Unconditional semiprimality and semiprimitivity transfer for partial smash products of semisimple Hopf algebra actions

For a semisimple Hopf algebra $H$ acting partially on an algebra $A$, Cavalheiro and Sant'Ana (Algebra Colloquium, 2018) proved that the partial smash product $\underline{A\#H}$ is semiprimitive when $A$ is $H$-semiprimitive under each of three finiteness hypotheses (all irreducible $A$-modules finite-dimensional; $A$ an affine PI-algebra over a perfect field; $A$ locally finite), and that $\underline{A\#H}$ is semiprime when $A$ is an $H$-semiprime PI-algebra. The unconditional questions remain open: without any of these extra hypotheses, is $\underline{A\#H}$ semiprimitive whenever $A$ is $H$-semiprimitive, and is $\underline{A\#H}$ semiprime whenever $A$ is $H$-semiprime? These are the partial-action analogues of the Cohen–Fishman semiprime smash product question (1986) and the Linchenko–Montgomery–Small semiprimitivity question (2005), both of which remain open for global actions; since every global action is a partial action, the partial questions contain the global ones.

Background

A Hopf algebra $H$ over a field $\mathbb{k}$ is simultaneously an algebra and a coalgebra whose comultiplication $\Delta$ and antipode $S$ make it compatible with the multiplication; the basic examples are group algebras $\mathbb{k}G$ and universal enveloping algebras of Lie algebras. $H$ is called semisimple when it is semisimple as an algebra (for instance $\mathbb{k}G$ with $\operatorname{char}\mathbb{k}\nmid|G|$). An algebra $A$ is a left $H$-module algebra if $H$ acts linearly on $A$ so that the multiplication and unit of $A$ are $H$-equivariant. From such a (global) action one builds the smash product $A\#H$: the vector space $A\otimes H$ with a crossed multiplication mixing the action of $H$ on $A$ with the product of $H$; for $H=\mathbb{k}G$ this recovers the skew group ring. A ring is semiprime when its prime radical (the intersection of all prime ideals) is zero, equivalently when it contains no nonzero nilpotent ideal, and semiprimitive when its Jacobson radical (the intersection of all left primitive ideals) is zero.

In 1986 Cohen and Fishman asked: if $H$ is semisimple and $A$ is a semiprime $H$-module algebra, is $A\#H$ semiprime? This question is still open in general. Positive cases include: $H=\mathbb{k}G$ with $|G|^{-1}\in\mathbb{k}$ (Fisher–Montgomery) and $H=(\mathbb{k}G)^*$ (Cohen–Montgomery); $A$ satisfying a polynomial identity, i.e. a PI-algebra, an algebra in which some noncommutative polynomial vanishes identically (Linchenko–Montgomery); and $A$ having an Artinian ring of quotients or $H$ a twist of a semisolvable Hopf algebra (Lomp). Studying the same problem through radicals, Linchenko, Montgomery and Small introduced $H$-stable versions: an ideal $I$ of $A$ is $H$-stable when $H\cdot I\subseteq I$; an $H$-stable ideal $P$ is $H$-prime when $IJ\subseteq P$ with $I,J$ $H$-stable implies $I\subseteq P$ or $J\subseteq P$; the $H$-prime radical $P_H(A)$ is the intersection of all $H$-prime ideals, and the $H$-Jacobson radical $J_H(A)=(J(A):H)=\{a\in A:H\cdot a\subseteq J(A)\}$. $A$ is $H$-semiprime when $P_H(A)=0$ and $H$-semiprimitive when $J_H(A)=0$; both weaken the plain notions in the same direction: for finite-dimensional $H$, $(I:H)$ is the largest $H$-stable ideal of $A$ contained in $I$, so $P_H(A)=(P(A):H)\subseteq P(A)$ and $J_H(A)\subseteq J(A)$; hence semiprimeness implies $H$-semiprimeness just as semiprimitivity implies $H$-semiprimitivity, $H$-semiprimeness only demanding that $P(A)$ contain no nonzero $H$-stable ideal and $H$-semiprimitivity only that $J(A)$ contain no nonzero $H$-stable ideal. They proved that $A\#H$ is semiprimitive when $A$ is $H$-semiprimitive under finiteness hypotheses (all irreducible $A$-modules finite-dimensional, $A$ a PI-algebra that is affine or algebraic over a field of characteristic zero, or $A$ locally finite, meaning every finitely generated subalgebra is finite-dimensional), and semiprime when $A$ is an $H$-semiprime PI-algebra.

A partial action of a Hopf algebra $H$ on $A$ (Caenepeel–Janssen, building on Exel's partial group actions) is a $\mathbb{k}$-linear map $H\otimes A\to A$, $h\otimes a\mapsto h\cdot a$, satisfying (PA1) $1_H\cdot a=a$ and (PA2) $h\cdot(a(g\cdot b))=(h_1\cdot a)((h_2g)\cdot b)$; for unital $A$ these are equivalent to (PA3) $h\cdot(ab)=\sum(h_1\cdot a)(h_2\cdot b)$ and (PA4) $h\cdot(g\cdot b)=(h_1\cdot 1_A)((h_2g)\cdot b)$. A global action is the special case in which additionally $h\cdot 1_A=\varepsilon(h)1_A$, equivalently $h\cdot(g\cdot a)=(hg)\cdot a$. From a partial action one forms the algebra $A\#H$ on the vector space $A\otimes H$ with crossed multiplication $(a\#h)(b\#g)=\sum a(h_1\cdot b)\#h_2g$; unless the action is global it has no identity element in general, so the partial smash product is the corner $\underline{A\#H}:=(A\#H)(1_A\#1_H)$, an algebra with unit $1_A\#1_H$ that coincides with $A\#H$ when the action is global. Every partial action admits a globalization (an enveloping global action, by Alves–Batista), though the globalization does not directly reduce radical questions about $\underline{A\#H}$ to the global theorems. Cavalheiro and Sant'Ana extended the $H$-radical machinery to partial actions via $H$-stable ideals and partial $(A,H)$-modules: $P_H(A)$ is the intersection of all $H$-prime ideals of $A$, $J_H(A)$ is the intersection of all right (or left) $H$-primitive ideals, i.e. annihilators of irreducible partial $(A,H)$-modules, with $J_H(A)=J(\underline{A\#H})\cap A$, and for finite-dimensional $H$, $P_H(A)=(P(A):H)$ and $J_H(A)=(J(A):H)$. They transferred the Linchenko–Montgomery–Small theorems to the partial setting: $\underline{A\#H}$ is semiprimitive if $A$ is $H$-semiprimitive and either all irreducible $A$-modules are finite-dimensional, or $A$ is an affine PI-algebra over a perfect field, or $A$ is locally finite; and $\underline{A\#H}$ is semiprime if $A$ is an $H$-semiprime PI-algebra, in which case moreover $P(\underline{A\#H})=P_H(A)\#H$. They also observed that semisimplicity of $H$ is necessary. The unconditional partial-action questions — the exact analogues of what remains open in the global theory — were left open by their paper and have not been addressed by later work.

Problem Statement

Let $\mathbb{k}$ be a field, $H$ a semisimple Hopf algebra over $\mathbb{k}$, and $A$ a unital left partial $H$-module algebra, with partial smash product $\underline{A\#H}$. Answer, in full generality and without the extra hypotheses used by Cavalheiro–Sant'Ana, the two linked transfer questions their results leave open:

1. (Semiprimitivity) If $A$ is $H$-semiprimitive, i.e. its partial $H$-Jacobson radical $J_H(A)$ (for finite-dimensional $H$, $J_H(A)=(J(A):H)=J(\underline{A\#H})\cap A$) is zero, and no assumption is made that all irreducible $A$-modules are finite-dimensional, nor that $A$ is an affine PI-algebra over a perfect field, nor that $A$ is locally finite — is $\underline{A\#H}$ necessarily semiprimitive (i.e. is $J(\underline{A\#H})=0$)?

2. (Semiprimality) If $A$ is $H$-semiprime, i.e. its $H$-prime radical $P_H(A)$ is zero, without assuming that $A$ satisfies a polynomial identity — is $\underline{A\#H}$ necessarily semiprime (i.e. is $P(\underline{A\#H})=0$)? The direct Cohen–Fishman analogue asks whether $\underline{A\#H}$ is semiprime when $A$ is only assumed semiprime; since $P_H(A)\subseteq P(A)$, every semiprime $A$ is $H$-semiprime, so Question 2 concerns a strictly larger class of algebras and would imply the Cohen–Fishman analogue.

Note that semisimplicity of $H$ cannot be dropped.

References

1. Rafael Cavalheiro, Alveri Sant'Ana, "On the Semiprimitivity and Semiprimality Problems for Partial Smash Products", Algebra Colloquium 25 (2018), no. 1, 1–30. DOI: 10.1142/S1005386718000020; https://doi.org/10.1142/S1005386718000020 (preprint: arXiv:1510.05013, https://arxiv.org/abs/1510.05013)
2. M. Cohen, D. Fishman, "Hopf algebra actions", Journal of Algebra 100 (1986), 363–379. DOI: 10.1016/0021-8693(86)90082-7; https://doi.org/10.1016/0021-8693(86)90082-7
3. V. Linchenko, S. Montgomery, L. W. Small, "Stable Jacobson radicals and semiprime smash products", Bulletin of the London Mathematical Society 37 (2005), 860–872. DOI: 10.1112/s0024609305004662; https://doi.org/10.1112/s0024609305004662
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