A conjecture of Bavula on homomorphisms of the Weyl algebras

Abstract

Bavula states [V. Bavula, The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic, J. Pure Appl. Algebra 212 (2008), pp. 2320–2337] the following conjecture: (BC) Let A _n be the Weyl algebra over a field K of characteristic p > 0. Then each K-algebra homomorphism is a monomorphism. The purpose of this article is to prove an analogue of BC for symplectic Poisson algebras, to check BC for A ₁ and to show that BC is wrong for A _n when n > 1.

Keywords:

• Weyl algebras
• finite characteristic
• finite-dimensional module/vector space
• Gelfand–Kirillov dimension
• Bavula conjecture

AMS Subject Classifications::

• 15A03
• 15A21
• 16P90
• 16332

Acknowledgements

The author is grateful to the CRM (Centre de Recerca Matemàtica) in Bellaterra, Spain, for the hospitality during the work on this project. This work was supported by an NSA grant H98230-09-1-0008, by an NSF grant DMS-0904713, and a Fulbright fellowship awarded by the United States–Israel Educational Foundation.