||γ|(αβ2(γ)⋅α2β(ε))⋅(α2β(w)⋅α3(δ))⏟4′−(−1)|w||γ|+|δ|(|ε|+|γ|)+|ε||γ|(αβ2(w)⋅α2β(δ))⋅(α2β(γ)⋅α3(ε)), |
(−1)|ε|(|γ|+|δ|)~asα,β(μ(β2(γ),αβ(w)),α2β(δ),α3(ε))=(−1)|ε|(|γ|+|δ|)((β2(γ)⋅αβ(w))⋅α2β(δ))⋅α3β(ε)+(−1)|ε||w|α2β2(ε)⋅((αβ(γ)⋅α2(w))⋅α3(δ))⏟5′+(−1)|ε|(|γ|+|δ|)+|δ|(|w|+|γ|)(αβ2(δ)⋅(αβ(γ)⋅α2(w)))⋅α3β(ε)+(−1)|ε||w|+|δ|(|γ|+|w|)α2β2(ε)⋅(α2β(δ)⋅(α2(γ)⋅α3β−1(w)))_⑦−(−1)|ε|(|γ|+|δ|)(αβ2(γ)⋅α2β(w))⋅(α2β(δ)⋅α3(ε))−(−1)|ε|(|δ|+|w|)+|δ|(|γ|+|w|)(αβ2(δ)⋅α2β(ε))⋅(α2β(γ)⋅α3(w))_⑧−(−1)|ε||γ|(αβ2(γ)⋅α2β(w))⋅(α2β(ε)⋅α3(δ))⏟6′−(−1)|ε||w|+|δ|(|γ|+|w|)(αβ2(ε)⋅α2β(δ))⋅(α2β(γ)⋅α3(w))_⑨+(−1)|ε|(|γ|+|δ|)+|γ||w|((β2(w)⋅αβ(γ))⋅α2β(δ))⋅α3β(ε)+(−1)|ε||w|+|γ||w|α2β2(ε)⋅((αβ(w)⋅α2(γ))⋅α3(δ))⏟7′+(−1)|ε|(|γ|+|δ|)+|δ|(|γ|+|w|)+|γ||w|(αβ2(δ)⋅(αβ(w)⋅α2(γ)))⋅α3β(ε)+(−1)|w||γ|+|δ|(|γ|+|w|)+|ε||w|α2β2(ε)⋅(α2β(δ)⋅(α2(w)⋅α3β−1(γ)))−(−1)|ε|(|γ|+|δ|)+|γ||w|(αβ2(w)⋅α2β(γ))⋅(α2β(δ)⋅α3(ε))−(−1)|ε|(|δ|+|w|)+|δ|(|γ|+|w|)+|γ||w|(αβ2(δ)⋅α2β(ε))⋅(α2β(w)⋅α3(γ))−(−1)|w||γ|+|ε||γ|(αβ2(w)⋅α2β(γ))⋅(α2β(ε)⋅α3(δ))⏟8′−(−1)|w||γ|+|δ|(|γ|+|w|)+|ε||w|(αβ2(ε)⋅α2β(δ))⋅(α2β(w)⋅α3(γ)), |
(−1)|γ|(|w|+|δ|)~asα,β(μ(β2(w),αβ(ε)),α2β(δ),α3(γ))=(−1)|γ|(|w|+|δ|)((β2(w)⋅αβ(ε))⋅α2β(δ))⋅α3β(γ)+(−1)|γ||ε|α2β2(γ)⋅((αβ(w)⋅α2(ε))⋅α3(δ))⏟9′+(−1)|γ|(|w|+|δ|)+|δ|(|w|+|ε|)(αβ2(δ)⋅(αβ(w)⋅α2(ε)))⋅α3β(γ)+(−1)|γ||ε|+|δ|(|w|+|ε|)α2β2(γ)⋅(α2β(δ)⋅(α2(w)⋅α3β−1(ε)))−(−1)|γ|(|w|+|δ|)(αβ2(w)⋅α2β(ε))⋅(α2β(δ)⋅α3(γ))−(−1)|γ|(|δ|+|ε|)+|δ|(|w|+|ε|)(αβ2(δ)⋅α2β(γ))⋅(α2β(w)⋅α3(ε))−(−1)|γ||w|(αβ2(w)⋅α2β(ε))⋅(α2β(γ)⋅α3(δ))⏟10′−(−1)|γ||ε|+|δ|(|w|+|ε|)(αβ2(γ)⋅α2β(δ))⋅(α2β(w)⋅α3(ε))+(−1)|γ|(|w|+|δ|)+|w||ε|((β2(ε)⋅αβ(w))⋅α2β(δ))⋅α3β(γ)+(−1)|w||ε|+|ε||γ|α2β2(γ)⋅((αβ(ε)⋅α2(w))⋅α3(δ))⏟11′+(−1)|γ|(|w|+|δ|)+|δ|(|w|+|ε|)+|w||ε|(αβ2(δ)⋅(αβ(ε)⋅α2(w)))⋅α3β(γ)+(−1)|w||ε|+|δ|(|w|+|ε|)+|ε||γ|α2β2(γ)⋅(α2β(δ)⋅(α2(ε)⋅α3β−1(w)))_⑩−(−1)|γ|(|w|+|δ|)+|w||ε|(αβ2(ε)⋅α2β(w))⋅(α2β(δ)⋅α3(γ))−(−1)|γ|(|δ|+|ε|)+|δ|(|w|+|ε|)+|w||ε|(αβ2(δ)⋅α2β(γ))⋅(α2β(ε)⋅α3(w))_⑪−(−1)|w||ε|+|γ||w|(αβ2(ε)⋅α2β(w))⋅(α2β(γ)⋅α3(δ))⏟12′−(−1)|w||ε|+|δ|(|w|+|ε|)+|ε||γ|(αβ2(γ)⋅α2β(δ))⋅(α2β(ε)⋅α3(w))_⑫, |
By Eq (4.4), we have ①+⋯+⑫=0, and by Eq (4.5), 1′+⋯+12′=0, Analogously, the conclusion that the sum is zero can be obtained by recombining the remaining unmarked formulas, which implies
↺ε,γ,w(−1)|w|(|ε|+|δ|)~asα,β(μ(β2(ε),αβ(γ)),α2β(δ),α3(w))=0. |
This completes the proof.
Definition 4.3. Let (J,μ,α,β) be a Bihom-Jordan superalgebra, and (V,ρ,ϕ,ψ) be its representation. If even the linear map T:J→V satisfies the following conditions: for all a,b∈H(V),
μ(T(a),T(b))=T(ρ(T(a))b+(−1)|a||b|ρ(T(ϕ−1ψ(b)))ϕψ−1(a)), |
T∘ϕ=α∘T,T∘ψ=β∘T, |
then T is called O-operator with respect to representation.
Definition 4.4. Let (J,μ,α,β) be a Bihom-Jordan superalgebra and α,β be reversible, R∈gl(J), R is called Rota–Baxter operator on J, if for all ε,γ∈H(J), the following conditions hold:
μ(R(ε),R(γ))=R(μ(R(ε),γ)+(−1)|ε||γ|μ(R(α−1β(γ)),αβ−1(ε))), |
R∘α=α∘R,R∘β=β∘R. |
Theorem 4.2. Let (J,μ,α,β) be a Bihom-Jordan superalgebra, (V,ρ,ϕ,ψ) be its representation, and T be an O-operator with respect to representation. Define bilinear operation ⋅ on V:
a⋅b=ρ(T(a))b,∀a,b∈H(V). |
Then (V,⋅,ϕ,ψ) is a Bihom-pre-Jordan superalgebra.
Proof. Actually, it can be calculated directly from Definition 4.1.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research is supported by NNSF of China (Nos. 12271085 and 12071405). The authors would like to thank the reviewers for valuable suggestions to improve the paper.
The authors declare there are no conflicts of interest.
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