First cohomology group of rank two Witt algebra to its Larsson modules

Abstract

Let U = ℂ2, Γ = ℤ2, and let ℂ[x 1 ±1 , x 2 ±1 ] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x 1 ±1 , x 2 ±1 ], which is spanned by elements of the form D(u, r) = x r (u 1 d 1 + u 2 d 2), u = (u 1, u 2) ∈ U, r ∈ Γ, where d 1 and d 2 are the degree derivations of ℂ[x 1 ±1 , x 2 ±1 ]. The image of gl 2-module V under Larsson functor F α , denoted by W = F α (V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H 1(L,W).