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-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 $P\left(A\right)$ 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 $\mathrm{SL}(2,K)$-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 $H_E^{•,•}\left(h\right)=H_E^{•}(h,{\bigwedge }^{•}h)$. This Poisson $\mathrm{SL}(2,K)$-algebra has dimension 18. From these $\mathrm{SL}(2,K)$-tables we deduce that the underlying Lie algebra of $H_E^{•,•}\left(h\right)$ is isomorphic to $\mathrm{gl}(3,K)⋉\mathrm{gl}{(3,K)}_{ab}$ with the first factor acting on the second (abelian) factor by the adjoint representation. It is notable that the Lie algebra structure on $H_E^{•,•}\left(h\right)$ contains a semisimple Lie subalgebra (in this case $\mathrm{sl}(3,K)$), that is strictly larger than the Levi factor of $\text{Der}\left(h\right)$, which in this case is $\mathrm{sl}(2,K)\subset H^1(h,h)$. This implies that the Levi factor of the Lie algebra $H_E^{•,•}\left(h\right)$ has nontrivial elements outside $H^1(h,h)$. Finally, this leads us to identify a family of commutative Poisson algebras whose underlying Lie structure are $\mathrm{gl}(n,K)⋉\mathrm{gl}{(n,K)}_{ab}$ for arbitrary n. In the special case $n=3$, it is isomorphic to $H_E^{•,•}\left(h\right)$.