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).

Keywords

Derivation / Larsson functor / first cohomology group

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