Mathematical Research Letters

Volume 22 (2015)

Number 4

Homology of analogues of Heisenberg Lie algebras

Pages: 1223 – 1241



Steven V Sam (Department of Mathematics, University of California at Berkeley)


We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work was to complete the calculations started by him. In all three cases, these algebras can be realized as the nilpotent radical of a parabolic subalgebra of a simple Lie algebra, and our first approach relies on a theorem of Kostant, but is otherwise elementary and involves combinatorics of Weyl groups and partitions which may be of independent interest. Our second approach is an application of (un)stable representation theory of the classical groups in the sense of recent joint work of the author with Snowden, which is shorter and more conceptual.

2010 Mathematics Subject Classification

05E10, 17B30, 17B56

Published 24 July 2015