For a finite-dimensional Hopf algebra HH acting on an algebra AA that satisfies a polynomial identity, one asks whether HH-semiprimeness of AA (absence of nonzero nilpotent HH-stable ideals) forces AA to be semiprime (absence of nonzero nilpotent ideals altogether). This transfer is known when HH is cosemisimple or when the base field has characteristic zero, by work of Linchenko and Montgomery, and the companion smash-product statement (HH-semiprime PI AA implies A#HA\#H semiprime for semisimple HH) was proved by them over arbitrary fields. The unresolved core is the positive-characteristic case in which HH is semisimple but not known to be cosemisimple: there every existing proof routes semiprimeness through the semisimplicity of the dual Hopf algebra H∗H^*, and the question sits at the same frontier as Kaplansky's fifth conjecture in positive characteristic (whether a semisimple finite-dimensional Hopf algebra must have involutory antipode, S2=idS^2=\mathrm{id}, and be cosemisimple), which is open below the Etingof–Gelaki bound p>(dim⁡H)φ(dim⁡H)/2p>(\dim H)^{\varphi(\dim H)/2}, where φ\varphi is Euler's totient function. A negative answer would necessarily exhibit the first known finite-dimensional semisimple Hopf algebra that is not cosemisimple.

Background

A Hopf algebra HH over a field kk is an associative unital algebra that is also a coalgebra, i.e. it carries a comultiplication Δ ⁣:H→H⊗H\Delta\colon H\to H\otimes H and a counit ε ⁣:H→k\varepsilon\colon H\to k compatible with multiplication, together with an antipode S ⁣:H→HS\colon H\to H (the convolution-inverse of the identity map) that plays the role of inversion. Group algebras kGkG of finite groups GG, with Δ(g)=g⊗g\Delta(g)=g\otimes g and S(g)=g−1S(g)=g^{-1}, and their linear duals kG∗kG^{*} are the basic finite-dimensional examples, but Hopf algebras mix these two pictures and need be neither commutative nor cocommutative. A finite-dimensional Hopf algebra HH acts on an algebra AA if AA is a left HH-module and the action respects products, h⋅(ab)=∑(h1⋅a)(h2⋅b)h\cdot(ab)=\sum (h_{1}\cdot a)(h_{2}\cdot b) for Δ(h)=∑h1⊗h2\Delta(h)=\sum h_{1}\otimes h_{2}; such an AA is called an HH-module algebra. From an action one builds the smash product A#HA\#H, the algebra on A⊗HA\otimes H with multiplication (a#h)(b#g)=∑a(h1⋅b)#h2g(a\#h)(b\#g)=\sum a(h_{1}\cdot b)\#h_{2}g, which for H=kGH=kG is the classical skew group ring. A crucial structural tool is Blattner–Montgomery duality: when dim⁡H<∞\dim H<\infty, the dual Hopf algebra H∗=Homk(H,k)H^{*}=\mathrm{Hom}_{k}(H,k) acts on A#HA\#H and (A#H)#H∗≅Mdim⁡H(A)(A\#H)\#H^{*}\cong M_{\dim H}(A), a full matrix algebra over AA; this lets one pass information between AA, A#HA\#H, and matrix rings, but the argument needs semisimplicity of the Hopf algebra acting on each side, so properties of HH and of H∗H^{*} enter symmetrically.

An algebra is semiprime if it has no nonzero nilpotent two-sided ideals, equivalently if its prime radical P(A)P(A) (the intersection of its prime ideals) is zero; a PI-algebra is one satisfying a polynomial identity — a nonzero polynomial in noncommuting variables that vanishes under every substitution of elements of the algebra; matrix algebras and their subalgebras are the standard examples. When HH acts on AA, an ideal II of AA is HH-stable if h⋅I⊆Ih\cdot I\subseteq I for all h∈Hh\in H, and AA is HH-semiprime if it has no nonzero nilpotent HH-stable ideals. Semiprime always implies HH-semiprime, and the converse asks how much of the nilpotent structure is forced to be HH-stable. Classical theorems for finite group actions and for group gradings established the analogous transfer for skew group rings and graded smash products, and in 1986 Cohen and Fishman posed the general question: if a finite-dimensional semisimple Hopf algebra HH acts on a semiprime algebra AA, is the smash product A#HA\#H semiprime? A finite-dimensional Hopf algebra is called cosemisimple when its dual algebra H∗H^{*} is semisimple, and involutory when S2=idS^{2}=\mathrm{id}. In characteristic zero the Larson–Radford theorem collapses these notions: semisimple, cosemisimple, and S2=idS^{2}=\mathrm{id} are equivalent. In positive characteristic they provably do not collapse for free: whether a semisimple finite-dimensional Hopf algebra must be cosemisimple or involutory is open (the positive-characteristic form of Kaplansky's fifth conjecture). Etingof and Gelaki proved that semisimple plus cosemisimple implies S2=idS^{2}=\mathrm{id} over any field (by lifting to characteristic zero), that for char k=p>(dim⁡H)φ(dim⁡H)/2\mathrm{char}\,k=p>(\dim H)^{\varphi(\dim H)/2} (φ\varphi the Euler function) semisimple, cosemisimple, and S2=idS^{2}=\mathrm{id} are equivalent, improving Sommerhäuser's earlier explicit bound; and Wang, Liu, and Li showed that, over an algebraically closed field of characteristic p>dim⁡Hp>\sqrt{\dim H}, the square of the antipode of a semisimple HH is inner, S2(h)=uhu−1S^{2}(h)=uhu^{-1} with u=S(Λ(2))Λ(1)u=S(\Lambda_{(2)})\Lambda_{(1)} built from a nonzero right integral Λ\Lambda (an element with Λh=ε(h)Λ\Lambda h=\varepsilon(h)\Lambda for all h∈Hh\in H; for a group algebra, Λ=∑g∈Gg\Lambda=\sum_{g\in G}g), and that Λ\Lambda is cocommutative if and only if S2=idS^{2}=\mathrm{id}. Below such bounds nothing is known, and no example of a finite-dimensional semisimple non-cosemisimple Hopf algebra is known in any positive characteristic.

This Hopf-algebra frontier directly gates the ring theory of Hopf actions. For HH-module PI-algebras, Linchenko and Montgomery proved in 2007, with no restriction on the field, that HH-semiprimeness of AA implies semiprimeness of A#HA\#H whenever HH is semisimple (settling the Cohen–Fishman question for PI-algebras in all characteristics, including finite fields), and that P(A)P(A) is HH-stable whenever HH is cosemisimple — the latter yielding, through duality, that an HH-semiprime PI-algebra is semiprime for cosemisimple HH. Skryabin and van Oystaeyen obtained the corresponding Noetherian statements, and Eryashkin developed the structure theory (quotient rings, integrality over invariants) of HH-semiprime PI module algebras under the standing hypothesis that HH is semisimple and cosemisimple. In every one of these arguments the implication from HH-semiprimeness to semiprimeness is extracted from semisimplicity of H∗H^{*}, i.e. from cosemisimplicity of HH; in characteristic zero this is automatic, but in positive characteristic it is exactly what is not known.

Problem Statement

Let kk be a field of positive characteristic and let HH be a finite-dimensional semisimple Hopf kk-algebra acting on an algebra AA that satisfies a polynomial identity. If AA is HH-semiprime (it has no nonzero nilpotent HH-stable two-sided ideals), must AA be semiprime? Equivalently, must the prime radical P(A)P(A) be HH-stable for every HH-module PI-algebra over such an HH? The answer is known to be yes when char k=0\mathrm{char}\,k=0 or when HH is cosemisimple; the problem is open precisely for semisimple HH over a positive-characteristic field for which cosemisimplicity is not known. A positive answer for all semisimple HH would follow from an affirmative resolution of the positive-characteristic form of Kaplansky's fifth conjecture (semisimple finite-dimensional Hopf algebras are cosemisimple, equivalently involutory); a negative answer would in particular produce the first known finite-dimensional semisimple Hopf algebra that is not cosemisimple. The problem is to be answered at this natural level of generality — no restriction on the characteristic beyond positivity, on the dimension of HH, on the field (finite or infinite), or on the polynomial identity satisfied by AA.

References

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