Cohomology of the infinite-dimensional Lie algebra $L_1$ with nontrivial coefficients

Cohomology and Deformations of the Infinite Dimensional Filiform Lie Algebra M0 Alice Fialowski and Friedrich Wagemann

Denote m0 the infinite dimensional N -graded Lie algebra defined by basis ei i ≥ 1 and relations [e1, ei] = ei+1 for all i ≥ 2. We compute in this article the bracket structure on H(m0, m0) , H (m0, m0) and in relation to this, we establish that there are only finitely many true deformations of m0 in each nonpositive weight by constructing them explicitely. It turns out that in weight 0 one get...

Cohomology and Deformations of the Infinite Dimensional Filiform Lie Algebra M2 Alice Fialowski and Friedrich Wagemann

Let m2 denote the infinite dimensional N-graded Lie algebra defined by the basis ei for i ≥ 1 and by relations [e1, ei] = ei+1 for all i ≥ 2, [e2, ej ] = ej+2 for all j ≥ 3. We compute in this article the bracket structure on H(m2;m2), H (m2;m2) and in relation to this, we establish that there are only finitely many true deformations of m2 in each weight by constructing them explicitely. It tur...

The Lie algebra cohomology of jets

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

Lie Algebra Cohomology and the Representations Ofsemisimple Lie Groups

Infinite Dimensional Lie Groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an ‘evolution operator’ exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and fla...