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

TL;DR

This paper introduces the concept of G-tables for G-(co)algebras, computes these tables for specific structures, and applies them to analyze the Poisson algebra of the cotangent bundle of the Heisenberg Lie group, revealing its Lie algebra structure.

Contribution

It defines G-tables for G-(co)algebras, computes them for certain cases, and uses these to analyze the Poisson structure of the cotangent bundle of the Heisenberg Lie group, uncovering new algebraic insights.

Findings

The G-table of a G-(co)algebra encodes detailed algebraic and G-module structure.

The Poisson algebra of the cotangent bundle of the Heisenberg group has an underlying Lie algebra isomorphic to gl(3)⋉gl(3)_{ab}.

The Lie algebra structure contains a larger semisimple subalgebra than the Levi factor, revealing complex internal structure.

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 $gl(3)\ltimes gl(3)_{ab}$ 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 $gl(n)\ltimes gl(n)_{ab}$ (arbitrary $n$) such that, for $n=3$, is isomorphic to $H_E(h)$.