G-tables and the Poisson structure of the even cohomology of cotangent bundle of the Heisenberg Lie group

Abstract

In the first part of the paper, we define the concept of a Gtable of a G-(co)algebra and we compute the G-table of some G-(co)algebras (here, a G-algebra is an algebra on which G acts, semisimply, by algebra automorphisms). The G-table of a G-algebra A is a set of scalars that provides precise and concise information about both the algebra structure and the G-module structure of A. In particular, the ordinary multiplication table of A can be derived from the G-table of A. Using the G-table of a G-algebra A, we define an associated plain algebra P(A) and present some basic functoriality results related to P. Obtaining the G-table of a given G-algebra A requires significant work, but the result is a very powerful tool, as shown in the second part of the paper. Here, we compute the SL(2, K)-tables of the Poisson algebra structure of the evendegree part of the cohomology associated to the cotangent bundle of the 3-dimensional Heisenberg Lie group with Lie algebra h, that is H•,• E (h) = H• E(h, • h). This Poisson SL(2, K)-algebra has dimension 18. From these SL(2, K)tables we deduce that the underlying Lie algebra of H•,• E (h) is isomorphic to gl(3, K) ⋉ gl(3, K)ab with the first factor acting on the second (abelian) factor by the adjoint representation. It is notable that the Lie algebra structure on H•,• E (h) contains a semisimple Lie subalgebra (in this case sl(3, K)), that is strictly larger than the Levi factor of Der(h), which in this case is sl(2, K) ⊂ H1(h, h). This implies that the Levi factor of the Lie algebra H•,• E (h) has nontrivial elements outside H1(h, h). Finally, this leads us to identify a family of commutative Poisson algebras whose underlying Lie structure are gl(n, K) ⋉ gl(n, K)ab for arbitrary n. In the special case n = 3, it is isomorphic to H•,• E (h).