Complex Lie algebras corresponding to weighted projective lines

TL;DR

This paper provides an alternative proof of Kac's theorem for weighted projective lines over the complex numbers, utilizing geometric realization of complex Lie algebras from derived categories.

Contribution

It offers a new proof approach for Kac's theorem in the context of weighted projective lines, connecting geometric realization and Lie algebra theory.

Findings

Alternative proof of Kac's theorem established

Geometric realization of Lie algebras applied

Connections between derived categories and Lie algebras elucidated

Abstract

The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines (\cite{W}) over the complex field. The geometric realization of complex Lie algebras arising from derived categories (\cite{XXZ}) is essentially used.