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)ab with the first factor acting on the second (abelian) one by the adjoint representation. We find it remarkable that the Lie algebra structure on HE(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)⊂ H¹(h,h). This means that the Levi factor of the Lie algebra HE(h) has nontrivial elements outside H¹(h,h). Finally, this leads us to find a family of commutative Poisson algebras whose underlying Lie structure is gl(n) gl(n)ab (arbitrary n) such that, for $n=3$, is isomorphic to HE(h).