模代数扩张

EXTENSION OF MODULE ALGEBRAS

设H是Hopf代数,A是左H-模代数.设_AM_A是H-模范畴中的A-A-双模.本文讨论了模代数A的通过双模M的奇异扩张,模代数的扩张既是代数扩张又是模扩张.为此,我们构作了一个融合代数结构和H-模结构的复形C_H~*(A,M),并且证明模代数的奇异扩张的等价类之集与这个复形的2阶上同调群H_H^2(A,M)是一一对应的.

Let H be a Hopf algebra,and A a left H-module algebra.Let _aM_a be an A-A-bimodule in the category of H-modules.In this paper,we discuss the singular extensions of the module algebra A by the bimodule M.An extension of a module algebra is a combination of an algebra extension and a module extension.So,we construct a complex C_H~*(A,M)which depends on the algebra structure of A and the H-module structures and prove that the equivalence class of the singular extensions of the module algebra A by the bimodule M is one-to-one corresponding to the second cohomology group H_H^2(A,M)of the complex C_H~*(A,M).