For a finite-dimensional involutory weak Hopf algebra H over a field of characteristic 0 acting on a semiprime module algebra A satisfying a polynomial identity, Borges and Lomp proved that the smash product A#H is again semiprime, transferring Linchenko and Montgomery's Hopf-theoretic theorem to quantum groupoid symmetries. Their proof uses these hypotheses essentially, and the 2015 survey by Lomp asks whether counterexamples to the Cohen–Fishman question exist for more general actions, in particular for the 13-dimensional weak Hopf algebra associated with the Lee–Yang fusion rule; whether the conclusion survives when (i) the polynomial identity assumption on A is dropped, (ii) H is finite-dimensional and semisimple but not involutory (antipode squared not the identity), (iii) the action has infinite-dimensional operator image, or (iv) the base field has positive characteristic while H is kept finite-dimensional and involutive and, in addition, semisimple, remains open. This problem asks for positive transfer theorems or explicit counterexamples in these regimes, including the survey's concrete question of whether the 13-dimensional Lee–Yang weak Hopf algebra admits a semiprime module algebra with non-semiprime smash product.

Background

A ring is called semiprime if it has no nonzero nilpotent two-sided ideal; semiprimeness is the ring-theoretic generalization of reducedness and is a basic regularity property in noncommutative ring theory. A Hopf algebra is a bialgebra (an algebra that is also a coalgebra with compatible structures) equipped with an antipode S, and it acts on an algebra A in a way compatible with multiplication, under which A is called an H-module algebra; one can then form the smash product A#H, a crossed-product-like algebra built on A ⊗ H that encodes the action. A classical question of Cohen and Fishman (1986) asks: if H is a finite-dimensional semisimple Hopf algebra acting on a semiprime algebra A, must A#H be semiprime? This remains open in general; Fisher and Montgomery settled the group-algebra case, and Linchenko and Montgomery (2007) proved the semiprime transfer under the additional hypothesis that A satisfies a polynomial identity (PI), i.e. A obeys some nonzero polynomial identity in noncommuting variables, over a characteristic-0 field. Their argument uses Blattner–Montgomery duality (A#H embeds in A#H#H*), semisimplicity of the dual H*, and H-stability of the Jacobson radical, the Jacobson radical being the intersection of all maximal left ideals.

Weak Hopf algebras (also called quantum groupoids), introduced by Böhm, Nill, and Szlachányi, generalize Hopf algebras by weakening the comultiplication and counit axioms: Δ(1) need not equal 1⊗1 and the counit need not be multiplicative. They arise naturally from groupoid algebras, from depth-2 subfactor inclusions (a class of inclusions of von Neumann algebras in operator algebra theory), and from fusion categories (semisimple monoidal categories with finitely many simple objects and well-behaved duality); they include the 13-dimensional weak Hopf algebra associated with the Lee–Yang fusion rule — the simplest nontrivial rank-2 fusion rule, originating in the Lee–Yang minimal model of two-dimensional conformal field theory — constructed by Böhm and Szlachányi. For a weak Hopf algebra H acting on a left H-module algebra A, the smash product A#H is built on the vector-space tensor product A ⊗_k H with multiplication twisted by the action, and its structure theory is governed by a Blattner–Montgomery-type duality due to Nikshych. A weak Hopf algebra is called involutive if its antipode is an involution (S² = id). For ordinary finite-dimensional Hopf algebras in characteristic 0, Larson and Radford proved that semisimplicity forces S² = id, but this rigidity fails for weak Hopf algebras, so involutivity is a genuinely additional hypothesis there. In this setting Borges and Lomp proved: over a field of characteristic 0, if a finite-dimensional involutive weak Hopf algebra H acts on a semiprime H-module algebra A that satisfies a polynomial identity, then A#H is semiprime. Their proof uses involutivity and finite-dimensionality at three load-bearing points: semisimplicity of the dual H*, the duality isomorphism A#H#H* ≃ eM_n(A)e, and H-stability of the Jacobson radical under an action of finite type (finite-dimensional image of the action map λ: H → End_k(A)).

Problem Statement

Let k be a field and let H be a weak Hopf algebra over k acting on a left H-module algebra A that is semiprime, with A#H the weak smash product. The theorem of Borges–Lomp (stated over characteristic 0) gives A#H semiprime when H is finite-dimensional and involutive and A satisfies a polynomial identity. Determine, by positive theorem or explicit counterexample, whether semiprimeness transfers to A#H in each of the regimes outside these hypotheses, each of which keeps the theorem's remaining hypotheses on H and A except those its description alters (so A is semiprime throughout and still satisfies a polynomial identity in (ii)–(iv)): (i) over a characteristic-0 field, with H finite-dimensional and involutive, as in the theorem, but A no longer assumed to satisfy a polynomial identity; (ii) over a characteristic-0 field, with H finite-dimensional semisimple but not involutive, i.e. S2≠idS^2\neq\mathrm{id} — note that non-involutive semisimple weak Hopf algebras exist, unlike the ordinary Hopf case in characteristic 0; (iii) over a characteristic-0 field, with H allowed to be infinite-dimensional and the action of infinite type, i.e. the image of the action map λ ⁣:H→End⁡k(A)\lambda\colon H\to\operatorname{End}_k(A) infinite-dimensional (if H itself is finite-dimensional then λ(H)\lambda(H) is automatically finite-dimensional, so an action of infinite type forces H to be infinite-dimensional), so that no finite-dimensionality or duality argument is available; and (iv) over base fields of positive characteristic, with H finite-dimensional and involutive as in the theorem and, in addition, semisimple. The problem concerns all four regimes, together with the concrete case posed in Lomp's survey: does there exist a semiprime module algebra A (with no polynomial-identity assumption) over the 13-dimensional weak Hopf algebra associated with the Lee–Yang fusion rule for which A#H fails to be semiprime, or can such failures be ruled out?

References

  1. 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
  2. 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
  3. Gabriella Böhm, Florian Nill and Kornél Szlachányi, 'Weak Hopf Algebras', Journal of Algebra 221 (1999), 385–438, DOI 10.1006/jabr.1999.7984, https://doi.org/10.1006/jabr.1999.7984
  4. Gabriella Böhm and Kornél Szlachányi, 'A coassociative C*-quantum group with nonintegral dimensions', Letters in Mathematical Physics 38 (1996), 437–456, DOI 10.1007/BF01815526, https://doi.org/10.1007/BF01815526
  5. Richard G. Larson and David E. Radford, 'Semisimple Cosemisimple Hopf Algebras', American Journal of Mathematics 110 (1988), 187–195, DOI 10.2307/2374545, https://doi.org/10.2307/2374545
  6. Inês Borges and Christian Lomp, 'Quantum Groupoids Acting on Semiprime Algebras', Advances in Mathematical Physics, vol. 2011, Article ID 546058, DOI 10.1155/2011/546058, https://doi.org/10.1155/2011/546058
  7. Christian Lomp, 'Ring theoretic aspects of weak Hopf actions', technical report (2009), Universidade do Porto, HDL 10216/25800, https://repositorio-aberto.up.pt/handle/10216/25800
  8. Christian Lomp, 'On the Semiprime Smash Product Question', Contemporary Mathematics 634 (2015), American Mathematical Society, 205–222, DOI 10.1090/conm/634/12700, arXiv:1401.1049, https://doi.org/10.1090/conm/634/12700
  9. Jason Gaddis, 'Prime and semiprime quantum linear space smash products', Glasgow Mathematical Journal 63 (2021), 503–514, DOI 10.1017/S0017089520000348, arXiv:1809.10713, https://doi.org/10.1017/S0017089520000348
  10. Martin Lorenz, Bach Nguyen and Ramy Yammine, 'Actions of cocommutative Hopf algebras', Journal of Algebra 546 (2020), 703–722, DOI 10.1016/j.jalgebra.2019.11.010, arXiv:1907.06958, https://doi.org/10.1016/j.jalgebra.2019.11.010