Superalgebras associated to Riemann surfaces: Jordan algebras of Krichever-Novikov type

TL;DR

This paper constructs and explores two superalgebras linked to punctured Riemann surfaces, revealing their algebraic, geometric, and representation-theoretic relationships, notably showing the Lie superalgebra as derivations of the Jordan superalgebra.

Contribution

It introduces a novel Jordan superalgebra associated with Riemann surfaces and establishes its connection with a Krichever-Novikov type Lie superalgebra.

Findings

The Lie superalgebra acts as derivations of the Jordan superalgebra.

The constructed superalgebras are related through algebraic and geometric structures.

The work advances understanding of superalgebra structures in complex geometry.

Abstract

We construct two superalgebras associated to a punctured Riemann surface. One of them is a Lie superalgebra of Krichever-Novikov type, the other one is a Jordan superalgebra. The constructed algebras are related in several ways (algebraic, geometric, representation theoretic). In particular, the Lie superalgebra is the algebra of derivations of the Jordan superalgebra.