Classifying finite-dimensional Hopf algebras whose actions force a nilpotent Jacobson radical on invariant rings

When a finite-dimensional Hopf algebra $H$ acts on an associative algebra $R$, the Jacobson radical of the invariant subring $R^H$ can behave very differently from $J(R)$. Bergen and Grzeszczuk (J. Algebra Appl. 2018) proved that $J(R^H)\subseteq J(R)$, or $J(R^H)^n\subseteq J(R)$ for some $n$, for large classes of semisimple and pointed Hopf algebras with a nondegenerate trace, but left open the structural question: exactly which finite-dimensional Hopf algebras $H$ have the property that $J(R^H)$ is nilpotent whenever $R$ is semiprimitive ($J(R)=0$)? Known positive cases (semisimple $H$ with usable trace, cyclic group actions, restricted enveloping algebras of one-dimensional restricted Lie algebras in characteristic $p$) and known counterexamples (higher-rank abelian group actions in dividing characteristic, i.e. $\operatorname{char} k$ divides the group order; two-dimensional restricted Lie algebras) delimit but do not settle the classification. No later work resolves the classification; the authors' own 2022 follow-up settled only the announced skew-derivation extension, and adjacent 2026 radical-invariance results of Chirvasitu give sufficient tracial conditions in a different finite-dimensional setting. The problem is to produce an intrinsic necessary-and-sufficient characterization of the Hopf algebras forcing nilpotency of invariant radicals.

Background

A Hopf algebra $H$ over a field $k$ is an algebra that is simultaneously a coalgebra (equipped with a comultiplication and a counit $\varepsilon$ compatible with the multiplication) together with an antipode $S$, the standard axiomatic framework generalizing group algebras, enveloping algebras of Lie algebras, and their quantized deformations. $H$ acts on an associative $k$-algebra $R$ as a module algebra if $R$ is a left $H$-module and the action respects multiplication and unit. The invariant subring is $R^H=\{r\in R:\ h\cdot r=\varepsilon(h)r \text{ for all } h\in H\}$, generalizing fixed rings $R^G$ of a group action and rings of constants $R^d=\{r: d(r)=0\}$ of a derivation. The action also determines the smash product $R\#H$, an associative algebra structure on $R\otimes_k H$ generalizing the semidirect product algebra of a group action, which links the invariant theory of $R$ to the ideal theory of $R\#H$. The Jacobson radical $J(R)$ is the intersection of all left primitive ideals of $R$; it contains every nil (in particular every nilpotent) one-sided ideal, and $R$ is called semiprimitive when $J(R)=0$. An ideal $I$ is nilpotent if $I^n=0$ for some $n$.

The classical lineage of the question: for a finite group $G$ with $\operatorname{char} k$ not dividing $|G|$, averaging over $G$ (a Reynolds operator) shows that invariants of a semiprime ring behave well, but in modular characteristic fixed rings can acquire nilpotent radical elements. For derivations, Grzeszczuk (J. Algebra 2014) studied when the Jacobson radical of the constants of a derivation is nilpotent, proving positive results for one-dimensional (algebraic) derivations. For general finite-dimensional Hopf algebras, semisimple $H$ acting on polynomial-identity (PI) algebras — algebras satisfying a polynomial identity, such as matrices over a commutative ring — are known to preserve good invariant-theoretic behavior (e.g. Eryashkin, Russian Mathematics 2016, and earlier work in the Lorenz school).

Bergen and Grzeszczuk (J. Algebra Appl. 17 (2018) 1850132) systematically compared $J(R)$ and $J(R^H)$. Their engine is the trace map $\hat t:R\to R^H$, $r\mapsto t\cdot r$, induced by a nonzero right integral $t$ in the dual Hopf algebra $H^{*}$ (a distinguished linear functional on $H$, unique up to scalars); nondegeneracy of $\hat t$, e.g. the existence of $r$ with $\hat t(r)=1$, is the Hopf-theoretic analogue of dividing by the group order in classical invariant theory. For large classes — semisimple $H$ with nondegenerate trace, and pointed Hopf algebras (those whose coradical, the sum of all simple subcoalgebras, is a group algebra) with suitable trace properties — acting on rings $R$ subject to hypotheses such as semiprimeness or primeness of $R^H$ and PI or finiteness conditions on $R$, they established $J(R^H)\subseteq J(R)$ or $J(R^H)^n\subseteq J(R)$ for some $n$. According to the paper's account, nilpotency of $J(R^H)$ holds for one-dimensional algebraic derivation actions (enveloping algebras $k[d]$) and for cyclic group actions, but fails for higher-rank abelian group actions in dividing characteristic and for restricted enveloping algebras $u(\mathfrak g)$ of two-dimensional restricted Lie algebras (a restricted Lie algebra in characteristic $p$ carries a $p$-power map; its restricted enveloping algebra $u(\mathfrak g)$ is the $p^{\dim\mathfrak g}$-dimensional quotient of the usual enveloping algebra). These results delimit but do not settle the classification question of which finite-dimensional Hopf algebras force nilpotency of invariant radicals. An extension to algebraic automorphisms and skew derivations (a $\sigma$-derivation $d$ satisfies $d(ab)=d(a)b+\sigma(a)d(b)$ for an automorphism $\sigma$) was later carried out by the same authors in Algebra Colloquium 29 (2022) 53–66 for algebraic skew derivations, including applications to the $n^2$-dimensional Taft Hopf algebra (generated by $g,x$ with $g^n=1$, $x^n=0$, $xg=qgx$ for a primitive $n$-th root of unity $q$) and the $q$-analogue of $U(\mathfrak{sl}_2)$. That follow-up, however, does not address the Hopf-algebra classification core; nor does the adjacent 2026 work of Chirvasitu (arXiv:2608.04851), which proves Jacobson-, solvable-, and nilpotent-radical (co)invariance for finite-dimensional algebras under trace-preserving (co)actions of Hopf algebras when the characteristic is large relative to the dimension, generalizing results of Gordienko, Linchenko, and Pagon–Repovš–Zaicev — sufficient conditions in a finite-dimensional, tracial setting, not a classification over all finite-dimensional $H$ and arbitrary semiprimitive $R$.

Problem Statement

Let $k$ be a field and $H$ a finite-dimensional Hopf algebra acting on associative $k$-algebras. Call $H$ radical-nilpotent for invariants if, for every $H$-module algebra $R$ with $J(R)=0$, the Jacobson radical $J(R^H)$ of the invariant subring is nilpotent, i.e. $J(R^H)^n=0$ for some finite $n$. The problem left open by the results of Bergen and Grzeszczuk (J. Algebra Appl. 17 (2018) 1850132) is to give an intrinsic structural classification of the finite-dimensional Hopf algebras that are radical-nilpotent for invariants — a necessary and sufficient condition on $H$ that provably accounts for the known positive cases (semisimple $H$ with a nondegenerate trace admitting $r$ with $\hat t(r)=1$; cyclic group algebras; and, in characteristic $p$, the restricted enveloping algebra $u(\mathfrak a)$ of a one-dimensional restricted Lie algebra $\mathfrak a$, the finite-dimensional analogue of the one-dimensional derivation actions $k[d]$), which the source obtains only under auxiliary hypotheses on $R$ (such as semiprimeness or primeness of $R^H$, and finiteness or PI conditions on $R$), and the known negative cases (higher-rank abelian group actions in dividing characteristic; restricted enveloping algebras of two-dimensional restricted Lie algebras).

References

1. Jeffrey Bergen and Piotr Grzeszczuk, "Jacobson radicals of invariants under Hopf algebra actions", Journal of Algebra and Its Applications 17 (2018), no. 7, 1850132. DOI: 10.1142/S0219498818501323. https://doi.org/10.1142/S0219498818501323
2. Jeffrey Bergen and Piotr Grzeszczuk, "Classical Radicals of Invariants of Algebraic Skew Derivations", Algebra Colloquium 29 (2022), no. 1, 53–66. DOI: 10.1142/S1005386722000050. https://doi.org/10.1142/S1005386722000050
3. Piotr Grzeszczuk, "On the Jacobson radical of constants of derivations", Journal of Algebra 401 (2014), 62–75. DOI: 10.1016/j.jalgebra.2013.11.024. https://doi.org/10.1016/j.jalgebra.2013.11.024
4. Jeffrey Bergen and Piotr Grzeszczuk, "On the semiprimitivity of free skew extensions of rings", Journal of Pure and Applied Algebra 224 (2020), no. 7, 106307. DOI: 10.1016/j.jpaa.2020.106307. https://doi.org/10.1016/j.jpaa.2020.106307 (preprint arXiv:1909.12712, https://arxiv.org/abs/1909.12712)
5. Piotr Grzeszczuk, "Semiprime algebras over finite fields with algebraic invariants of bounded degree", Communications in Algebra 53 (2025), no. 10, 4280–4286. DOI: 10.1080/00927872.2025.2481493. https://doi.org/10.1080/00927872.2025.2481493
6. Alexandru Chirvasitu, "Associative/Lie radical (co)invariance under tracial (co)actions", preprint (2026). arXiv:2608.04851. https://arxiv.org/abs/2608.04851
7. M. S. Eryashkin, "Invariants of the action of a semisimple Hopf algebra on PI-algebra", Russian Mathematics 60 (2016), no. 8, 17–28. DOI: 10.3103/S1066369X1608003X. https://doi.org/10.3103/S1066369X1608003X