Low-dimensional cohomology of current Lie algebras

We deal with low-dimensional homology and cohomology of current Lie algebras, i.e., Lie algebras which are tensor products of a Lie algebra L and an associative commutative algebra A. We derive, in two different ways, a general formula expressing the second cohomology of current Lie algebra with coefficients in the trivial module through cohomology of L, cyclic cohomology of A, and other invariants of L and A. The first proof is achieved by using the Hopf formula expressing the second homology of a Lie algebra in terms of its presentation. The second proof employs a certain linear-algebraic technique, ideologically similar to “separation of variables” of differential equations. We also obtain formulas for the first and, in some particular cases, for the second cohomology of the current Lie algebra with coefficients in the “current” module, and the second cohomology with coefficients in the adjoint module in the case where L is the modular Zassenhaus algebra. Applications of these results include: description of modular semi-simple Lie algebras with a solvable maximal subalgebra; computations of structure functions for manifolds of loops in compact Hermitian symmetric spaces; a unified treatment of periodizations of semi-simple Lie algebras, derivation algebras (with prescribed semi-simple part) of nilpotent Lie algebras, and presentations of affine Kac-Moody algebras