Given a $k$-bialgebra $H$, and a left Hopf action $\triangleright$ of $H$ on a $k$-algebra $A$, one defines the crossed product algebra $A\sharp H$ (in Hopf algebra literature also called the smash product algebra or Hopf smash product; distinguish from the rather different smash product in topology) as the $k$-algebra whose underlying vector space is $A\otimes H$ and the product is given by
The idea is that if the bialgebra $H$ is in fact a Hopf algebra embedded as $1\otimes H\subset A\sharp H$ – whatever the product in the latter is (but assumed to satisfy $(a\otimes 1)(1\otimes h) = a\otimes h$) – and if the action is inner within $A\sharp H$, i.e. $h\triangleright a = \sum h_{(1)} a S(h_{(2)})$, then we have
and hence the formula for the product above is a tautology: $a h a' h' = a(h_{(1)}\triangleright a') h_{(2)} h'$.
Similarly, given a right Hopf action of $H$ on $A$, one defines the crossed product algebra $H\sharp A$ whose underlying space is $H\otimes A$. The left and right versions are isomorphic if $H$ has an invertible antipode; this extends the correspondence between the left and right actions obtained by composing with the antipode map.
Every smash product algebra of the form $A\sharp H$ is naturally equipped with a monomorphism $A\mapsto A\sharp 1\hookrightarrow A\sharp H$ of algebras and with a right $H$-coaction $a\otimes h\mapsto a\otimes \Delta(h)\in (A\sharp H)\otimes H$ making $A\sharp H$ into a right $H$-comodule algebra. Map $\gamma: h\mapsto 1\otimes h$, $H\hookrightarrow A\sharp H$ is then a map of right $H$-comodule algebra (where the coaction on $H$ is $\Delta$), and $A\otimes 1\subset A\sharp H$ is the subalgebra of $H$-coinvariants.
If $H$ is a Hopf algebra, then the homomorphism $\gamma$ is a convolution invertible linear map with convolution inverse $\gamma^{-1}$ defined by $\gamma^{-1}(h)=\gamma(Sh)$ for $h\in H$, where $S$ is the antipode of $H$. Conversely,
Proposition Let $H$ be a Hopf algebra, $E$ a right $H$-comodule algebra, and $\gamma:H\to E$ a map of right $H$-comodule algebra. Clearly $H$ acts on $E^{co H}$ by $h\triangleright a = \sum \gamma(h_{(1)}) a\gamma(Sh_{(2)})$ for $a\in E^{co H}$ and $h\in H$, where the product on the right-hand side is in $E$. Conclusion: $E\cong E^{co H}\sharp H$ where the smash product is with respect to that action.
There is also a more general cocycled crossed product. For a bialgebra $H$ and an algebra $U$, if we consider the category $C(U,H)$ of extensions $U\hookrightarrow E$ which are compatibly left $U$-modules and right $H$-comodules, and where $U=E^{\mathrm{co}H}$, then the crossed product algebras are the canonical representatives of cleft Hopf-Galois extensions which are a more invariant concept.
Let $U$ be an algebra, $H$ a Hopf algebra, $\triangleright : H\otimes U\to U$ a measuring, i.e. a $k$-linear map satisfying $h\triangleright(u v)=\sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright v)$ for all $h\in H$, $u,v\in U$, and which we assume unital, i.e. $h\triangleright 1 = \epsilon(h)1$ for all $h\in H$. We do not assume that $\triangleright$ is an action.
Let further a (convolution) invertible $k$-linear map $\sigma \in Hom_k(H\otimes H,U)$ be given.
We say that $\sigma$ is a 2-cocycle (relative to the measuring $\triangleright$) if the following two cocycle identities hold
These identities clearly generalize the classical factor system?s in group theory (linearly extended to the case of group algebras, for the finite groups at least). Therefore it is an example of a nonabelian cocycle in Hopf algebra theory. However, its role in the general theory is less well understood than the group case.
Define the cocycled crossed product on $U\otimes H$ by
for all $h,k\in H$, $u,v\in U$. The cocycled crossed product is an associative algebra iff $\sigma$ is a cocycle.
If so, we call $U\sharp_\sigma H$ cocycled crossed product algebra. Map $1\otimes\Delta_H:U\sharp_\sigma H\to (U\sharp_\sigma H)\otimes H$ is a right $H$-coaction, making $U\sharp_\sigma H$ into a right $H$-comodule algebra, which is cleft extensions]] are always isomorphic (as $H$-extensions) to the cocycled crossed product algebras.
If $\sigma(h,k)=\epsilon(h)\epsilon(k)1_U$ then we say that $\sigma$ is a trivial cocycle and then the compatibility conditions above reduce to demanding that the measuring $\triangleright$ is an action. The cocycled crossed product then reduces to the usual smash product algebra.
Theorem. Suppose we are given two measurings $\triangleright,\triangleright':H\otimes U\to A$ with cocycles $\sigma, \tau$ respectively. Then there exists an isomorphism of $H$-extensions of $U$, $i: U\sharp_\sigma H\cong U\sharp_\tau H$ (i.e. an isomorphism of $k$-algebras, left $U$-modules and right $H$-comodules) iff there is an invertible element $f\in Hom_k(H,U)$ such that for all $u\in U$, $h,k\in H$
The isomorphism $i$ is then given by
Related $n$Lab entres include crossed product C*-algebra, noncommutative torsor, Hopf-Galois extension
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Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. Alg. 17 (1989), 3053–3085, MR91k:16027, doi
S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics 82, AMS 1993.
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