Degenerations of Generalized Krichever Novikov algebras on Tori

TL;DR

This paper investigates how Lie algebras of meromorphic vector fields on complex tori degenerate into subalgebras on the Riemann sphere, revealing how the number of marked points influences the algebraic structure.

Contribution

It introduces an algebraic geometric degeneration process for generalized Krichever-Novikov algebras on tori, explicitly determining subalgebras and their dependence on markings.

Findings

Degeneration process produces subalgebras on P^1 from elliptic curve algebras.

Number of markings affects the structure and classification of subalgebras.

Explicit descriptions provided for certain natural marking configurations.

Abstract

Degenerations of Lie algebras of meromorphic vector fields on elliptic curves (i.e. complex tori) which are holomorphic outside a certain set of points (markings) are studied. By an algebraic geometric degeneration process certain subalgebras of Lie algebras of meromorphic vector fields on P^1 the Riemann sphere are obtained. In case of some natural choices of the markings these subalgebras are explicitly determined. It is shown that the number of markings can change. AMS subject classification (1991): 17B66, 17B90, 14F10, 14H52, 30F30, 81T40