Zoran Skoda Hopf smash product bimodule

Preliminaries: Hopf smash product algebras

If $H$ is a $k$-bialgebra (or, in particular, $k$-Hopf algebra) an action $\triangleright: H\otimes U\to U$ of $H$ (as a $k$-algebra) on a $k$-algebra $U$ is a (left) Hopf action if it is also a measuring (of $U$ by an underlying coalgebra of $H$); recall that an algebra measuring? of $U$ by a coalgebra $C$ is a bilinear map $\triangleright: C\otimes U\to U$ such that $h\triangleright 1_U = \epsilon(h) 1_U$ and $h\triangleright (u \cdot u') = \sum (h_{(1)}\triangleright u)\cdot (h_{(2)}\triangleright u')$ for all $h\in C$, $u,u'\in U$. If $H$ acts on $U$ by a Hopf action, one also says that $U$ is an $H$-module algebra (equivalently, a monoid in the monoidal category of $H$-modules).

Bimodule measurings

If $U$ and $V$ are $k$-algebras measured by a $k$-coalgebra $C$ and $M$ is a $U$-$V$-bimodule, then we say that $M$ is measured by $C$, if there is a $k$-bilinear map $\triangleright_M:C\otimes M\to M$ such that for all $u\in U$, $v\in V$, $m\in M$, $c\in C$,

$$c\triangleright (u m v) = \sum (c_{(1)}\triangleright_U u) (c_{(2)} \triangleright_M m)(c_{(3)}\triangleright v)$$

If $C$ is in addition a $k$-bialgebra and $\triangleright_U,\triangleright_V,\triangleright_M$ actions of $H$, we say that a measuring is Hopf action of $C$ on the $U$-$V$-bimodule $M$.

Hopf smash product bimodule

…