Normal Smash Products

Shilin Yang and Dingguo Wang

Abstract: Let $H$ be a co-Frobenius Hopf algebra over a field $k$ and $A$ a right $H$-comodule algebra. It is shown that $A$ is $H$-faithful and $A_N\cardinal N^*\in\Phi$ iff $A\cardinal H^{*\rat}\in\Phi$, where $N$ is a subgroup of $G(H)=\{g\in H\,| \Delta (g)=g\otimes g\}$ and $A_N$ is $N$-coinvariants, $\Phi$ denotes a normal class. It is also shown that if $A/A_1$ is right $H$-Galois and $A_1$ is central simple, then so is $A\cardinal H^{*\rat}$. In particular, if $A_1$ is a divisible ring, then $A\cardinal H^{*\rat}$ is a dense ring of linear transformations of the vector space $A$ over $A_1$. Let $H$ be a finite dimensional Hopf algebra over the field $k$ and $A$ an $H$-module algebra, $K$ is a unimodular and normal subHopfalgebra and $\overline H=H/K^+H$, it is obtained that $A^K\cardinal \overline H\in\Phi$ and $A$ is $H^*$-faithful iff $A\cardinal H\in \Phi$.

Keywords: Smash product; normal class; central simple.

Classification (MSC2000): 16S40, 16W30

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