Semiprimitivity and semiprimality of partial smash products: the general partial-action analogue of the Cohen–Fishman–Linchenko–Montgomery question

Let $H$ be a finite-dimensional semisimple Hopf algebra over a field $\mathbb{k}$ acting partially on a unital algebra $A$, and let $\underline{A\#H}$ be the partial smash product. Cavalheiro and Sant'Ana (2015, published 2018) proved that $\underline{A\#H}$ is semiprimitive when $A$ is $H$-semiprimitive and all irreducible $A$-modules are finite-dimensional, or $A$ is an affine PI-algebra over a perfect field, or $A$ is locally finite; and that $\underline{A\#H}$ is semiprime when $A$ is an $H$-semiprime PI-algebra. The problem is whether these implications hold without any of those auxiliary hypotheses — the partial-action analogue of the classical Cohen–Fishman and Linchenko–Montgomery smash-product question, which itself has been open since 1986 — and, if not, what necessary and sufficient conditions on $A$ and the partial action characterize semiprimitivity and semiprimality of $\underline{A\#H}$. No resolution in either direction is known to date.

Background

A Hopf algebra $H$ over a field $\mathbb{k}$ is simultaneously an algebra and a coalgebra (with comultiplication $\Delta$ and counit $\varepsilon$) whose structures are compatible; the group algebra $\mathbb{k}G$ and its linear dual are the basic examples. A (left) $H$-module algebra is an algebra $A$ on which $H$ acts by algebra endomorphisms in the sense that $h\cdot(ab)=\sum(h_1\cdot a)(h_2\cdot b)$ and $h\cdot 1_A=\varepsilon(h)1_A$, where $\Delta(h)=\sum h_1\otimes h_2$ is written in Sweedler's notation (the summation being implicit); to such an action one associates the smash product $A\#H$, the vector space $A\otimes H$ with multiplication twisted by the action, which contains $A$ as a subring and is a basic object of Hopf-action ring theory. A ring is semiprime if its prime radical (the intersection of its prime ideals) is zero, and semiprimitive if its Jacobson radical (the intersection of its maximal left ideals) is zero. A PI-algebra is one satisfying a polynomial identity, such as commutativity or the matrix identities; an algebra is affine if finitely generated as algebra, and locally finite if every finite subset generates a finite-dimensional subalgebra. A finite-dimensional Hopf algebra is called semisimple when it is semisimple as an algebra.

A long-standing question, raised by Cohen and Fishman in 1986, asks: if a semisimple Hopf algebra $H$ acts on a semisimple, or more generally semiprime, algebra $A$, must $A\#H$ be semiprime? The question is open in general; positive answers are known under extra hypotheses on $H$ (e.g., group algebras with $|G|^{-1}\in\mathbb{k}$, dually $H=(\mathbb{k}G)^*$) or on $A$ (Lomp 2004; Linchenko–Montgomery–Small 2005 via stability of the Jacobson radical in characteristic 0 for PI-algebras that are affine or algebraic, for algebras whose irreducible representations are all finite-dimensional, and for locally finite algebras; Linchenko–Montgomery 2007 for the semiprimality of PI-algebras). This suggested the refinement known as Question 2 in Linchenko–Montgomery–Small's framework: for a semisimple Hopf algebra $H$ and an $H$-semiprimitive (resp. $H$-semiprime) $H$-module algebra $A$, is $A\#H$ semiprimitive (resp. semiprime)? Here $A$ is $H$-semiprimitive ($H$-semiprime) when its $H$-Jacobson radical $J_H(A)$ (resp. $H$-prime radical $P_H(A)$) vanishes — these are the Hopf-action analogues of the ordinary radicals built from $H$-stable ideals, i.e., ideals of $A$ stable under the action: $P_H(A)$ is the intersection of the $H$-prime $H$-stable ideals of $A$, and $J_H(A)$ is the intersection of the $H$-primitive ideals, i.e., the annihilators of the irreducible equivariant modules (in the partial-action setting used below, the irreducible partial $(A,H)$-modules).

Partial actions of Hopf algebras, introduced by Caenepeel and Janssen (2008), model actions defined only on parts of the algebra: a (left) partial action of $H$ on a unital algebra $A$ is a linear map $\xi:H\otimes A\to A$, $h\otimes a\mapsto h\cdot a$, with $1_H\cdot a=a$ and $h\cdot(a(g\cdot b))=\sum(h_1\cdot a)((h_2 g)\cdot b)$; unlike a global action, $h\cdot 1_A$ need not equal $\varepsilon(h)1_A$, and every global action is partial. On the vector space $A\otimes H$ one considers the multiplication $(a\#h)(b\#g)=\sum a(h_1\cdot b)\#h_2 g$, which makes it an associative algebra denoted $A\#H$; in general $A\#H$ has no identity element unless the action of $H$ on $A$ is global. The partial smash product is the subalgebra $\underline{A\#H}:=(A\#H)(1_A\#1_H)$, generated by the elements $\underline{a\#h}:=(a\#h)(1_A\#1_H)=\sum a(h_1\cdot 1_A)\#h_2$; it is an associative algebra with identity element $1_A\#1_H$, it contains $A$ as a subalgebra via $a\mapsto a\#1_H$, and it coincides with $A\#H$ when the action is global. Cavalheiro and Sant'Ana (2015; published in Algebra Colloquium 2018) adapted the notions of $H$-stable ideal and of a partial $(A,H)$-module (an abelian group on which $A$ and the partial action cooperate, the partial analogue of the modules over a smash product) and defined the partial $H$-Jacobson radical $J_H(A)$ and $H$-prime radical $P_H(A)$; $A$ is $H$-semiprimitive when $J_H(A)=0$ and $H$-semiprime when $P_H(A)=0$. Their key duality result states that $\underline{A\#H}$ carries a genuine global action of the dual Hopf algebra $H^*$ and that $A$ is $H$-semiprimitive (resp. $H$-semiprime) if and only if $\underline{A\#H}$ is $H^*$-semiprimitive (resp. $H^*$-semiprime), so the partial problem embeds into the classical global Question 2. Using this, they generalized the Linchenko–Montgomery–Small and Linchenko–Montgomery theorems to partial actions: $\underline{A\#H}$ is semiprimitive when $A$ is $H$-semiprimitive and either all irreducible $A$-modules are finite-dimensional, or $A$ is an affine PI-algebra over a perfect field (no extra hypothesis in positive characteristic beyond perfectness), or $A$ is locally finite; and $\underline{A\#H}$ is semiprime when $A$ is an $H$-semiprime PI-algebra. Their final remarks show the $H$-semiprimitive/$H$-semiprime hypotheses are essential and that semisimplicity of $H$ is necessary, since for a trivial action $A\#H=A\otimes H$ is not semiprime when $H$ is not semisimple. The above sufficient conditions all impose PI, finiteness, or local-finiteness restrictions on $A$, and no partial-action result without such restrictions is known.

Problem Statement

Let $\mathbb{k}$ be a field, $H$ a finite-dimensional semisimple Hopf $\mathbb{k}$-algebra, $A$ a unital left partial $H$-module algebra in the sense of Caenepeel–Janssen, and $\underline{A\#H}$ the partial smash product. Consider the partial-action analogue of the classical Linchenko–Montgomery–Small Question 2:

1. (Semiprimitivity) If $A$ is $H$-semiprimitive, i.e., its partial $H$-Jacobson radical $J_H(A)$ (the intersection of the annihilators of all irreducible partial $(A,H)$-modules) is zero, must $\underline{A\#H}$ be semiprimitive, i.e., must $J(\underline{A\#H})=0$ — with no hypothesis on $A$ beyond the partial action (in particular, without assuming that all irreducible $A$-modules are finite-dimensional, that $A$ is an affine PI-algebra over a perfect field, or that $A$ is locally finite)?
2. (Semiprimality) If $A$ is $H$-semiprime, i.e., $P_H(A)=0$, must $\underline{A\#H}$ be semiprime, without assuming that $A$ satisfies a polynomial identity?
3. (Characterization) If either implication fails, determine necessary and sufficient conditions on $A$ and on the partial action under which $\underline{A\#H}$ is semiprimitive (resp. semiprime), including deciding whether any of the known sufficient hypotheses can be weakened and exhibiting the exact obstruction in any counterexample.

Items 1–3 constitute the open core.

References

1. Rafael Cavalheiro and Alveri Sant'Ana, "On the semiprimitivity and the semiprimality problems for partial smash products", Algebra Colloquium 25 (2018), no. 1, 1–30, DOI: 10.1142/S1005386718000020; preprint arXiv:1510.05013 (2015), https://arxiv.org/abs/1510.05013
2. Miriam Cohen and Davida 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. Christian Lomp, "When is a smash product semiprime? A partial answer", Journal of Algebra 275 (2004), 339–355, DOI: 10.1016/j.jalgebra.2003.07.017, https://doi.org/10.1016/j.jalgebra.2003.07.017
4. V. Linchenko, S. Montgomery and 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
5. V. Linchenko and S. Montgomery, "Semiprime smash products and H-stable prime radicals for PI-algebras", Proceedings of the American Mathematical Society 135 (2007), 3091–3098, DOI: 10.1090/s0002-9939-07-08849-1, https://doi.org/10.1090/s0002-9939-07-08849-1
6. S. Caenepeel and K. Janssen, "Partial (co)actions of Hopf algebras and partial Hopf-Galois theory", Communications in Algebra 36 (2008), 2923–2946, DOI: 10.1080/00927870802110334, https://doi.org/10.1080/00927870802110334
7. Christian Lomp, "On the semiprime smash product question", Contemporary Mathematics 634 (2015), 205–222, DOI: 10.1090/conm/634/12700; preprint arXiv:1401.1049, https://arxiv.org/abs/1401.1049
8. M. Dokuchaev, "Recent developments around partial actions", São Paulo Journal of Mathematical Sciences 13 (2018), no. 1, 195–247, DOI: 10.1007/s40863-018-0087-y; preprint arXiv:1801.09105, https://arxiv.org/abs/1801.09105
9. Rafael Cavalheiro and Alveri Sant'Ana, "On partial H-radicals of Jacobson type", São Paulo Journal of Mathematical Sciences 10 (2016), 140–163, DOI: 10.1007/s40863-015-0036-y, https://doi.org/10.1007/s40863-015-0036-y