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: ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \mathrm {U} ({\mathfrak {g}})}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}\to \mathrm {U} ({\mathfrak {g}})}$

If a unitary associative algebra, then the Lie algebra homomorphisms are in bijection with the unitary algebra homomorphisms . This bijection is mediated by homomorphism . ${\displaystyle A}$ ${\displaystyle {\mathfrak {g}}\to A}$ ${\displaystyle \mathrm {U} ({\mathfrak {g}})\to A}$ ${\displaystyle {\mathfrak {g}}\to \mathrm {U} ({\mathfrak {g}})}$

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 ${\displaystyle X_{1},\ldots ,X_{n}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle i\colon {\mathfrak {g}}\to \mathrm {U} ({\mathfrak {g}})}$

${\displaystyle i(X_{i_{1}})i(X_{i_{2}})\cdots i(X_{i_{k}})}$ With ${\displaystyle i_{1}\leq i_{2}\leq \ldots \leq i_{k}}$

a base of . ${\displaystyle \mathrm {U} ({\mathfrak {g}})}$

• In particular, is injective , and every Lie algebra is a subalgebra of an associative algebra. ${\displaystyle i}$
• 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 ${\displaystyle \mathrm {T} {\mathfrak {g}}}$

${\displaystyle X\otimes Y-Y\otimes X-[X,Y]}$

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. ${\displaystyle X,Y\in {\mathfrak {g}}}$ ${\displaystyle \mathrm {U} ({\mathfrak {g}})}$

Examples

• If Abelian, the universal enveloping algebra is isomorphic to the symmetric algebra over . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$