Universal enveloping algebra
The universal enveloping algebra (also universal enveloping ) is a term from the mathematical branch of the theory of Lie algebras . It is an associative algebra that shows that the Lie bracket can always be understood as a commutator , even with Lie algebras that do not come from an associative algebra.
definition
Let it be a Lie algebra (over a field). A universal enveloping algebra of consists of a unitary associative algebra and a Lie algebra homomorphism (the Lie algebra structure on associative algebras is given by the commutator ), so that:
- If a unitary associative algebra, then the Lie algebra homomorphisms are in bijection with the unitary algebra homomorphisms . This bijection is mediated by homomorphism .
properties
- The most important statement about universal enveloping algebras is the Poincaré-Birkhoff-Witt theorem (after Henri Poincaré , Garrett Birkhoff and Ernst Witt ; also abbreviated as PBW ): If a base of and is the canonical mapping, then the monomials form
- With
- a base of .
- In particular, is injective , and every Lie algebra is a subalgebra of an associative algebra.
- Modules under a Lie algebra are the same as modules under its universal enveloping algebra .
construction
The universal envelope can be stated explicitly as the quotient of the tensor algebra according to the two-sided ideal, that of elements of form
for is generated. Note: In contrast to the corresponding constructions of external algebra or symmetrical algebra , this ideal is not homogeneous, so it does not have an induced graduation.
Examples
- If Abelian, the universal enveloping algebra is isomorphic to the symmetric algebra over .