zbMATH — the first resource for mathematics

The fermion model of representations of affine Krichever-Novikov algebras. (English. Russian original) Zbl 1013.17020
Funct. Anal. Appl. 35, No. 3, 209-219 (2001); translation from Funkts. Anal. Prilozh. 35, No. 3, 60-72 (2001).
Affine Krichever-Novikov algebras are generalizations of affine Kac-Moody algebras for algebraic curves of higher genus. The affine Kac-Moody case corresponds to the genus zero situation. These algebras are almost-graded. The concept of almost-gradedness allows to define highest weight representations, Verma modules, etc. In this article explicit realizations of such representations are constructed. To a generic holomorphic vector bundle over the curve and an irreducible finite-dimensional representation of a semi-simple Lie algebra the author assigns a representation of the corresponding affine Krichever-Novikov algebra in the space of semi-infinite exterior forms (fermionic forms, wedge forms). They are defined with respect to a certain basis of a space of functions which is associated to the (framed) bundle. These bases generalize the Krichever-Novikov bases for the functions and the forms of arbitrary weights. Equivalent pairs of data determine equivalent representations and vice versa.

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
17B68 Virasoro and related algebras
14H60 Vector bundles on curves and their moduli
22E67 Loop groups and related constructions, group-theoretic treatment
Full Text: DOI arXiv