We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jaco...We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative Mgebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.展开更多
The concept of weak Hopf group coalgebras is a natural generalization of the notions of both weak Hopf algebras(quantum groupoids) and Hopf group coalgebras.Let π be a group.The Morita context is considered in the ...The concept of weak Hopf group coalgebras is a natural generalization of the notions of both weak Hopf algebras(quantum groupoids) and Hopf group coalgebras.Let π be a group.The Morita context is considered in the sense of weak Hopf π-coalgebras.Let H be a finite type weak Hopf π-coalgebra,and A a weak right π-H-comodule algebra.It is constructed that a Morita context connects A#H* which is a weak smash product and the ring of coinvariants AcoH.This result is the generalization of that of Wang's in the paper "Morita contexts,π-Galois extensions for Hopf π-coalgebras" in 2006.Furthermore,the result is important for constructing weak π-Galois extensions.展开更多
文摘We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F 〉 dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative Mgebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.
基金The Scientific Research Innovation Project for College Graduates in Jiangsu Province(No.CXLX_0094)
文摘The concept of weak Hopf group coalgebras is a natural generalization of the notions of both weak Hopf algebras(quantum groupoids) and Hopf group coalgebras.Let π be a group.The Morita context is considered in the sense of weak Hopf π-coalgebras.Let H be a finite type weak Hopf π-coalgebra,and A a weak right π-H-comodule algebra.It is constructed that a Morita context connects A#H* which is a weak smash product and the ring of coinvariants AcoH.This result is the generalization of that of Wang's in the paper "Morita contexts,π-Galois extensions for Hopf π-coalgebras" in 2006.Furthermore,the result is important for constructing weak π-Galois extensions.