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Journal of Lie Theory 32 (2022), No. 1, 267--279
Copyright Heldermann Verlag 2022

Partial Classification of Irreducible Modules for Loop-Witt Algebras

Priyanshu Chakraborty
School of Mathematics, Harish-Chandra Research Institute, Prayagraj-Allahabad, Uttar Pradesh, India

S. Eswara Rao
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India


Consider the Lie algebra of the group of diffeomorphisms of a $n$-dimensional torus which is also known as the derivation algebra of the Laurent polynomial algebra $A$ over $n$ commuting variables, denoted by $Der\,A$. In this paper we consider the Lie algebra $(A\rtimes Der\,A)\otimes B$ for some commutative associative unital algebra $B$ over $\mathbb C$ and classify all irreducible modules for $(A\rtimes Der\,A) \otimes B$ with finite dimensional weight spaces under some natural conditions. In particularly, we show that Larsson's constructed modules of tensor fields exhaust all such irreducible modules for $(A\rtimes Der\,A)\otimes B$.

Keywords: Witt algebra, Virasoro algebra, current algebra.

MSC: 17B65,17B68,17B67.

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