In this paper, for n≥2 and n≠3, the module algebra structures of Xq(A1) on the quantum n-space were discussed, where the quantum n-space is denoted by Aq(n). In particular, a complete list of Xq(A1)-module algebra structures on the quantum plane Aq(2) was produced and the isomorphism classes of these structures were described.
Citation: Dong Su, Fengxia Gao, Zhenzhen Gao. Module algebra structures of nonstandard quantum group Xq(A1) on the quantum plane[J]. Electronic Research Archive, 2025, 33(6): 3543-3560. doi: 10.3934/era.2025157
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In this paper, for n≥2 and n≠3, the module algebra structures of Xq(A1) on the quantum n-space were discussed, where the quantum n-space is denoted by Aq(n). In particular, a complete list of Xq(A1)-module algebra structures on the quantum plane Aq(2) was produced and the isomorphism classes of these structures were described.
The nonstandard quantum groups were studied in [1], where Ge et al. [1] obtained new solutions of the Yang-Baxter equations. For these new solutions, they followed the Faddeev-Reshetikhin-Takhtajan [2] method to establish the related quantum group structure, which, in general, may not be the same as the standard ones. In [3] one class of nonstandard quantum deformation corresponding to simple Lie algebra sln was given, which is denoted by Xq(An−1). For each vertex i(i=1,⋯,n−1) of the Dynkin diagram, the parameter qi is equal to q or −q−1, and if qi=q for all i, then Xq(An−1) is just Uq(sln). However, if qi≠qi+1 for some 1≤i≤n−1, it has the relation E2i=F2i=0 in Xq(An−1), such that Xq(An−1) is different from Uq(sln). For more results for nonstandard quantum groups, one can refer to [4,5,6].
The notion of Hopf algebra actions on algebras was introduced by Sweedler [7] in 1969. The Brauer groups of H-module and H-dimodule algebras were researched by Beattie [8]. A duality theorem for Hopf module algebras was studied by Blattner and Montgomery [9] in 1985. Moreover, the actions of Hopf algebras and their generalizations [10,11] play an important role in quantum group theory [12,13], and the actions of Hopf algebras have various applications in physics [14]. Duplij and Sinel'shchikov [15,16] used a general form of the automorphism of the quantum plane to render the notion of weight for Uq(sl2)-actions, and they completely classified Uq(sl2)-module algebra structures on the quantum plane, which consist of 6 non-isomorphic cases. Moreover, in [17] the authors used the method of weights [15,16] to study the module algebra structures of Uq(slm+1) on the coordinate algebra of quantum vector spaces. More relevant research can be found at [18,19]. However, the module algebras of nonstandard quantum groups have not yet achieved research results. Consequently, based on the above research results, we consider here the actions of the nonstandard quantum group Xq(A1) on the quantum n-space Aq(n). In particular, a complete list of Xq(A1)-module algebra structures on the quantum plane Aq(2) is produced and the isomorphism classes of these structures are described.
This paper is organized as follows. In Section 1, we introduce some necessary notations and concepts, as well as prove a lemma about actions on generators and any elements of Aq(n). In Section 2, using the method of weights [15,16,17], the 0-th homogeneous component and 1-st homogeneous component of the action matrix are given. We have 2n+1 cases for the 0-th homogeneous component (MEF)0, and 2n(n−1)+1 cases for the 1-st homogeneous component (MEF)1. In Section 3, we study the actions of Xq(A1) on Aq(2), and characterize all module algebra structures of Xq(A1) on the quantum plane Aq(2), which rely upon considering the 0-th and 1-st homogeneous components of an action.
Throughout, we work over the complex field C unless otherwise stated. All algebras, Hopf algebras, and modules are defined over C; all maps are C-linear.
Let (H,m,η,Δ,ε,S) be a Hopf algebra, where Δ, ε, and S are the comultiplication, counit, and antipode of H, respectively. Let A be a unital algebra with unit 1. Sweedler's notations [7] are used in the sequel. For example, for h∈H, we denote
Δ(h)=∑(h)h(1)⊗h(2). |
Definition 2.1. By a structure of an H-module algebra on A, we mean a homomorphism π:H→EndCA such that:
1) for all h∈H,a,b∈A, π(h)(ab)=∑(h)π(h(1))(a)π(h(2))(b);
2) for all h∈H,π(h)(1)=ε(h)1.
Let π1 and π2 be two H-module algebras on A, and the structures π1,π2 are said to be isomorphic, if there exists an automorphism Ψ of the algebra A, such that Ψπ1(h)Ψ−1=π2(h) for all h∈H.
Throughout the paper we assume that q∈C∗=C∖{0} is not a root of the unit (qn≠1 for all non-zero integers n). A class of the nonstandard quantum group Xq(A1) was studied by the authors of [3,4]. Now, we recall the definition of Xq(A1).
Definition 2.2. The nonstandard quantum group Xq(A1) is a unital associative C-algebra generated by E,F,K1,K2,K−11,K−12 subject to the relations:
K1K−11=K−11K1=1,K2K−12=K−12K2=1,K1K2=K2K1, | (2.1) |
K1E=q−1EK1, | (2.2) |
K1F=qFK1, | (2.3) |
K2E=−q−1EK2, | (2.4) |
K2F=−qFK2, | (2.5) |
EF−FE=K2K−11−K−12K1q−q−1, | (2.6) |
E2=F2=0. | (2.7) |
The algebra Xq(A1) is also a Hopf algebra, and the comultiplication Δ, counit ε, and antipode S are given as the following:
Δ(K1)=K1⊗K1,Δ(K2)=K2⊗K2, | (2.8) |
Δ(E)=E⊗1+K2K−11⊗E, | (2.9) |
Δ(F)=1⊗F+F⊗K−12K1, | (2.10) |
ε(K1)=1,ε(K2)=1,ε(E)=0,ε(F)=0, | (2.11) |
S(K1)=K−11,S(K2)=K−12,S(E)=−K1K−12E,S(F)=−FK2K−11. | (2.12) |
Let us review the definition of the quantum n-space (see [20,21]).
Definition 2.3. The quantum n-space Aq(n) is a unital algebra, generated by n generators xi for i∈{1,2,⋯,n}, and for any i>j it satisfies the relation:
xixj=qxjxi. | (2.13) |
The quantum n-space Aq(n) is also called a coordinate algebra of quantum n-dimensional vector space. If n=2, Aq(2) is called a quantum plane.
For all n≥2 and n≠3, by [22,23,24], one has a description of automorphisms of the algebra Aq(n), as follows. Let Ψ be an automorphism of Aq(n), and then there exist nonzero constants αi for i∈{1,2,4,⋯,n}, such that
Ψ:xi→αixi. |
All such automorphisms form the automorphism group of Aq(n), which we denote by Aut(Aq(n)), and in addition, one can get
Aut(Aq(n))≅(C∗)n. |
It should be pointed out that there are more automorphisms of Aq(3). Let σ be an automorphism of Aq(3), and then there exist nonzero constants α,β,γ∈C∗ and t∈C, such that
σ:x1→αx1,x2→βx2+tx1x3,x3→γx3, |
and Aut(Aq(3))≅C⋊(C∗)3. Obviously, the automorphism group of Aq(3) is more complex, and therefore, we separately discussed the module algebra structures of nonstandard quantum group Xq(A1) on Aq(3), as detailed in [25].
Unless otherwise specified, in the following text, we fix the integers n≥2 and n≠3.
Next, we give a lemma which will be useful for checking the module algebra structures of Xq(A1) on Aq(n).
Lemma 2.4. Given the module algebra actions of the generators E,F,K1,K2 of Xq(A1) on Aq(n), if an element in the ideal generated by the relations (2.1)–(2.7) of Xq(A1), which acting on the generators xi of Aq(n) produces zero for all i=1,2,4,⋯,n, then this element acting on any v∈Aq(n) produces zero.
Proof. Here, we only prove that, if
[(EF−FE)−K2K−11−K−12K1q−q−1](xi)=0,[(EF−FE)−K2K−11−K−12K1q−q−1](xj)=0, |
where xi,xj are arbitrary generators of Aq(n), then
[(EF−FE)−K2K−11−K−12K1q−q−1](xixj)=0. |
The other relationships can be proven similarly. Indeed, by (2.9) and (2.10), we have
Δ(E)Δ(F)−Δ(F)Δ(E)=(E⊗1+K2K−11⊗E)(1⊗F+F⊗K−12K1)=(EF−FE)⊗K−12K1+K2K−11⊗(EF−FE), |
and by Definition (2.1), then,
(EF−FE)(xixj)=π(EF−FE)(xi)π(K−12K1))(xj)+π((K2K−11)(xi)π((EF−FE))(xj)=K2K−11−K−12K1q−q−1(xi)K2K−11(xj)+K2K−11(xi)K2K−11−K−12K1q−q−1(xj)=K2K−11−K−12K1q−q−1(xixj). |
Thus, [(EF−FE)−K2K−11−K−12K1q−q−1](xixj)=0, and the lemma holds.
Therefore, by Lemma 2.4, in checking whether the relations of Xq(A1), acting on any v∈Aq(n), produces zero, we only need to check whether they produce zero when they act on the generators x1,x2,⋯,xn.
In this section, we will study the module algebra structures of Xq(A1) on Aq(n), where K1,K2∈Aut(Aq(n)), n≥2, and n≠3.
The s-th homogeneous component of Aq(n) is denoted by Aq(n)s, which is linear spanned by the monomials xm11xm22⋯xmnn with m1+m2+⋯+mn=s. Also, given a polynomial p∈Aq(n), the s-th homogeneous component of p is denote by (p)s, which is the projection of p onto Aq(n)s parallel to the direct sum of all other homogeneous components of Aq(n).
By the definition of module algebra, it is easy to see that any action of Xq(A1) on Aq(n) is determined by the following 4×n matrix with entries from Aq(n):
Mdefinition=(K1(x1)K1(x2)⋯K1(xn)K2(x1)K2(x2)⋯K2(xn)E(x1)E(x2)⋯E(xn)F(x1)F(x2)⋯F(xn)), | (3.1) |
which is called the full action matrix, see[22]. Given a Xq(A1)-module algebra structure on Aq(n), obviously, the action of K1 or K2 determines an automorphism of Aq(n). Therefore, by the assumption K1,K2∈Aut(Aq(n), we can set
MK1K2definition=(K1(x1)K1(x2)⋯K1(xn)K2(x1)K2(x2)⋯K2(xn))=(α1x1α2x2⋯αnxnβ1x1β2x2⋯βnxn), | (3.2) |
where αi,βi∈C∗ for i∈{1,2,⋯,n}.
It is easy to see that every monomial xm11xm22⋯xmnn∈Aq(n) is an eigenvector of K1 and K2, and the associated eigenvalues αm11αm22⋯αmnn and βm11βm22⋯βmnn are called the K1-weight and K2-weight of this monomial, respectively, which will be written as
wtK1(xm11xm22⋯xmnn)=αm11αm22⋯αmnn,wtK2(xm11xm22⋯xmnn)=βm11βm22⋯βmnn. |
We will also need another matrix MEF as follows:
MEFdefinition=(E(x1)E(x2)⋯E(xn)F(x1)F(x2)⋯F(xn)), | (3.3) |
and we call MK1K2 and MEF the action K1K2-matrix and EF-matrix, respectively. It follows from relations (2.2)–(2.5) that all entries of M are weight vectors for K1 and K2, and we have
wtK1(M)definition=(wtK1(K1(x1))wtK1(K1(x2))⋯wtK1(K1(xn))wtK1(K2(x1))wtK1(K2(x2))⋯wtK1(K2(xn))wtK1(E(x1))wtK1(E(x2))⋯wtK1(E(xn))wtK1(F(x1))wtK1(F(x2))⋯wtK1(F(xn)))⋈(wtK1(x1)wtK1(x2)⋯wtK1(xn)wtK1(x1)wtK1(x2)⋯wtK1(xn)q−1wtK1(x1)q−1wtK1(x2)⋯q−1wtK1(xn)qwtK1(x1)qwtK1(x2)⋯qwtK1(xn))=(α1α2⋯αnα1α2⋯αnq−1α1q−1α2⋯q−1αnqα1qα2⋯qαn), | (3.4) |
wtK2(M)definition=(wtK2(K1(x1))wtK2(K1(x2))⋯wtK2(K1(xn))wtK2(K2(x1))wtK2(K2(x2))⋯wtK2(K2(xn))wtK2(E(x1))wtK2(E(x2))⋯wtK2(E(xn))wtK2(F(x1))wtK2(F(x2))⋯wtK2(F(xn)))⋈(wtK2(x1)wtK2(x2)⋯wtK2(xn)wtK2(x1)wtK2(x2)⋯wtK2(xn)−q−1wtK2(x1)−q−1wtK2(x2)⋯−q−1wtK2(xn)−qwtK2(x1)−qwtK2(x2)⋯−qwtK2(xn))=(β1β2⋯βnβ1β2⋯βn−q−1β1−q−1β2⋯−q−1βn−qβ1−qβ2⋯−qβn), | (3.5) |
where the relation (ast)⋈(bst) means that for every pair of indices s,t such that both ast and bst are nonzero, one has ast=bst.
In the following, we denote the j-th homogeneous component of M, whose elements are just the j-th homogeneous components of the corresponding entries of M, by (M)j. Set
(M)0=(00⋯000⋯0a1a2⋯anb1b2⋯bn)0, |
where ai,bi∈C for all i∈{1,2,⋯,n}. Then, we obtain
wtK1((MEF)0)⋈(q−1α1q−1α2⋯q−1αnqα1qα2⋯qαn)⋈(ε(K1)ε(K1)⋯ε(K1)ε(K1)ε(K1)⋯ε(K1))=(11⋯111⋯1), | (3.6) |
wtK2((MEF)0)⋈(−q−1β1−q−1β2⋯−q−1βn−qβ1−qβ2⋯−qβn)⋈(ε(K2)ε(K2)⋯ε(K2)ε(K2)ε(K2)⋯ε(K2))=(11⋯111⋯1). | (3.7) |
Therefore, the relations (3.6) and (3.7) imply that ai and bi are at most one nonzero for any i∈{1,2,⋯,n}, and
ai≠0⇒αi=q,βi=−q, | (3.8) |
bi≠0⇒αi=q−1,βi=−q−1. | (3.9) |
An application of E and F to the relation (2.13) and by Eq (3.2), one has the following equalities:
E(xi)xj+α−1iβixiE(xj)=qE(xj)xi+qα−1jβjxjE(xi),fori>j, | (3.10) |
xiF(xj)+β−1jαjF(xi)xj=qxjF(xi)+qβ−1iαiF(xj)xi,fori>j. | (3.11) |
After projecting (3.10) and (3.11) to Aq(n)1, we obtain
ai(1−qα−1jβj)xj+aj(α−1iβi−q)xi=0,fori>j,bj(1−qβ−1iαi)xi+bi(β−1jαj−q)xj=0,fori>j, |
which certainly implies
ai(1−qα−1jβj)=aj(α−1iβi−q)=bj(1−qβ−1iαi)=bi(β−1jαj−q)=0. |
For any i,j∈{1,2,⋯,n} and i>j, we will determine the weight constants αi and βi as follows:
ai≠0⇒βjα−1j=q−1, | (3.12) |
aj≠0⇒βiα−1i=q, | (3.13) |
bi≠0⇒αjβ−1j=q, | (3.14) |
bj≠0⇒αiβ−1i=q−1. | (3.15) |
Lemma 3.1. For any i,j,s,t∈{1,2,⋯,n}, ai, aj, bs, and bt are at most one nonzero.
Proof. For any i,j∈{1,2,⋯,n}, and i>j, we only prove that ai and aj are at most one nonzero. Assume ai≠0 and aj≠0, and then
ai≠0⇒αi=q,βi=−q,βjα−1j=q−1,aj≠0⇒αj=q,βj=−q,βiα−1i=q, |
by Eqs (3.8), (3.12), and (3.13). However
βjα−1j=−qq−1=−1=q−1andβiα−1i=−qq−1=−1=q, |
which are impossible, since it is contradictory to q not being a root of the unit. Therefore at least one of ai and aj is zero for i,j∈1,2,⋯,n.
The remaining statements can be proven in a similar way.
In summary, we have obtained the following results for the 0-th homogeneous component (MEF)0 of MEF.
Theorem 3.2. There are 2n+1 cases for the 0-th homogeneous component (MEF)0 of MEF, as follows:
1) ai≠0,aj=0 for i≠j and all bs=0 for any i,j,s∈{1,2,⋯,n}, i.e.,
(a10⋯000⋯0)0,(0a2⋯000⋯0)0,⋯,(00⋯an00⋯0)0, |
and we have
ai≠0⇒αi=q,βi=−q,β1α−11=β2α−12=⋯=βi−1α−1i−1=q−1,βi+1α−1i+1=βi+2α−1i+2=⋯=βnα−1n=q; | (3.16) |
2) bi≠0,bj=0 for i≠j and all as=0 for any i,j,s∈{1,2,⋯,n}, i.e.,
(00⋯0b10⋯0)0,(00⋯00b2⋯0)0,⋯,(00⋯000⋯bn)0, |
and we have
bi≠0⇒αi=q−1,βi=−q−1,β−11α1=β−12α2=⋯=β−1i−1αi−1=q,β−1i+1αi+1=β−1i+2αi+2=⋯=β−1nαn=q−1; | (3.17) |
3) all ai=bi=0 for any i∈{1,2,⋯,n}, i.e.,
(00⋯000⋯0)0. |
Therefore, it does not determine the weight constants at all.
Next, for the 1-st homogeneous component (MEF)1, due to q not being a root of the unit, one has
wtK1(E(xi))=q−1αi=q−1wtK1(xi)≠wtK1(xi),wtK2(E(xi))=−q−1βi=−q−1wtK2(xi)≠wtK2(xi), |
which implies
(E(xi))1=i−1∑s=1cisxs+n∑s=i+1cisxs, |
for some cis∈C. In a similar way, we have
(F(xi))1=i−1∑s=1disxs+n∑s=i+1disxs, |
for some dis∈C. Hence
(MEF)1=(n∑s=2c1sxsc21x1+n∑s=3c2sxs⋯i−1∑s=1cisxs+n∑s=i+1cisxs⋯n−1∑s=1cnsxsn∑s=2disxsd21x1+n∑s=3d2sxs⋯i−1∑s=1disxs+n∑s=i+1disxs⋯n−1∑s=1dnsxs)1 |
where cis,dis∈C.
Now project (3.10) and (3.11) to Aq(n)2, and we can obtain
cij(1−qβjα−1j)x2j+cji(βiα−1i−q)x2i+j−1∑s=1cis(1−q2βjα−1j)xsxj+i−1∑s=j+1cisq(1−βjα−1j)xjxs+n∑s=i+1cisq(1−βjα−1j)xjxs+j−1∑t=1cjtq(βiα−1i−1)xtxi+i−1∑t=j+1cjtq(βiα−1i−1)xtxi+n∑t=i+1cjt(βiα−1i−q2)xixt=0,dji(1−qβ−1iαi)x2i+dij(β−1jαj−q)x2j+j−1∑t=1djtq(1−β−1iαi)xtxi+i−1∑t=j+1djtq(1−β−1iαi)xtxi+n∑t=i+1djt(1−q2β−1iαi)xixt+j−1∑s=1dis(β−1jαj−q2)xsxj+i−1∑s=j+1disq(β−1jαj−1)xjxs+n∑s=i+1disq(β−1jαj−1)xjxs=0. |
for any i,j∈{1,2,⋯,n} and i>j, Where
cij(1−qβjα−1j)=0,fori>j,cji(βiα−1i−q)=0,fori>j,cis(1−q2βjα−1j)=0,for1≤s≤j−1,cisq(1−βjα−1j)=0,forj+1≤s≤i−1,cisq(1−βjα−1j)=0,fori+1≤s≤n,cjtq(βiα−1i−1)=0,for1≤t≤j−1,cjtq(βiα−1i−1)=0,forj+1≤t≤i−1,cjt(βiα−1i−q2)=0,fori+1≤t≤n. |
dji(1−qβ−1iαi)=0,fori>j,dij(β−1jαj−q)=0,fori>j,djtq(1−β−1iαi)=0,for1≤t≤j−1,djtq(1−β−1iαi)=0,forj+1≤t≤i−1,djt(1−q2β−1iαi)=0,fori+1≤t≤n,dis(β−1jαj−q2)=0,for1≤s≤j−1,disq(β−1jαj−1)=0,forj+1≤s≤i−1,disq(β−1jαj−1)=0,fori+1≤s≤n. |
As a consequence, for any i,j∈{1,2,⋯,n} and i>j, we have
cij≠0⇒βjα−1j=q−1,cji≠0⇒βiα−1i=q, |
cis≠0⇒βjα−1j=q−2,for1≤s≤j−1,cis≠0⇒βjα−1j=1,forj+1≤s≤i−1,cis≠0⇒βjα−1j=1,fori+1≤s≤n,cjt≠0⇒βiα−1i=1,for1≤t≤j−1,cjt≠0⇒βiα−1i=1,forj+1≤t≤i−1,cjt≠0⇒βiα−1i=q2,fori+1≤t≤n. | (3.18) |
dji≠0⇒β−1iαi=q−1,dij≠0⇒β−1jαj=q,djt≠0⇒β−1iαi=1,for1≤t≤j−1,djt≠0⇒β−1iαi=1,forj+1≤t≤i−1,djt≠0⇒β−1iαi=q−2,fori+1≤t≤n,dis≠0⇒β−1jαj=q2,for1≤s≤j−1,dis≠0⇒β−1jαj=1,forj+1≤s≤i−1,dis≠0⇒β−1jαj=1,fori+1≤s≤n. | (3.19) |
Lemma 3.3. For any i∈{1,2,⋯,n}, every 1-st homogeneous component (E(xi))1 and (F(xi))1, if nonzero, reduces to a monomial.
Proof. We assume that
E(xi)1=i−1∑s=1cisxs+n∑s=i+1cisxs, |
and cis≠0,cis′≠0(s≠s′) for some s,s′∈{1,2,⋯,i−1,i+1,⋯,n}. Without loss of generality, we stipulate that s<s′.
If s,s′<i, then
cis≠0⇒βsα−1s=q−1,cis′≠0⇒β1α−11=β2α−12=⋯=βs′−1α−1s′−1=1. |
However, s must be one of the {1,2,⋯,s′−1}, and one gets q−1=1, which is impossible. Hence, cis and cis′ are at most one nonzero, and (E(xi))1 is equal to zero or a monomial. The remaining situations can be proven in a similar way.
Similarly, (F(xi))1 is equal to zero or a monomial.
Additionally, since
wtK1((MEF)1)⋈(q−1α1q−1α2⋯q−1αnqα1qα2⋯qαn), | (3.20) |
wtK2((MEF)1)⋈(−q−1β1−q−1β2⋯−q−1βn−qβ1−qβ2⋯−qβn). | (3.21) |
We obtain the following result.
Lemma 3.4. For any i,j,s,t∈{1,2,⋯,n}, (E(xi))1, (E(xj))1, (F(xs))1, (F(xt))1 are at most one nonzero.
Proof. Here, we only prove that (E(xi))1 and (E(xj))1 are at most one nonzero. The other statements can be proven similarly.
By Lemma 3.3, we get that if (E(xi))1 and (E(xj))1 are nonzero, then they are a monomial for any i,j∈{1,2,⋯,n}. Assume
E(xi)1=cisxs≠0andE(xj)1=cjs′xs′≠0. |
Without loss of generality, we stipulate that i>j. According to the Eqs (3.20) and (3.21), we have
wtK1(E(xi)1)=q−1αi,wtK2(E(xi)1)=−q−1βi, |
wtK1(E(xj)1)=q−1αj,wtK2(E(xj)1)=−q−1βj. |
In addition,
wtK1(E(xi)1)=αs,wtK2(E(xi)1)=βs, |
wtK1(E(xj)1)=αs′,wtK2(E(xj)1)=βs′. |
So, αi=qαs,βi=−qβs,αj=qαs′,βj=−qβs′.
On the other hand, since cis≠0 and cjs′≠0, it follows that
βjα−1j={q−1s=j,q−21≤s≤j−1,1j+1≤s≤n, |
βiα−1i={qs′=i,11≤s′≤j−1,q2j+1≤s′≤n, |
by (3.18). Then q−1=−q−2 or q−1=−1, and q=−q2 or q=−1, which are impossible. Hence, (E(xi))1 and (E(xj))1 are at most one nonzero.
From the above discussion, we have the following result for the 1-st homogeneous component (MEF)1 of MEF.
Theorem 3.5. There are 2n(n−1)+1 cases for the 1-st homogeneous component (MEF)1 of MEF, as follows:
1) cis≠0(i≠s), and otherwise ci′s′=0 and all djt=0 for any i,s,j,t,i′,s′∈{1,2,⋯,n}, i.e.,
(0⋯0cisxs0⋯00⋯000⋯0)1, |
and we have αi=qαs,βi=−qβs, and
ifi>s,thenβsα−1s=q−1,βi+1α−1i+1=βi+2α−1i+2=⋯=βnα−1n=1,βi−1α−1i−1=βi−2α−1i−2=⋯=βs+1α−1s+1=q−2,βs−1α−1s−1=βs−2α−1s−2=⋯=β1α−11=1; | (3.22) |
ifi<s,thenβsα−1s=q,βi−1α−1i−1=βi−2α−1i−2=⋯=β1α−11=1,βi+1α−1i+1=βi+2α−1i+2=⋯=βs−1α−1s−1=q2,βs+1α−1s+1=βs+2α−1s+2=⋯=βnα−1n=1; | (3.23) |
2) dis≠0(i≠s), and otherwise di′s′=0 and all cjt=0 for any i,s,j,t,i′,s′∈{1,2,⋯,n}, i.e.,
(0⋯000⋯00⋯0disxs0⋯0)1, |
and we have αi=q−1αs,βi=−q−1βs, and
ifi>s,thenβ−1sαs=q,β−1i+1αi+1=β−1i+2αi+2=⋯=β−1nαn=1,β−1i−1αi−1=β−1i−2αi−2=⋯=β−1s+1αs+1=q2,β−1s−1αs−1=β−1s−2αs−2=⋯=β−11α1=1; | (3.24) |
ifi<s,thenβ−1sαs=q−1,β−1i−1αi−1=β−1i−2αi−2=⋯=β−11α1=1,β−1i+1αi+1=β−1i+2αi+2=⋯=β−1s−1αs−1=q−2,β−1s+1αs+1=β−1s+2αs+2=⋯=β−1nαn=1; | (3.25) |
3) all cis=0 and di′s′=0, for any i,s,i′,s′∈{1,2,⋯,n}, i.e.,
(00⋯000⋯0)1. |
Therefore, it does not determine the weight constants at all.
In this section, our aim is to describe the concrete Xq(A1)-module algebra structures on the quantum plane Aq(2), where K1,K2∈Aut(Aq(2))≅(C∗)2.
By Theorems 3.2 and 3.5, it follows that if both the 0-th homogeneous component and the 1-st homogeneous component of MEF are nonzero, it is easy to see that these series are empty, so we only need to consider 9 possibilities.
[(a1000)0,(0000)1],[(0a200)0,(0000)1],[(00b10)0,(0000)1], |
[(000b2)0,(0000)1],[(0000)0,(c12x2000)1],[(0000)0,(0c21x100)1], |
[(0000)0,(00d12x20)1],[(0000)0,(000d21x1)1],[(0000)0,(0000)1] |
where ai≠0,bi≠0 for i=1,2 and c12,c21,d12,d21 are not zero.
Lemma 4.1. If the 0-th homogeneous component of MEF is zero and the 1-st homogeneous component of MEF is nonzero, then these series are empty.
Proof. Now we show that the [(0000)0,(c12x2000)1]-series is empty. If we suppose the contrary, then it follows from
EF−FE=K2K−11−K−12K1q−q−1 |
that within this series, one can have
K2K−11−K−12K1q−q−1(x1)=β1α−11−β−11α1q−q−1x1. |
By c12≠0, one can get α1=qα2,β1=−qβ2, and β2α−12=q. Hence, β1α−11=−q, and
K2K−11−K−12K1q−q−1(x1)=−x1. |
On the other hand, projecting (EF−FE)(x1) to Aq(2)1, we obtain
(EF−FE)(x1)=E(F(x1))−F(E(x1))=E(0)−F(c12x2)=0. |
However, 0≠−xi. We get the contradiction, and prove our claim.
In a similar way, one can prove that all other series where the 0-st homogeneous component of MEF is zero and the 1-st homogeneous component of MEF is nonzero are empty.
Lemma 4.2. If the 0-th homogeneous component of MEF is nonzero and the 1-st homogeneous component of MEF is zero, then these series are empty.
Proof. We only show that the [(a1000)0,(0000)1]-series is empty, and in a similar way, one can prove that all other series are empty.
Consider this series and we obtain that
a1≠0⇒α1=q,β1=−q,β2α−12=q, |
and suppose that it is not empty. We set
K1(x1)=α1x1=qx,K2(x1)=β1x1=−qx1,K2(x2)=α2x2,K2(x2)=β2x2,E(x1)=a1+∑m1+m2≥2ρm1m2xm11xm22form1,m2∈N,E(x2)=∑l1+l2≥2θl1l2xl11xl22forl1,l2∈N,F(x1)=∑t1+t2≥2σt1t2xt11xt22fort1,t2∈N,F(x2)=∑h1+h2≥2τh1h2xh11xh22forh1,h2∈N, |
where α2,β2∈C∗, and ρm1m2,θl1l2,σt1t2,τh1h2∈C.
Then we apply the relations (2.1)–(2.7) to the generators of Aq(2). It is easy to see that the application of relation (2.1) to the generators of Aq(2) produces zero. So, we consider the residue, as follows.
(K1E−q−1EK1)(x1)=K1(E(x1))−q−1E(K1(x1))=K1(a1+∑m1+m2≥2ρm1m2xm11xm22)−q−1qE(x1)=a1+∑m1+m2≥2ρm1m2αm11αm22xm11xm22−E(x1)=∑m1+m2≥2ρm1m2(αm11αm22−1)xm11xm22=0, |
and then ρm1m2=0 for all m1,m2∈N with m1+m2≥2, or αm22=q−m1 for some m1,m2∈N with m1+m2≥2.
(K2E+q−1EK2)(x1)=K2(E(x1))+q−1E(K2(x1))=K2(a1+∑m1+m2≥2ρm1m2xm11xm22)−q−1qE(x1)=a1+∑m1+m2≥2ρm1m2βm11βm22xm11xm22−E(x1)=∑m1+m2≥2ρm1m2(βm11βm22−1)xm11xm22=0, |
and then ρm1m2=0 for all m1,m2∈N with m1+m2≥2, or βm22=(−q)−m1 for some m1,m2∈N with m1+m2≥2.
If some ρm1m2≠0, and it meets the conditions, i.e.,
{αm22=q−m1,βm22=(−q)−m1, |
and β2α−12=q, one can get qm2=(−1)m1, since q is not a unit root, which is impossible. Therefore, we have E(x1)=a1.
Similar to the discussion above, we can obtain that
E(x2)=0, |
F(x1)=0orF(x1)=σ20x21, |
F(x2)=0orF(x2)=τ11x1x2, |
where σ20,τ11∈C.
From EF−FE=K2K−11−K−12K1q−q−1, we have
K2K−11−K−12K1q−q−1(x1)=β1α−11−β−11α1q−q−1=0,K2K−11−K−12K1q−q−1(x2)=β2α−12−β−12α2q−q−1=x2. |
If F(x2)=0, then
(EF−FE)(x2)=0≠x2; |
if F(x2)=τ11x1x2≠0, then
(EF−FE)(x2)=τ11a1x2=x2. |
Hence, we have τ11=1a1 and F(x2)=1a1x1x2.
By F2=0, one has that
F2(x2)=1a1F(x1x2)=1a1(x1F(x2)+F(x1)K−12K1(x2))=1a1(1a1x21x2+q−1F(x1)x2). |
If F(x1)=0, then
F2(x2)=1a21x21x2≠0; |
if F(x1)=σ20x21, then
F2(x2)=1a21x21x2+q−11a1σ20x21x2=0. |
So σ20=−qa1 and F(x1)=−qa1x21.
With an application of F to x2x1=qx1x2, we have
F(x2x1−qx1x2)=x2F(x1)−F(x2)x1−qx1F(x2)−F(x1)x2=−qa1x2x21−1a1x1x2x1−qa1x21x2+qa1x21x2=−qa1(1+q2)x21x2≠0. |
In summary, this series is empty.
In a similar way, one can prove that all other series where the 0-th homogeneous component of MEF is nonzero and the 1-st homogeneous component of MEF is zero are empty.
Theorem 4.3. The [(0000)0,(0000)1] -series has Xq(A1)-module algebra structures on the quantum plane Aq(2) given by
K1(x1)=λ1x1,K2(x1)=±λ1x1, | (4.1) |
K1(x2)=λ2x2,K2(x2)=±λ2x2, | (4.2) |
E(x1)=F(x1)=E(x2)=F(x2)=0, | (4.3) |
where λ1,λ2∈C∗, and therefore, they are pairwise nonisomorphic.
Proof. It is easy to check that (4.1)–(4.3) determine a well-defined Xq(A1)-action consistent with the multiplication in Xq(A1) and in the quantum plane Aq(2), as well as with comultiplication in Xq(A1). We prove that there are no other Xq(A1)-actions here. Note that an application of (2.6) to x1 or x2 has zero projection to Aq(2)1, i.e., (EF−FE)(xi)=0,(i=1,2), because in this series E and F send any monomial to a sum of the monomials of higher degree. Therefore,
K2K−11−K−12K1q−q−1(x1)=β1α−11−β−11α1q−q−1x1=0,K2K−11−K−12K1q−q−1(x2)=β2α−12−β−12α2q−q−1x2=0, |
and we have
β1α−11−β−11α1=β2α−12−β−12α2=0, |
which leads to β21=α21 and β22=α22. Let α1=λ1 and α2=λ2, and we have β1=±λ1 and β2=±λ2. To prove (4.3), note that if E(xi)≠0 or F(xi)≠0, for i=1,2, then they are a sum of the monomials with degrees greater than 1. Similar to the proof of Lemma 4.2, we get that this is impossible, because they cannot satisfy the conditions of Xq(A1)-module algebra on Aq(2).
To see that the Xq(A1)-module algebra structures are pairwise nonisomorphic, observe that all the automorphisms of the quantum plane commute with the actions of K1 and K2.
Next, our immediate intention is to describe the composition series for these representations.
Proposition 4.4. The representations corresponding to the [(0000)0,(0000)1]-series described in (4.1)–(4.3) split into the direct sum Aq(2)=⊕∞m=0⊕∞n=0Cxm1xn2 of one-dimensional subrepresentations. These subrepresentations may belong to two isomorphism classes, depending on the weights of a specific monomial xm1xn2 which can be K1(xm1xn2)=λm1λn2xm1xn2 and K2(xm1xn2)=(±1)m+nλm1λn2xm1xn2.
Proof. Since E and F are represented by zero operators and the monomials xm1xn2 are eigenvectors for K1 and K2, then every direct summand is Xq(A1)-invariant.
In this paper, we discuss the module algebra structures of Xq(A1) on the quantum n-space Aq(n) for n≥2 and n≠3. However, we have presented only a complete list of Xq(A1)-module algebra structures on the quantum plane Aq(2), and described the isomorphism classes of these structures. For all n≥4, it is complicated to give the solutions of (3.7) and (3.8). We will continue to classify the module algebra structures of Xq(A1) on the quantum n-space Aq(n) for n≥4 in the future.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the National Natural Science Foundation of China (Grant No. 12201187), Natural Science Foundation of Henan Province (Grant No. 222300420156), and Key Research Project Plan of Henan Province Higher Education Institutions (Grant No. 23B110006).
The authors declare there is no conflicts of interest.
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