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Higher genus affine algebras of Krichever-Novikov type. (English) Zbl 1115.17010
Summary: For higher genus multi-point current algebras of Krichever-Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are studied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a simple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the currents algebras.

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
14H55 Riemann surfaces; Weierstrass points; gap sequences
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. in quantum mechanics