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

In the first part of the paper, we define the concept of a G-table 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- (co) algebra A is a set of scalars that provides very 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. From the G-table of a G-algebra A we define a plain algebra P (A) associated to it and we present some basic functoriality results about P. Obtaining the G-table of a given G-algebra A requires a considerable amount of work but, the result, is a very powerful tool as shown in the second part of the paper. Here we compute the SL (2) -tables of the Poisson algebra structure of the even-degree part of the cohomology associated to the cotangent bundle of the 3-dimensional Heisenberg Lie group with Lie algebra h, that is HE (h) =HE^ (h, ^h). This Poisson SL (2) -algebra has dimension 18. From these SL (2) -tables we deduce that the underlying Lie algebra of HE (h) is isomorphic to gl (3) gl (3) ₀₁ with the first factor acting on the second (abelian) one by the adjoint representation. We find it remarkable that the Lie algebra structure on H₄ (h) contains a semisimple Lie subalgebra (in this case sl (3) ) strictly larger than the Levi factor of Der (h), which in this case is sl (2) H^1 (h, h). This means that the Levi factor of the Lie algebra H₄ (h) has nontrivial elements outside H^1 (h, h). Finally, this leads us to find a family of commutative Poisson algebras whose underlying Lie structure is gl (n) gl (n) ₀₁ (arbitrary n) such that, for n=3, is isomorphic to HE (h).