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 21 is a set of scalars that provides precise and concise information about both the algebra structure and the G-module structure of 21. In particular, the ordinary multiplication table of 21 can be derived from the G-table of 21. Using the G-table of a G-algebra 21, we define an associated plain algebra P(21) 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 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(center dot,center dot)(h)=H-E(center dot)(h,boolean AND(center dot)h). This Poisson SL(2,K)-algebra has dimension 18. From these SL(2,K)-tables we deduce that the underlying Lie algebra of H-E(center dot,center dot)(h) is isomorphic to gl(3,K) (sic) 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(center dot,center dot)(h) contains a semisimple Lie subalgebra (in this case sl(3,K)), that is strictly larger than the Levi factor of Der(h), which in this case is sl(2,K)subset of H-1(h,h). This implies that the Levi factor of the Lie algebra H-E(center dot,center dot)(h) 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 gl(n,K) (sic) gl(n,K)(ab) for arbitrary n. In the special case n=3, it is isomorphic to H-E(center dot,center dot)(h). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
