Smash products and differential identities

Abstract

Let U be the universal enveloping algebra of a Lie algebra and R a U-module algebra, where U is considered as a Hopf algebra canonically. We determine the centralizer of R in R#U with its associated graded algebra. We then apply this to the Ore extension R[X; φ], where φ : X → Der(R). With the help of PBW-bases, the following is proved for a prime ring R: Let Q be the symmetric Martindale quotient ring of R. For f i , g i ∈ Q[X; φ], i f i rg i = 0 for all r ∈ R iff i f i ⊗ g i = 0, where ⊗ is over the centralizer of R in Q[X; φ]. Finally, we deduce from this Kharchenko's theorem on differential identities.

Symmetric Martindale Quotient Rings of Ore Extensions with More than One Indeterminate

“…The condition (3) comes in because the Hopf algebra in the context of Ore extension R X D is a smash product of R with k X , the free k-algebra considered as the universal enveloping algebra of the free Lie k-algebra X generated by X ( [7,8]). We refer the details to [5,6,8]. Each of the three conditions (1), (2), and (3) does not seem easy to verify for a particular application.…”