Gaddis (2020) gave complete, checkable criteria for when the smash product R#BR\#B of an associative algebra RR with a bosonization BB of a quantum linear space is prime or semiprime, extending Bergen's earlier work for Taft algebras. The source paper explicitly expresses the hope that these results are a step toward criteria for actions of all finite-dimensional pointed Hopf algebras, and this extension is not known: the existing criteria depend essentially on the nilpotence of the skew-primitive generators, the monomial basis of the quantum linear space, and induction on the rank. The open problem is to establish verifiable prime/semiprime criteria for R#HR\#H when the finite-dimensional pointed Hopf algebra HH need not be a bosonization of a quantum linear space — for example liftings of quantum linear spaces, pointed Hopf algebras whose infinitesimal braiding is not of diagonal type, and small quantum groups.

Background

A Hopf algebra is a vector space that is simultaneously an algebra and a coalgebra with compatible multiplication and comultiplication; a finite-dimensional Hopf algebra HH can act on an associative algebra RR by h⋅(rr′)=∑(h1⋅r)(h2⋅r′)h\cdot(rr')=\sum (h_1\cdot r)(h_2\cdot r'), in which case RR is called a left HH-module algebra. For such an action one forms the smash product R#HR\#H, which is R⊗HR\otimes H as a vector space with multiplication twisted by the action; it simultaneously generalizes skew group rings (when HH is a group algebra) and differential-operator rings. A ring is prime if the product of two nonzero two-sided ideals is nonzero, and semiprime if it has no nonzero nilpotent ideal; these properties control the ring's ideal structure and its representation theory. Classical results describe how primeness and semiprimeness behave for skew group rings (Fisher–Montgomery) and for actions of semisimple Hopf algebras (the Cohen–Fishman semiprime smash product question surveyed by Lomp), but much less is known when the acting Hopf algebra is finite-dimensional and non-semisimple. A Hopf algebra is pointed if every simple subcoalgebra is one-dimensional; equivalently it is generated by grouplike elements gg (with Δ(g)=g⊗g\Delta(g)=g\otimes g) and skew-primitive elements xx (with Δ(x)=g⊗x+x⊗h\Delta(x)=g\otimes x+x\otimes h). By the classification program of Andruskiewitsch and Schneider, finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero are built from a finite abelian group GG of grouplikes and a braided Hopf algebra called a Nichols algebra, via a construction called bosonization. The simplest case is a quantum linear space: the Nichols algebra on generators x1,…,xθx_1,\dots,x_\theta with gxi=χi(g)xiggx_i=\chi_i(g)x_ig, ximi=0x_i^{m_i}=0, and xixj=χj(gi)xjxix_ix_j=\chi_j(g_i)x_jx_i; its bosonization B(G,g,χ)B(G,\mathbf g,\boldsymbol\chi) contains the Taft algebras, the smallest non-semisimple Hopf algebras, and their generalizations. Liftings are deformations of these bosonizations in which the nilpotence relation ximi=0x_i^{m_i}=0 is replaced by ximi=μi(gimi−1)x_i^{m_i}=\mu_i(g_i^{m_i}-1), and pointed Hopf algebras of non-diagonal type (for example most small quantum groups uq(g)u_q(\mathfrak g), the finite-dimensional Hopf quotients of the quantum enveloping algebra of a semisimple Lie algebra g\mathfrak g at a root of unity) have Nichols algebras without a monomial basis of this kind. Bergen determined when R#TR\#T is (semi)prime for TT a Taft algebra; Gaddis extended this to arbitrary quantum linear space bosonizations BB acting on a BB-module algebra RR: R#BR\#B is semiprime if and only if, after possibly reordering generators, the joint kernel ⋂iR⟨xi⟩\bigcap_i R^{\langle x_i\rangle} of the skew-derivations is semiprime and certain iterated nonvanishing conditions xk(L)≠0x_k(L)\neq 0 hold for all nonzero left ideals LL at each rank (Theorem 2.4 of the source), and R#BR\#B is prime if and only if tχ(L)t_\chi(L) has zero left annihilator for every nonzero BB-stable left ideal LL and every character χ\chi of GG, where tχ=eχxt_\chi=e_\chi x is built from the central idempotent eχe_\chi and the product x=∏iximi−1x=\prod_i x_i^{m_i-1} (Theorem 3.6). For RR a domain this reduces to nonvanishing of xk+1(Rk)x_{k+1}(R_k) plus the existence of simultaneous GG-eigenvectors in Rx=⋂iR⟨xi⟩R^x=\bigcap_i R^{\langle x_i\rangle} for every character (Corollary 3.8). The proofs use the nilpotence of the xix_i, the resulting polynomial (PBW) filtration of the subalgebra generated by RR and the xix_i, and induction on the rank θ\theta; the source paper gives examples where the hypotheses fail and the smash product is not semiprime (Example 2.5), or semiprime but not prime because the required eigenvectors cannot exist (Example 3.10), but no classification of such failure behaviors. The paper closes its introduction by expressing the hope that these results are a crucial step toward criteria for actions of all finite-dimensional pointed Hopf algebras.

Problem Statement

Over an algebraically closed field of characteristic zero, establish verifiable criteria — in the spirit of Theorems 2.4 and 3.6 of Gaddis (2020) — for when the smash product R#HR\#H is prime and when it is semiprime, where HH ranges over finite-dimensional pointed Hopf algebras that are not bosonizations of quantum linear spaces. The single scientific objective is a prime/semiprime theory of smash products for general finite-dimensional pointed Hopf algebra actions, replacing the quantum-linear-space hypotheses by structure available on all such HH. The criteria sought are for finite-dimensional pointed Hopf algebras in general, not for one class among them. These classes lie within the single objective in increasing generality and are not alternatives that each resolve it: (1) liftings of quantum linear spaces, whose skew-primitive generators satisfy ximi=μi(gimi−1)x_i^{m_i}=\mu_i(g_i^{m_i}-1) instead of ximi=0x_i^{m_i}=0; (2) pointed Hopf algebras whose infinitesimal braiding (Nichols algebra) is finite-dimensional of diagonal type but not a quantum linear space; (3) finite-dimensional pointed Hopf algebras beyond diagonal type, including small quantum groups. Any proposed criterion must specialize back to the known criteria when HH is a quantum linear space bosonization (and thence to Bergen's Taft-algebra criteria), and must correctly predict the source paper's examples, including the non-semiprime Example 2.5 and the semiprime-but-not-prime Example 3.10.

References

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