半单弱Hopf代数的作用

Actions of Semisimple Weak Hopf Algebras

中文关键词:  弱Hopf代数  Gorenstein代数  内射维数.

英文关键词:Weak Hopf algebras  Gorenstein algebras  Injective dimension.

中文摘要:

      令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.

英文摘要:

      Let $H$ be a semisimple weak Hopf algebra, $A$ a left $H$-module algebra. We prove that the injective dimension of the regular $A$-module is equal to the one of $A$ as the module over the smash product $A\#H$, and equal to the one of the regular $A\#H$-module. Also, we give a necessary and sufficient condition for $A$ being a Gorenstein algebra, in terms of the fixed subalgebra of $A$ under the action of $H$ on $A$.