Deformation of Koszul algebras and the Duflo Isomorphism theorem

2016/07/04 by Gerstenhaber, Murray
#16S30 (secondary) #16S37 #16S80 (primary) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1607.00917

Abstract

Let \mathfrak g be a finite dimensional Lie algebra over a field \mathbf k, U\mathfrak g be its enveloping algebra and S\mathfrak g be the symmetric algebra on \mathfrak g. Extending the work of Braverman and Gaitsgory on the deformation of Koszul algebras and the Poincaré-Birkhoff-Witt theorem we obtain a generalized Duflo isomorphism which is valid also over fields of finite characteristic: H_Lieⁿ(\mathfrak g, S\mathfrak g) ≅ H_Hochⁿ(U\mathfrak g,U\mathfrak g) for all n < char\mathbf k. This implies, in particular, that Duflo's classic theorem, which is the special case in characteristic zero of dimension zero, in fact holds in all characteristics and the generalized theorem holds whenever dim \mathfrak g < char \mathbf k.

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