The Cohen–Fishman question: is the smash product of a semisimple Hopf algebra with a semiprime module algebra always semiprime?

A longstanding open problem posed by M. Cohen and D. Fishman in 1986 asks whether the smash product $A\#H$ of a finite-dimensional semisimple Hopf algebra $H$ acting on a semiprime left $H$ -module algebra $A$ over a field of characteristic $0$ is necessarily semiprime — equivalently, whether every semisimple Hopf algebra is strongly semisimple in the sense that semiprimeness of $A$ is always inherited by $A\#H$ . The question is the noncommutative analogue of Maschke's theorem and of the classical semiprimeness theorems for skew group rings, and it remains open in its unrestricted form: all published positive results assume either extra structure on $A$ (polynomial identity, commutativity, Noetherianity, an Artinian classical quotient ring) or that $H$ is a twist of a semisolvable Hopf algebra.

Background

A Hopf algebra $H$ over a field $k$ is an associative algebra that is at the same time a coalgebra, i.e. it carries a comultiplication $\Delta:H\to H\otimes H$ and a counit $\varepsilon:H\to k$ compatible with multiplication, together with an antipode $S$ (the inverse of the identity map for the convolution product on $\mathrm{End}(H)$ ). The basic finite-dimensional examples are group algebras $k[G]$ of finite groups and their duals $k[G]^*$ . A left $H$ -module algebra is a $k$ -algebra $A$ with a linear action of $H$ satisfying $h\cdot(ab)=\sum(h_1\cdot a)(h_2\cdot b)$ and $h\cdot 1_A=\varepsilon(h)1_A$ ; for $H=k[G]$ this is exactly an action of the group $G$ by algebra automorphisms, and for $H=k[G]^*$ it is a $G$ -grading of $A$ . The smash product $A\#H$ is the algebra on the vector space $A\otimes H$ with multiplication $(a\#h)(b\#g)=\sum a(h_1\cdot b)\#h_2g,$ generalizing the skew group ring $A*G$ of a group action. A ring is semiprime if it has no nonzero nilpotent two-sided ideals, equivalently if its prime radical is zero; for commutative rings this is reducedness. A Hopf algebra is called semisimple if it is semisimple as an algebra (hence finite-dimensional); by the Larson–Radford theorem, over a field of characteristic $0$ this is equivalent to the antipode being involutive, $S^2=\mathrm{id}$ , and to the dual Hopf algebra $H^*$ being semisimple.

The question originates in classical results on finite group actions: Fisher–Montgomery (1978) and Lorenz–Passman (1980) proved that the skew group ring of a finite group action is semiprime whenever the underlying ring has no nonzero nilpotent group-stable ideals, and Cohen–Montgomery (1984) proved the analogous statement for group gradings, i.e. for actions of the dual group algebra. Since group algebras and their duals are precisely the semisimple Hopf algebras that are commutative or cocommutative (the trivial ones), Cohen and Fishman asked in 1986 whether semiprimeness is preserved for actions of arbitrary semisimple Hopf algebras, proving the case where $A$ is semisimple Artinian. It is useful to call $H$ strongly semisimple if $A$ semiprime implies $A\#H$ semiprime for every left $H$ -module algebra $A$ ; the question is exactly whether every semisimple Hopf algebra is strongly semisimple. By Blattner–Montgomery duality, $A\#H\#H^*\cong M_{\dim H}(A)$ , so the question is symmetric in $H$ and $H^*$ .

The known partial answers split into two families. Under hypotheses on $A$ : the answer is positive when $A$ is von Neumann regular (every principal left ideal is generated by an idempotent) or semisimple Artinian (the extension $A\subseteq A\#H$ is separable and has a nonzero integral), when $A$ is commutative semiprime (Lomp 2004, via the $H$ -central closure and integrality of the invariants), when $A$ has an Artinian classical ring of quotients — in particular when $A$ is semiprime Noetherian, i.e. satisfies the ascending chain condition on left ideals (Skryabin–van Oystaeyen 2006, who showed the Hopf action extends to the classical quotient ring) — and when $A$ satisfies a polynomial identity, i.e. is a PI-algebra (Linchenko–Montgomery 2007, via stability of the Jacobson radical $\mathrm{Jac}(A\#H)$ under the dual action; a PI-algebra is one satisfying a nonzero polynomial identity identically, like commutative algebras or matrix algebras). Under hypotheses on $H$ : Montgomery–Schneider (1999) proved that every semisimple semisolvable Hopf algebra is strongly semisimple, where semisolvable means admitting a normal series whose subquotients are commutative or cocommutative Hopf algebras; Lomp showed the class of strongly semisimple Hopf algebras is closed under Drinfeld twists (deformations of the comultiplication leaving the algebra structure unchanged), which covers all semisimple triangular Hopf algebras. Etingof–Walton (2014) proved that over an algebraically closed field of characteristic $0$ , every action of a semisimple Hopf algebra on a commutative integral domain factors through a group action, so any counterexample must involve a noncommutative $A$ .

What remains uncovered are pairs where $H$ is semisimple but not a twist of a semisolvable Hopf algebra and where $A$ is semiprime but neither PI, Noetherian, commutative, nor quotient-regular. Weak Hopf algebras (quantum groupoids), where the comultiplication is an algebra map only into a suitable quotient of $H\otimes H$ , generalize Hopf algebras; Borges–Lomp (2011) extended the Linchenko–Montgomery theorem to finite-dimensional involutive weak Hopf algebras acting on semiprime PI-algebras, and recalled the Cohen–Fishman question as the motivating unresolved problem, with the removal of the PI hypothesis, non-involutive antipodes, and non-finite actions listed as open regimes.

Problem Statement

Let $k$ be a field of characteristic $0$ , let $H$ be a finite-dimensional semisimple Hopf $k$ -algebra, and let $A$ be a semiprime left $H$ -module algebra. Is the smash product $A\#H$ necessarily semiprime? Equivalently, is every semisimple Hopf algebra strongly semisimple, i.e. does $A$ semiprime imply $A\#H$ semiprime for every left $H$ -module algebra $A$ ? The question is posed in its original unrestricted generality: no additional hypothesis on $A$ (such as satisfying a polynomial identity, being commutative, Noetherian, von Neumann regular, or having an Artinian classical quotient ring) and no restriction on $H$ beyond being a semisimple Hopf algebra. A complete answer is either a general proof of the implication, or an explicit counterexample consisting of a semisimple Hopf algebra $H$ , a semiprime left $H$ -module algebra $A$ , and a nonzero nilpotent ideal of $A\#H$ .

References

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