As is well known,the classical nonsingular Whittaker modules over quantum groups cannot be defined for the non-A1 type.In this paper,by choosing different generators for the quantum group U_(q)(gl_(n+1)),we introduce ...As is well known,the classical nonsingular Whittaker modules over quantum groups cannot be defined for the non-A1 type.In this paper,by choosing different generators for the quantum group U_(q)(gl_(n+1)),we introduce and study the twisted Whittaker modules over U_(q)(gl_(n+1)).We classify all the simple twisted Whittaker modules with nonsingular Whittaker functions.This agrees with Kostant’s results on Whittaker modules for the simple complex Lie algebras sln+1 as q approaches 1.展开更多
It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should ...It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.展开更多
Let g be the finite dimensional simple Lie algebra of type A_n, and let U = U_q(g,Λ)and U= U_q(g,Q)be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find...Let g be the finite dimensional simple Lie algebra of type A_n, and let U = U_q(g,Λ)and U= U_q(g,Q)be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U = U_q(g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U = U_q(g,Λ) and U = U_q(g,Q).展开更多
Based on the n-fold tensor product version of the generalized double-bosonization construction,we prove the Majid conjecture of the quantum Kac-Moody algebras version.Particularly,we give explicitly the double-bosoniz...Based on the n-fold tensor product version of the generalized double-bosonization construction,we prove the Majid conjecture of the quantum Kac-Moody algebras version.Particularly,we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types G(1)2,E(1)6,and Tp,q,r,and in this way,we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category.This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.展开更多
We define the Whittaker modules over the simply-connected quantum group U_(q)(sl3,∧),where A is the weight lattice of Lie algebra sl3.Then we completely classify all those simple ones.Explicitly,a simple Whittaker mo...We define the Whittaker modules over the simply-connected quantum group U_(q)(sl3,∧),where A is the weight lattice of Lie algebra sl3.Then we completely classify all those simple ones.Explicitly,a simple Whittaker module over U_(q)(sl3,∧)is either a highest weight module,or determined by two parameters z∈C andγ∈C^(*)(up to a Hopf automorphism).展开更多
Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebraη.Then C[η]has a g-module structure if and only if g is of type A or of type C;this is called the polynomial module of rank one,In the q...Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebraη.Then C[η]has a g-module structure if and only if g is of type A or of type C;this is called the polynomial module of rank one,In the quantum version,the rank one polynomial modules over U_(q)(sl_(2))have been classified.In this paper,we prove that the quantum group U_(q)(sl_(3))has no rank one polynomial module.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11871249,11871190 and 12171155)Natural Sciences and Engineering Research Council of Canada(Grant No.311907-2015).
文摘As is well known,the classical nonsingular Whittaker modules over quantum groups cannot be defined for the non-A1 type.In this paper,by choosing different generators for the quantum group U_(q)(gl_(n+1)),we introduce and study the twisted Whittaker modules over U_(q)(gl_(n+1)).We classify all the simple twisted Whittaker modules with nonsingular Whittaker functions.This agrees with Kostant’s results on Whittaker modules for the simple complex Lie algebras sln+1 as q approaches 1.
基金Project supported by the National Natural Science Foundation of China (No. 11001110)
文摘It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.
基金supported by National Natural Science Foundation of China(Grant No.11471282)
文摘Let g be the finite dimensional simple Lie algebra of type A_n, and let U = U_q(g,Λ)and U= U_q(g,Q)be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U = U_q(g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U = U_q(g,Λ) and U = U_q(g,Q).
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11801394,11771142,11871249)and the Science and Technology Commission of Shanghai Municipality(No.18dz2271000).
文摘Based on the n-fold tensor product version of the generalized double-bosonization construction,we prove the Majid conjecture of the quantum Kac-Moody algebras version.Particularly,we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types G(1)2,E(1)6,and Tp,q,r,and in this way,we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category.This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.
基金the National Natural Science Foundation of China(Grant Nos.11971440,11871249,11771142,11931009)the Jiangsu Natural Science Foundation(No.BK20171294).
文摘We define the Whittaker modules over the simply-connected quantum group U_(q)(sl3,∧),where A is the weight lattice of Lie algebra sl3.Then we completely classify all those simple ones.Explicitly,a simple Whittaker module over U_(q)(sl3,∧)is either a highest weight module,or determined by two parameters z∈C andγ∈C^(*)(up to a Hopf automorphism).
基金support from the NNSF(Nos.11971440,11871249,11771142,11931009,11871326).
文摘Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebraη.Then C[η]has a g-module structure if and only if g is of type A or of type C;this is called the polynomial module of rank one,In the quantum version,the rank one polynomial modules over U_(q)(sl_(2))have been classified.In this paper,we prove that the quantum group U_(q)(sl_(3))has no rank one polynomial module.
基金This work was supported in part the National Natural Science Foundation of China(Grant Nos.11871249,11771142)the Jiangsu Natural Science Foundation(No.BK20171294).
文摘For all generic q∈C^∗,when g is not of type A1;we prove that the quantum toroidal algebra Uq(gtor)has no nontrivial finite dimensional simple module.