Kac-Moody algebras and Lie algebras of regular vector fields on tori

Details Kac-Moody algebras and Lie algebras of regular vector fields on tori Kac-Moody algebras and Lie algebras of regular vector fields on tori

2001-09-14

arxiv.org/abs/math/0109090v2

J.Algebra253:189-208,2002

Mathematics

Representation Theory

20 pages, typo corrected in the title

Scientific paper

10.1016/S0021-8693(02)00054-6

We consider the problem of representing the Kac-Moody algebra $\mathfrak{g}(N)$ specified by an $r\times r$ indecomposable generalised Cartan matrix $N$ as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if the representations are of a certain form, this is possible if and only if $\mathfrak{g}(N)\cong sl(r+1,{\bb C})$ or $\tilde{sl}(r,{\bb C})$. For $sl(r+1,{\bb C})$ and $\tilde{sl}(r,{\bb C})$, discrete families of representations are constructed. These generalise the well-known discrete families of representations of $sl(2,{\bb C})$ as regular vector fields on ${\bb C}^*$

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