[Paper Review] Finite dimensional Hopf actions on Weyl algebras

Juan Cuadra, Pavel Etingof+1 more|arXiv (Cornell University)|Sep 3, 2015

Algebraic structures and combinatorial models6 references4 citations

TL;DR

This paper proves that any finite-dimensional Hopf algebra action on a Weyl algebra over an algebraically closed field of characteristic zero factors through a group action, implying Weyl algebras admit no genuine finite quantum symmetries. The proof uses reduction modulo prime powers rather than primes, extending the result to algebras of differential operators on smooth affine irreducible varieties.

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.

Motivation & Objective

• To establish that finite-dimensional Hopf actions on Weyl algebras over algebraically closed fields of characteristic zero factor through group actions.
• To resolve an open question by proving that such actions do not admit genuine finite quantum symmetries.
• To generalize the result from semisimple Hopf algebras to all finite-dimensional Hopf algebras, removing prior restrictions.
• To extend the result to algebras of differential operators on smooth affine irreducible varieties over the same field.
• To provide a new proof technique using reduction modulo prime powers instead of modulo primes, enabling stronger conclusions.

Proposed method

• Use reduction modulo prime powers (e.g., $ p^m $) rather than just modulo primes to analyze the structure of Hopf actions.
• Construct a filtration on the Weyl algebra and its reduction modulo $ p^m $, analyzing invariants under the Hopf action.
• Prove that the center of the division ring of quotients of the reduced algebra is stable under the induced Hopf action.
• Show that the centralizer of the fixed subalgebra in the reduced division algebra coincides with the $ p^m $-th power center.
• Apply a key result on Hopf actions on fields in positive characteristic to deduce cocommutativity of the reduced Hopf algebra.
• Lift the cocommutativity of the reduced algebra back to the original algebra using injectivity of the product of reduction maps.

Experimental results

Research questions

• RQ1 Can finite-dimensional Hopf actions on Weyl algebras over characteristic zero fields be factored through group actions?
• RQ2 Do Weyl algebras admit genuine finite quantum symmetries beyond group actions?
• RQ3 Can the result on Weyl algebras be extended to algebras of differential operators on smooth varieties?
• RQ4 Does reduction modulo prime powers provide a stronger tool than reduction modulo primes for analyzing Hopf actions?
• RQ5 Is the cocommutativity of the reduced Hopf algebra $ H_p $ sufficient to conclude that the original $ H $ is a group algebra?

Key findings

• Any finite-dimensional Hopf algebra action on the $ n $-th Weyl algebra $ π_n(k) $ over an algebraically closed field of characteristic zero factors through a group action.
• The result implies that Weyl algebras do not admit genuine finite quantum symmetries, as no non-group-like finite-dimensional Hopf algebras can act faithfully.
• The proof technique using reduction modulo prime powers allows the removal of the semisimplicity assumption required in earlier work.
• The result extends to the algebra of differential operators $ D(X) $ on any smooth affine irreducible variety $ X $, where such actions also factor through group actions.
• The center of the division ring of quotients of the reduced Weyl algebra is preserved under the induced Hopf action, and its $ p^m $-th power subring is stable.
• The cocommutativity of the reduced Hopf algebra $ H_p $ for large primes $ p $ implies the cocommutativity of the original $ H $, hence $ H o kG $ for a finite group $ G $.