The Lie algebra of sl(2)-valued automorphic functions on a torus

TL;DR

This paper characterizes the Lie algebra of automorphic meromorphic sl(2,C)-valued functions on a torus, showing it as a geometric realization of an infinite-dimensional finitely generated Lie algebra, and explores its limits.

Contribution

It introduces a geometric realization of the Lie algebra of automorphic functions on a torus and analyzes its relation to loop algebras in the trigonometric limit.

Findings

The Lie algebra is a geometric realization of a finitely generated infinite-dimensional algebra.

In the trigonometric limit, it reduces to the sl(2,C) loop algebra.

The algebra transitions to a quotient of the affine sl(2) algebra in the limit.

Abstract

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, C) -valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2,C) -valued loop algebra, while the latter one - into the Lie algebra (sl(2)^)'/(centre) .