$G$-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 $G$-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 $H_E(h)=H_E^{\bullet }(h,{\bigwedge }^{\bullet }h)$. This Poisson $SL(2)$-algebra has dimension 18. From these $SL(2)$-tables we deduce that the underlying Lie algebra of $H_E(h)$ 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 $H_E(h)$ contains a semisimple Lie subalgebra (in this case $sl(3)$) strictly larger than the Levi factor of $\text{Der}(h)$, which in this case is $sl(2)\subset H^1(h,h)$. This means that the Levi factor of the Lie algebra $H_E(h)$ 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 $gl(n)⋉gl(n{)}_{ab}$ (arbitrary $n$) such that, for $n=3$, is isomorphic to $H_E(h)$.