Heat semigroups on Weyl algebra

Abstract

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators ∇i± forming the Lie algebra [∇j±,∇k±]=iRjk± and [∇j+,∇k−]=i12(Rjk++Rjk−) with some anti-symmetric matrices Rij± and define the corresponding Laplacians Δ±=g±ij∇i±∇j± with some positive matrices g±ij. We show that the heat semigroups exp(tΔ±) can be represented as a Gaussian average of the operators expξ,∇± and use these representations to compute the product of the semigroups, exp(tΔ+)exp(sΔ−) and the corresponding heat kernel.