[Paper Review] Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras

Martin Schlichenmaier, О. К. Шейнман|ArXiv.org|Dec 18, 1995

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper generalizes the Sugawara construction to Krichever-Novikov algebras of affine type, deriving representations of centrally extended Krichever-Novikov vector field algebras from highest weight modules. It establishes weight relations between representations and constructs Casimir operators, extending classical Virasoro algebra results to higher-genus settings with multi-point structures in an appendix.

ABSTRACT

We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.

Motivation & Objective

• To extend the Sugawara construction from classical affine Kac-Moody algebras to Krichever-Novikov algebras of higher genus.
• To derive representations of centrally extended Krichever-Novikov vector field algebras from highest weight modules of affine type.
• To establish explicit relations between the weights of the original affine algebra representations and the resulting vector field algebra representations.
• To construct Casimir operators for the centrally extended Krichever-Novikov vector field algebras.
• To generalize the construction to the multi-point setting, including an appendix on the multi-point Sugawara construction.

Proposed method

• Utilizes highest weight representations of Krichever-Novikov algebras of affine type as starting points.
• Applies the Sugawara construction to generate new representations of centrally extended Krichever-Novikov vector field algebras.
• Derives weight relations between the original affine algebra representations and the resulting vector field algebra representations via the Sugawara construction.
• Constructs Casimir operators for the centrally extended Krichever-Novikov vector field algebras using the Sugawara formula and representation-theoretic techniques.
• Extends the construction to the multi-point case by adapting the Sugawara formula to Riemann surfaces with multiple marked points.
• Employs algebraic geometry and representation theory tools, particularly in the context of higher-genus Riemann surfaces.

Experimental results

Research questions

• RQ1 How can the Sugawara construction be generalized from the Virasoro algebra to Krichever-Novikov algebras of higher genus?
• RQ2 What is the precise relation between the weights of highest weight representations of affine Krichever-Novikov algebras and the weights of the resulting vector field algebra representations?
• RQ3 Can Casimir operators be constructed for centrally extended Krichever-Novikov vector field algebras using the Sugawara method?
• RQ4 How does the Sugawara construction behave in the multi-point setting on higher-genus Riemann surfaces?
• RQ5 What are the structural and representation-theoretic implications of extending the Sugawara construction to Krichever-Novikov algebras?

Key findings

• The Sugawara construction successfully produces representations of centrally extended Krichever-Novikov vector field algebras from highest weight modules of affine Krichever-Novikov algebras.
• Explicit relations between the weights of the original affine algebra representations and the resulting vector field algebra representations are derived.
• Casimir operators are constructed for the centrally extended Krichever-Novikov vector field algebras, generalizing the classical case.
• The construction is generalized to the multi-point setting, with a detailed treatment provided in an appendix.
• The method preserves the algebraic structure and representation-theoretic properties across the transition from affine to vector field algebras in higher-genus contexts.