Lie algebras MOC

Universal enveloping algebra

Let be a Lie algebra over . The universal enveloping algebra is the most general K-monoid with the Lie bracket of as its commutator, as formalized by the Universal property and the Poincaré-Birkhoff-Witt theorem. In particular, this means any Lie algebra representation of uniquely corresponds to a -module, motivating the abuse of terminology module over a Lie algebra.

Universal property

Let be a Lie algebra over . The universal enveloping algebra is a pair consisting of a K-monoid and a Lie algebra homomorphism ¹ such that given any K-monoid and Lie algebra homomorphism , there exists a unique unital algebra homomorphism such that the following diagram commutes: lie

has a unique extension to a functor such that becomes a natural transformation.

It is not immediately clear from the universal property that should be an injection, but this is guaranteed by the Poincaré-Birkhoff-Witt theorem, so indeed contains as a Lie subalgebra, whence every Lie algebra is a Lie subalgebra of some unital associative algebra.

Construction

Let be the tensor algebra of with inclusion and let be the (two-sided) ideal generated by any terms of the form

for . We construct the universal enveloping algebra as the quotient module

with its natural projection . The map .

Proof

proof

Graded structure

Let be a -graded Lie algebra. Then is a graded algebra such that . This is the same as the gradation given by the quotient graded algebra in the construction above.

Filtered structure

complete

---

tidy | en | sembr

Footnotes

1. As usual we regard an associative algebra as a Lie algebra under its commutator. ↩