Cohomology of the infinite-dimensional Lie algebra 𝐿₁ with nontrivial coefficients

Abstract

with the bracket [ei, ej ] = (j − i)ei+j . The Lie algebra L is naturally graded, the degree of ei being i. The most natural modules over L are the so-called tensor field modules. A tensor field on the circle is of the form g(φ)(d/dφ). A vector field acts on this by infinitesimally changing the coordinate φ, where g(φ) is a section of some line bundle on the circle S with a flat connection. In the space of tensor fields we choose a basis fi, i ∈ Z, such that ei(fj) = (−λ(i + 1) + μ + j) · fi+j . Here λ, μ ∈ C are the invariants characterizing the module, i.e., the power of d/dφ and the logarithm of the monodromy of the flat connection. We denote such a module by Fλ,μ (see [3]). Denote by L1 the subalgebra of L with basis (e1, e2, e3, . . . ). It is easy to see that L1 is isomorphic to the Lie algebra of vector fields on the line, with polynomial coefficients, having a two-fold zero at the origin. The strategy of the cohomology computation for L1 with coefficients in the adjoint module is the following: we first compute the cohomology of L1 with coefficients in Fλ,μ, and then remark that the adjoint representation of L1 is a submodule of such an Fλ,μ. After this the spaces H(L1, L1) can easily be determined. The computations of H(L1, L1) and H (L1, L1) are contained in [3]. Deformations of L1 are studied in [5]. In this paper we shall describe a more general method for the computation of H∗(L1,Fλ,μ). It will be more convenient for us to deal with homology instead of cohomology. It is easy to see that H ∗ (L1,Fλ,μ) is dual to H∗(L1,F−1−λ,−μ). Then, using the fact that L∗1 is the factor of some Fλ,μ, we can compute Hi(L1, L ∗ 1). Notice that for almost every (λ, μ) the module Fλ,μ is an irreducible representation of L, and L1 is the maximal nilpotent subalgebra in L. That means that the problem of determining H∗(L1,Fλ,μ) is analogous to that of determining the cohomology of the maximal nilpotent subalgebra of a complex semisimple Lie