Electrical Lie algebra

In my research, I build upon the results of my previous year's coursework. In it, I considered a special electrical Lie algebra and its representation in n x n matrices. It turned out that this representation is also a non-standard representation of the standard symplectic Lie algebra. The main results of the work were the description of the basis of the special electrical Lie algebra, the Pieri formula, and the description of the center in the odd case. In my thesis, I investigate the symplectic electrical Lie algebra. I prove that this algebra lies in the Lie algebra G, witch preserving two symplectic forms. The main result of the study is to prove the existence of a decomposition of the bi-symplectic algebra G into a direct sum of two special electrical Lie algebras. Moreover, it turns out that G is semisimple for any n. I compare my results with those obtained by Daniel Goldstein and Robert M. Guralnick, who studied a group preserving two symplectic forms.