Differential cohomology of \(C^\infty ({\mathfrak G}^*)\)
Let \({\mathfrak g}\) be a finite dimensional Lie algebra. It is well known that the dual \({\mathfrak g}^*\) of \({\mathfrak g}\) carries a Poisson structure, and consequently on the vector space N of all smooth functions on \({\mathfrak g}^*\) we can introduce a Lie algebra structure. Motivated by the problem of existence and classification of differential *-products on \({\mathfrak g}^*\), the author investigates the Chevalley cohomology of N, i.e. the differentiable cohomology of the Lie algebra N with coefficients in N with respect to the adjoint representation. We shall denote it by \(H^*_{diff}(N,\partial)\). He expresses \(H^ 1_{diff}(N,\partial)\) and \(H^ 2_{diff}(N,\partial)\) in terms of \(H^ 1({\mathfrak g},N)\) and \(H^ 2({\mathfrak g},N)\) respectively, where \(H^*({\mathfrak g},N)\) denotes the cohomology of the Lie algebra \({\mathfrak g}\) with coefficients in N with respect to the natural representation of \({\mathfrak g}\) in N. The author obtains the same results when replacing N by the vector space of all polynomial functions on \({\mathfrak g}^*.\) The methods applied in the paper represent a nontrivial modification of the methods used for symplectic manifolds by \textit{S. Gutt} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 33, 1-31 (1980; Zbl 0476.53021)].
scientific article; zbMATH DE number 4120360
scientific article