ABSTRACT
[1] Definition of the universal enveloping algebra of a Lie algebra: Let g be a Lie algebra and let (C, π) be a pair such that (a) C is an associative algebra, (b) π: g → C is a linear mapping satisfying π( [X, Y]) = π(X)π(Y) -π(Y)π(X)∀X, Y ∈ g , (c) π(g) generates C And (d) if U any associative algebra and ξ : g→U it is a linear map satisfying ξ([X, Y]) = ξ(X)ξ(Y)–ξ(Y)ξ(X)∀X,Y ∈ g, then there exists an algebra homomorphism ξ': C→U such that ξ'(π(X)) =ξ (X)∀X ∈ g. Then, (C, π) is called a universal enveloping algebra of g. Theoremml: If (πk, Ck), k = 1,2 are two universal enveloping algebras of a Lie algebra g, then they are isomorphic in the sense that there exists an algebra isomorphism ξ: C1 →C2 such thatξ (π1 (X)) = π2(X)∀X ∈ g.
