PlanetMath: universal enveloping algebra

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universal enveloping algebra

(Definition)

A universal enveloping algebra of a Lie algebra $\mathfrak{g}$ over a field is an associative algebra (with unity) over , together with a Lie algebra homomorphism $\iota:\mathfrak{g} \rightarrow U$ (where the Lie algebra structure on is given by the commutator), such that if is a another associative algebra over and $\phi:\mathfrak{g} \rightarrow A$ is another Lie algebra homomorphism, then there exists a unique homomorphism $\psi:U \rightarrow A$ of associative algebras such that the diagram

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ \mathfrak{g} \ar[dr]_\phi \ar[r]^\iota & U \ar[d]^\psi\ & A} } \end{xy}$

commutes. Any $\mathfrak{g}$ has a universal enveloping algebra: let

be the associative tensor algebra generated by the vector space $\mathfrak{g}$ , and let

be the two-sided ideal of

generated by elements of the form

$\displaystyle xy-yx-[x,y]$ for $\displaystyle x,y \in \mathfrak{g};$

then

is a universal enveloping algebra of $\mathfrak{g}$ . Moreover, the universal property above ensures that all universal enveloping algebras of $\mathfrak{g}$ are canonically isomorphic; this justifies the standard notation $U(\mathfrak{g})$ .

Some remarks:

By the Poincaré-Birkhoff-Witt theorem, the map $\iota$ is injective; usually $\mathfrak{g}$ is identified with $\iota(\mathfrak{g})$ . From the construction above it is clear that this space generates $U(\mathfrak{g})$ as an associative algebra with unity.
By definition, the (left) representation theory of $U(\mathfrak{g})$ is identical to that of $\mathfrak{g}$ . In particular, any irreducible $\mathfrak{g}$ -module corresponds to a maximal left ideal of $U(\mathfrak{g})$ .

Example: let $\mathfrak{g}$ be the Lie algebra generated by the elements

and

with Lie bracket determined by

and

. Then

(where

denotes the two-sided ideal generated by

) is isomorphic to the skew polynomial algebra $k[x,\frac{\partial}{\partial x}]$ , the isomorphism being determined by

$\displaystyle p + (e-1)$	$\displaystyle \mapsto \frac{\partial}{\partial x}$ and
$\displaystyle q + (e-1)$	$\displaystyle \mapsto x.$

"universal enveloping algebra" is owned by draisma.

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Attachments:: Lie element (Definition) by Algeboy

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Cross-references: polynomial algebra, Lie bracket, left ideal, irreducible, theory, representation, generates, clear, injective, map, Poincaré-Birkhoff-Witt theorem, isomorphic, universal property, two-sided ideal, vector space, generated by, tensor algebra, algebras, algebra, commutator, structure, homomorphism, unity, associative, field, Lie algebra
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This is version 4 of universal enveloping algebra, born on 2002-09-18, modified 2006-03-22.
Object id is 3466, canonical name is UniversalEnvelopingAlgebra.
Accessed 5889 times total.

Classification:

AMS MSC:	17B35 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Universal enveloping algebras)
	16S30 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Universal enveloping algebras of Lie algebras)

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