Universal Enveloping Algebra - Direct Construction

Direct Construction

From this universal property, one can prove that if a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by L (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of adjoint functors), we establish that indeed every Lie algebra does have a universal enveloping algebra.

Starting with the tensor algebra T(L) on the vector space underlying L, we take U(L) to be the quotient of T(L) made by imposing the relations

for all a and b in (the image in T(L) of) L, where the bracket on the RHS means the given Lie algebra product, in L.

Formally, we define

where I is the two-sided ideal of T(L) generated by elements of the form

The natural map L → T(L) descends to a map h : L → U(L), and this is the Lie algebra homomorphism used in the universal property given above.

The analogous construction for Lie superalgebras is straightforward.