Operators in the cohomology of nilpotent Lie algebras.

Abstract

Operators in the cohomology of Lie algebras are defined, and fundamental results are proven. The central representation is shown to be useful, in particular cases, for proving the Toral Rank Conjecture (TRC), which states the logarithm of the total dimension of any nilpotent Lie algebra's cohomology space is greater than or equal to the dimension of the centre. Central representation and secondary operators are used to find hypercube-like structures in the cohomology of the free two-step nilpotent Lie algebras with two, three and four generators. Also, a theorem about the new operators acting on the cohomology of the Heisenberg Lie algebras, and how these operators interact with Poincare duality for this case, is proven.