Let An be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over An. By further exploring the relationship between the Poincare series and the dimension and the multiplicity of M...Let An be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over An. By further exploring the relationship between the Poincare series and the dimension and the multiplicity of M, we are able to prove that the tensor product of two finitely generated modules over An has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of An.展开更多
We study a family of ＂symmetric＂ multiparameter quantized Weyl alge- bras A-q,A n（K） and some related algebras. We compute the Nakayama automorphism of A-q,A n（K）, give a necessary and sufficient condition for A...We study a family of ＂symmetric＂ multiparameter quantized Weyl alge- bras A-q,A n（K） and some related algebras. We compute the Nakayama automorphism of A-q,A n（K）, give a necessary and sufficient condition for A-q,A n（K） to be Calabi-Yau, and prove that A-q,A n（K） is cancellative. We study the automorphisms and isomorphism problem for A-q,A n（K） and .A-q,A n（K[t]）. Similar results are established for the Maltsiniotis multiparam- eter quantized Weyl algebraA-q,A n（K） and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization （A-q,A n（K））z and determine its automorphism group.展开更多
In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided pow...In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.展开更多
The concern of this paper is to derive formulas for the injective dimension of the n-th Weyl algebra A_n(R)in case k is a field of characteristic zero and R is a commutative affine k-algebra of finite injective dimens...The concern of this paper is to derive formulas for the injective dimension of the n-th Weyl algebra A_n(R)in case k is a field of characteristic zero and R is a commutative affine k-algebra of finite injective dimension.For the case n=1 we prove a more general result from which the above result follows.Such formulas can be viewed as generalizations of the corresponding results given by J.C.McConnell in the case R has finite global dimension.展开更多
Let L be an n-dimensional nilpotent Lie algebra with a basis {x1,…,xn}, and every xiacts as a locally nilpotent derivation on algebra A. This paper shows that there exists a setof derivations {y1,…,yn} on U(L) such ...Let L be an n-dimensional nilpotent Lie algebra with a basis {x1,…,xn}, and every xiacts as a locally nilpotent derivation on algebra A. This paper shows that there exists a setof derivations {y1,…,yn} on U(L) such that (A#U(L))#k[yi,…,yn] is isomorphic to theWeyl algebra An(A). The author also uses the derivations to obtain a necessary and sufficientcondition for a finite dimensional Lie algebra to be nilpotent.展开更多
We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras.We first establish a Van den Bergh duality at the level of complex.Then...We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras.We first establish a Van den Bergh duality at the level of complex.Then based on the results of Solotar et al.,we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras,and translate the homological information into cohomological one by virtue of the Van den Bergh duality,obtaining the desired Batalin–Vilkovisky algebra structures.Finally,we apply our results to quantum weighted projective lines and Podleśquantum spheres,and the Batalin–Vilkovisky algebra structures for them are described completely.展开更多
We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient condit...We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.展开更多
A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe pol...A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A suchthat A is D-simple. The derivations of these associative and Lie algebras are precisely determined.展开更多
Using a method stemming from the representation theory of finite-dimensional algebra’s,results about a category of Artinian modules over the first Weyl algebra consisting of objects having composition series with com...Using a method stemming from the representation theory of finite-dimensional algebra’s,results about a category of Artinian modules over the first Weyl algebra consisting of objects having composition series with composition factors isomorphic to prescribed ones have been obtained.This category corresponds to a tube of Z/2Z type and it is serial and standard.The Auslander-Reiten formula in this category describing the functor Ext,,(-,-) in terms of the AR-quiver has been derived.The dimensions of Ext4(M,N) for M and N in different tubes are calculated.展开更多
基金Supported by the National Natural Science Foundation of China (10771182)
文摘Let An be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over An. By further exploring the relationship between the Poincare series and the dimension and the multiplicity of M, we are able to prove that the tensor product of two finitely generated modules over An has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of An.
文摘We study a family of ＂symmetric＂ multiparameter quantized Weyl alge- bras A-q,A n（K） and some related algebras. We compute the Nakayama automorphism of A-q,A n（K）, give a necessary and sufficient condition for A-q,A n（K） to be Calabi-Yau, and prove that A-q,A n（K） is cancellative. We study the automorphisms and isomorphism problem for A-q,A n（K） and .A-q,A n（K[t]）. Similar results are established for the Maltsiniotis multiparam- eter quantized Weyl algebraA-q,A n（K） and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization （A-q,A n（K））z and determine its automorphism group.
基金Supported by National Natural Science Foundation of China(Grant No.11271131)
文摘In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋1）. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.
基金Project supported in part by the National Natural Science Foundation for Youth
文摘The concern of this paper is to derive formulas for the injective dimension of the n-th Weyl algebra A_n(R)in case k is a field of characteristic zero and R is a commutative affine k-algebra of finite injective dimension.For the case n=1 we prove a more general result from which the above result follows.Such formulas can be viewed as generalizations of the corresponding results given by J.C.McConnell in the case R has finite global dimension.
基金Project supported by the National Natural Science Foundation of Chin
文摘Let L be an n-dimensional nilpotent Lie algebra with a basis {x1,…,xn}, and every xiacts as a locally nilpotent derivation on algebra A. This paper shows that there exists a setof derivations {y1,…,yn} on U(L) such that (A#U(L))#k[yi,…,yn] is isomorphic to theWeyl algebra An(A). The author also uses the derivations to obtain a necessary and sufficientcondition for a finite dimensional Lie algebra to be nilpotent.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11971418).
文摘We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras.We first establish a Van den Bergh duality at the level of complex.Then based on the results of Solotar et al.,we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras,and translate the homological information into cohomological one by virtue of the Van den Bergh duality,obtaining the desired Batalin–Vilkovisky algebra structures.Finally,we apply our results to quantum weighted projective lines and Podleśquantum spheres,and the Batalin–Vilkovisky algebra structures for them are described completely.
基金supported by Huizhou University(Grant Nos.hzu202001 and 2021JB022)the Guangdong Provincial Department of Education(Grant Nos.2020KTSCX145 and 2021ZDJS080)。
文摘We study the growth and the Gelfand-Kirillov dimension(GK-dimension)of the generalized Weyl algebra(GWA)A=D(σ,a),where D is a polynomial algebra or a Laurent polynomial algebra.Several necessary and sufficient conditions for GKdim(A)=GKdim(D)+1 are given.In particular,we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates,i.e.,GKdim(A)is either 3 or∞in this case.Our results generalize several existing results in the literature and can be applied to determine the growth,GK-dimension,simplicity and cancellation properties of some GWAs.
基金This work was supported by the Natronal Natural Science Foundation of China(Grant No.10171064)and an EYTP grant of MOE of China.
文摘A class of the associative and Lie algebras A[D] = A × F[D] of Weyl type are studied, where Ais a commutative associative algebra with an identity element over a field F of characteristic zero, and F[D] isthe polynomial algebra of a finite dimensional commutative subalgebra of locally finite derivations of A suchthat A is D-simple. The derivations of these associative and Lie algebras are precisely determined.
文摘Using a method stemming from the representation theory of finite-dimensional algebra’s,results about a category of Artinian modules over the first Weyl algebra consisting of objects having composition series with composition factors isomorphic to prescribed ones have been obtained.This category corresponds to a tube of Z/2Z type and it is serial and standard.The Auslander-Reiten formula in this category describing the functor Ext,,(-,-) in terms of the AR-quiver has been derived.The dimensions of Ext4(M,N) for M and N in different tubes are calculated.
