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

Abstract

Let U={ℂ}^{\mathrm{2}}, \mathrm{\Gamma }={\mathbb{Z}}^{\mathrm{2}}, and let \mathrm{ℂ}[x_{\mathrm{1}}^{\pm \mathrm{1}},x_{\mathrm{2}}^{\pm \mathrm{1}}] be the ring of Laurent polynomials. The Witt algebra ℒ is the Lie algebra of derivations over \mathrm{ℂ}[x_{\mathrm{1}}^{\pm \mathrm{1}},x_{\mathrm{2}}^{\pm \mathrm{1}}], which is spanned by elements of the form D(u, r) = x^r(u₁d₁ + u₂d₂), u=(u_{\mathrm{1}},u_{\mathrm{2}})\in U, r\in \mathrm{\Gamma }, where d₁ and d₂ are the degree derivations of \mathrm{ℂ}[x_{\mathrm{1}}^{\pm \mathrm{1}},x_{\mathrm{2}}^{\pm \mathrm{1}}]. The image of gl₂-module V under Larsson functor F^{\alpha }, denoted by W=F^{\alpha }(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 H¹(ℒ, W).

Keywords

Derivation / Larsson functor / first cohomology group

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