When a finite-dimensional Hopf algebra H acts on an associative algebra R, the smash product R#H is the ring extension encoding the action, and its primeness and semiprimeness control the ideal theory of the extension. Bergen (2015) gave necessary and sufficient conditions for R#H to be prime or semiprime when H is a Taft algebra, in terms of pseudo-trace elements built from the top power of the skew-primitive generator; Gaddis (2021) extended both criteria to bosonizations of quantum linear spaces over arbitrary finite abelian groups and expressed the hope that this was a step toward criteria for actions of all finite-dimensional pointed Hopf algebras. The surviving open core is the following: extend these criteria to finite-dimensional pointed Hopf algebras beyond quantum linear spaces — non-abelian grouplike groups, Nichols algebras of non-Cartan type, small quantum groups.

Background

A Hopf algebra H over a field k is an associative algebra with a compatible coalgebra structure (comultiplication ΔΔ, counit εε) and an antipode; the model cases are group algebras, with Δ(g)=g⊗gΔ(g)=g⊗g for group elements, and enveloping algebras of Lie algebras. H acts on an associative unital k-algebra R if R is a left H-module with h⋅(rr′)=Σ(h1⋅r)(h2⋅r′)h·(rr') = Σ (h_1·r)(h_2·r') and h⋅1=ε(h)1h·1 = ε(h)1; one then calls R an H-module algebra. The action is encoded by the smash product R#H: the vector space R⊗HR⊗H with multiplication (r#h)(r'#h') = Σ r(h_1·r')#h_2, an algebra containing both R and H whose ideal structure reflects the H-stable ideals of R. A ring is prime if the product of two nonzero ideals is nonzero, and semiprime if it has no nonzero nilpotent ideal — the basic gateway to ideal theory, localization, and representation theory. For a finite group G acting by automorphisms, the classical theory of skew group algebras settles semiprimeness of R#kG completely (in characteristic 0 it is equivalent to semiprimeness of R), but for non-semisimple Hopf algebras no general theory exists; the semiprime smash product question of whether semiprimeness of R implies that of R#H is open in general (Lomp, 2015).

The simplest non-semisimple, non-cocommutative Hopf algebras are the Taft algebras Hn(λ)H_n(λ): for a primitive n-th root of unity λ, Hn(λ)H_n(λ) is generated by a grouplike element g (with Δ(g)=g⊗gΔ(g)=g⊗g, gn=1g^n=1) and a (g,1)-skew-primitive element x (with Δ(x)=g⊗x+x⊗1Δ(x)=g⊗x+x⊗1) subject to xn=0x^n=0 and xg=λgxxg=λgx. A Hopf algebra is pointed if every simple subcoalgebra is one-dimensional, spanned by a grouplike; the finite-dimensional pointed Hopf algebras are the class accessible to classification, and Taft algebras are their rank-one building blocks. When Hn(λ)H_n(λ) acts on R, the grouplike g acts as an automorphism σ of order dividing n, giving a decomposition of R into σ-eigenspaces, and x acts as a σ-skew derivation: x(rr′)=σ(r)x(r′)+x(r)r′x(rr') = σ(r)x(r') + x(r)r'.

Bergen (2015) determined exactly when the smash product of a Taft algebra with an associative algebra is prime or semiprime. Writing R⟨x⟩R⟨x⟩ for the kernel of x, his semiprimeness criterion (Theorem 4 of his paper, in the form recorded by Gaddis) states that R#H_n(λ) is semiprime if and only if R⟨x⟩R⟨x⟩ is semiprime and x(L)≠0x(L) ≠ 0 for every nonzero left ideal L of R. His primeness criterion (Theorem 8) is phrased through pseudo-traces: the elements tχ=eχ⋅xn−1t_χ = e_χ·x^{n-1} of H, where eχ=(1/n)⋅Σjχ(gj)−1⋅gje_χ = (1/n)·Σ_j χ(g^j)^{-1}·g^j are the central idempotents of the group algebra of ⟨g⟩ indexed by characters χ of ⟨g⟩. The criterion reads: R#H_n(λ) is prime if and only if tχ(L)t_χ(L) has zero left annihilator in R for every nonzero H-stable left ideal L and every character χ. Bergen worked out the consequences for fields, reduced rings, domains, and the 4-dimensional Taft algebra; the criteria themselves hold for arbitrary R, but the left-ideal conditions are hard to verify in practice.

Gaddis (2021) extended both criteria from Taft algebras to bosonizations of quantum linear spaces. For a finite abelian group G, grouplikes g1,…,gθg_1,…,g_θ and characters χ1,…,χθχ_1,…,χ_θ with mi=ord(χi(gi))≥2m_i = ord(χ_i(g_i)) ≥ 2, this pointed Hopf algebra B(G,g,χ)B(G,g,χ) is generated by G and skew-primitives xix_i with gxi=χi(g)xigg x_i = χ_i(g) x_i g, ximi=0x_i^{m_i} = 0, and xixj=χj(gi)xjxix_i x_j = χ_j(g_i) x_j x_i; the Taft algebra is the case θ=1θ=1 with G cyclic. Gaddis's semiprimeness criterion iterates Bergen's along the rank, and the primeness criterion has the same shape: R#B is prime if and only if tχ(L)t_χ(L) has zero left annihilator for every nonzero B-stable left ideal L and every character χ, where now tχ=eχ⋅Πiximi−1t_χ = e_χ·Π_i x_i^{m_i-1} and the eχe_χ are the character idempotents of kGkG. The proof rests on the character-idempotent decomposition of the abelian group algebra and on the single top monomial Πiximi−1Π_i x_i^{m_i-1}; Gaddis states the hope that this is a crucial step toward criteria for actions of all finite-dimensional pointed Hopf algebras.

Both statement and technique stop at quantum linear spaces. For non-abelian grouplike groups there is no character-idempotent decomposition of kGkG; for Nichols algebras beyond quantum linear spaces (the braided Hopf algebras that constitute the non-grouplike part of a pointed Hopf algebra — for example the positive-root parts of small quantum groups uq(g)u_q(g), or braided vector spaces of non-Cartan type) the skew-primitives satisfy higher-order relations and no analogue of the top monomial is available. In parallel, the criteria quantify over all nonzero (H-stable) left ideals of R, and no algorithmic procedure is known that decides them even for a finite-dimensional R given by structure constants. On the division-algebra side of the source paper — which inputs R produce simple or prime smash products — Etingof and Negron (2022) proved that the standard families of finite-dimensional pointed Hopf algebras, including small quantum groups and generalized Taft algebras with non-nilpotent skew primitives, do admit faithful actions on central simple division algebras, but a systematic description of such inputs is missing. An adjacent program classifies Taft and quantum-linear-space actions on concrete algebras (quantum affine spaces, down-up algebras, preprojective algebras of extended Dynkin type A) and computes their invariants; it does not address the general (semi)primeness criteria.

Problem Statement

Throughout, k is an algebraically closed field of characteristic 0. Let H be a finite-dimensional pointed Hopf algebra acting on an associative unital k-algebra R, and let R#H denote the smash product.

Determine necessary and sufficient conditions, stated in terms of the action data of H on R (the automorphisms induced by grouplike elements and the skew-derivations induced by skew-primitive generators), for R#H to be prime, and for R#H to be semiprime, valid for finite-dimensional pointed Hopf algebras H beyond bosonizations of quantum linear spaces over finite abelian groups — including grouplike groups that are non-abelian, Nichols algebras outside the quantum-linear-space class, such as the positive-root parts of small quantum groups uq(g)u_q(g), and pointed Hopf algebras with skew-primitive generators not covered by the quantum-linear-space relations. Such criteria are sought for the class of finite-dimensional pointed Hopf algebras that are not bosonizations of quantum linear spaces over finite abelian groups, reducing to Bergen's Taft criteria and Gaddis's quantum-linear-space criteria on those known classes; the families enumerated above illustrate this class, and a solution must cover the class as a whole.

References

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