# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hopf algebra actions. (English) Zbl 0591.16005

Let H be a finite dimensional Hopf algebra over the field k and let A be an H-module algebra. Then the smash product A#H is a ring extension of A and the ring ${A}^{H}$ of H-invariants defined by ${A}^{H}=\left\{a\in A|$ $h·a=ϵ\left(h\right)a\right\}$ is a subring of A. This paper studies the connection between these rings using the key fact that if H is semisimple, then it has a left integral, namely an element x with $hx=ϵ\left(h\right)x$ for all $h\in H$, satisfying $ϵ$ (x)$\ne 0·$

Section 1 obtains a Maschke-type theorem for smash products. Indeed it is shown that if H is semisimple and if $W\subseteq V$ are A#H-modules, then W is an A-direct summand of V if and only if it is an A#H-direct summand.

Section 2 introduces a Morita context $\left[B{,}_{B}{A}_{A#H}{,}_{A#H}{A}_{B},A#H\right]$ where $B={A}^{H}$ and with [, ] nondegenerate. Then criteria are given for (, ) to be nondegenerate and the context is used to relate ${A}^{H}$ to A#H. The paper also proposes an interesting question. Namely if H is a finite dimensional semisimple Hopf algebra and if A is semiprime, must A#H also be semiprime? Some partial results are given.

Reviewer: D.S.Passman

##### MSC:
 16W30 Hopf algebras (assoc. rings and algebras) (MSC2000) 16D70 Structure and classification of associative ring and algebras