Almost-graded central extensions of Lax operator algebras
Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for 𝔤𝔩(n), with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. Some results are joint work with Oleg Sheinman.
- 17B65: Infinite-dimensional Lie (super)algebras
- 14H60: Vector bundles on curves and their moduli
- 14H55: Riemann surfaces; Weierstrass points; gap sequences
- 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
- 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, W -algebras and other current algebras and their representations
- 81T40: Two-dimensional field theories, conformal field theories, etc.
- 14H70: Relationships with integrable systems
- 30F30: Differentials on Riemann surfaces
- 14D20: Algebraic moduli problems, moduli of vector bundles
- 17B80: Applications to integrable systems