Koszul dual of a commutative algebra

General observation:

Consider the case when is a commutative algebra. This means that includes all antisymmetric tensors.

⇒

is a universal enveloping of a Lie algebra.

Example: If is the algebra of polynomials of variables , then the dual Lie algebra is the algebra of antisymmetric polynomials . This is the universal enveloping of the Lie superalgebra .