Cohomology of Infinite-Dimensional Lie Algebras

Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach

“…The Lie algebra of the Virasoro-Bott Lie group is thus the central extension R × ω g of g induced by this cocycle. We have H 2 (X c (M ), R) = 0 for each finite dimensional manifold of dimension ≥ 2 (see [21]), which blocks the way to find a higher dimensional analog of the Korteweg -de Vries equation in a way similar to that sketched below. For further use we also note the expression for the adjoint action on the Virasoro-Bott groups which we computed along the way.…”

On the Projective Geometry of the Supercircle: A Unified Construction of the Super Cross-Ratio and Schwarzian Derivative

“…From these invariants we can obtain, using Cartan-like formulae, three 1-cocycles of Diff + (S 1 ) with coefficients in some tensorial density modules F λ (S 1 ) with λ ∈ R; see [9]. They are the generators of the three nontrivial cohomology spaces H 1 (Diff + (S 1 ), F λ ), with λ = 0, 1, 2, as proved in [12].…”

“…From these invariants we can obtain, using Cartan-like formulae, three 1-cocycles of Diff + (S 1 ) with coefficients in some tensorial density modules F λ (S 1 ) with λ ∈ R; see [9]. They are the generators of the three nontrivial cohomology spaces H 1 (Diff + (S 1 ), F λ ), with λ = 0, 1, 2, as proved in [12].The purpose of this article is to extend these results to the supercircle, S 1|N , en- Quite independently, and from a more mathematical point of view, Manin [23] introduced the even and odd cross-ratios, for N = 1, 2, by resorting to linear super- …”

On the First aff(1)-Relative Cohomology of the Lie Algebra of Vector Fields and Differential Operators

“…that form a Lie subalgebra of Vect() isomorphic to the Lie algebra aff (1). This realization of aff(1) is understood throughout this paper.…”

“…0→→.→→0 are classified by the first cohomology group H 1 (g; Hom(,)) [1]. Any 1-cocycle  generates a new action on ⊕ as follows: for all g∈g and for all (a,b)∈⊕, we define g …”