Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

Abstract

We compute the homology of the Lie algebra W1 of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W1-valued currents on the line.