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Second order Casimirs for the affine Krichever-Novikov algebras \(\widehat{\mathfrak {gl}}_{g,2}\) and \(\widehat{\mathfrak {sl}}_{g,2}\). (English) Zbl 1123.17300
Summary: The second order Casimirs for the affine Krichever-Novikov algebras \(\widehat{\mathfrak{gl}}g,2\) and \(\widehat{\mathfrak{sl}}g,2\) are described. More general operators which we call semi-Casimirs are introduced. It is proven that the semi-Casimirs induce well-defined operators on conformal blocks and, for a certain moduli space of Riemann surfaces with two marked points and fixed jets of local coordinates, there is a natural projection of its tangent space onto the space of these operators.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie 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
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