“Universal” Lie algebra extensions and their Casimir invariants via commutative structures

We consider a special kind of Lie algebra extension, called “universal” extension, introduced recently. We show that there is one-to-one correspondence with commutative associative algebras. We also investigate the Casimir invariants for the “universal” extensions and consider in detail the so-called coextension structures, arising in the calculation of the Casimir functions.