2014/07/02 by Vincent E. Coll, Vincent E. Coll Jr., Coll, Vincent E. +2
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.RA #msc:17B56 #msc:32G05
paper · pdf · doi:10.48550/arxiv.1407.0428
Revised, errors corrected, 16 pages
arxiv created 2015/01/22 · arxiv updated 2015/01/23
When \mathfrak h is a toral subalgebra of a Lie algebra \mathfrak g over a field \mathbf k, and M a \mathfrak g-module on which \mathfrak h also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product \mathfrak g = \mathfrak h \ltimes \mathfrak k with \mathfrak h toral one has H^*(\mathfrak g, M) ≅ \bigwedge\mathfrak h\vee \bigotimes H^*(\mathfrak k,M)\mathfrak h = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)\mathfrak h, and for a Lie poset algebra \mathfrak g, that H^*(\mathfrak g, \mathfrak g), which controls the deformations of \mathfrak g, can be computed from the nerve of the underlying poset. The deformation theory of Lie poset algebras, analogous to that of complex analytic manifolds for which it is a small model, is illustrated by examples.