Finite dimensional Hopf actions on Weyl algebras

Abstract

We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors.

Finite dimensional Hopf actions on algebraic quantizations

“…Instead, note that if a ∈ D p m is central modulo p m−1 for some m ≥ 2, then a p is central. Hence Z(m) ⊃ Z(m − 1) p , implying that Z(m) ⊃ Z p m−1 and therefore [Z : Z(m)] is finite (by Lemma 3.4) and is a power of p. Now the proof proceeds by invoking Theorem 3.2, whose assumptions are satisfied by the nondegeneracy property of B and using a straightforward generalization of [CEW2,Lemma 4.6].…”

“…Another generalization of Theorem 2.4 concerns nondegenerate quantizations, defined in Definition 1.6. To obtain it, we will first need to generalize [CEW2,Theorem 3.2]. Let H be a finite dimensional Hopf algebra over an algebraically closed field F of characteristic p > 0, and let Z be a finitely generated field extension of F .…”

“…. , L r of B, and the number 2n in the proof of [CEW2,Lemma 4.3] should be replaced by r (cf. [CEW2, proof of Theorem 1.2]).…”

“…Moreover, it turns out that even without the Hopf-Galois assumption, Theorem 1.2 extends to non-semisimple Hopf actions for a somewhat more restrictive class of quantizations. To see this, let us recall some algebras introduced in [CEW2].…”

“…These algebras are defined over the truncated Witt ring W m,p of F p (cf. [CEW2,Subsections 2.1,2.3,2.4]). Let Z be the center of the central division algebra D p .…”