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universal enveloping algebra
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(Definition)
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A universal enveloping algebra of a Lie algebra $\mathfrak{g}$ over a field $k$ is an associative algebra $U$ (with unity) over $k$ , together with a Lie algebra homomorphism $\iota:\mathfrak{g} \rightarrow U$ (where the Lie algebra structure on $U$ is given by the commutator), such that if $A$ is a another associative algebra over $k$ 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
commutes. Any $\mathfrak{g}$ has a universal enveloping algebra: let $T$ be the associative tensor algebra generated by the vector space $\mathfrak{g}$ , and let $I$ be the two-sided ideal of $T$ generated by elements of the form$$ xy-yx-[x,y] \text{ for } x,y \in \mathfrak{g};$$ then $U=T/I$ 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 $p,q,$ and $e$ with Lie bracket determined by $[p,q]=e$ and $[p,e]=[q,e]=0$ . Then $U(g)/(e-1)$ (where $(e-1)$ denotes the two-sided ideal generated by $e-1$ ) is isomorphic to the skew polynomial algebra $k[x,\frac{\partial}{\partial x}]$ , the isomorphism being determined by
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"universal enveloping algebra" is owned by draisma.
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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, diagram, algebras, algebra, commutator, structure, homomorphism, unity, associative, field, Lie algebra
There are 10 references to this entry.
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 7069 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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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ \mathfrak{g} \ar[dr]_\phi \ar[r]^\iota & U \ar[d]^\psi\ & A}$](http://images.planetmath.org:8080/cache/objects/3466/js/img1.png "$\displaystyle \xymatrix{ \mathfrak{g} \ar[dr]_\phi \ar[r]^\iota & U \ar[d]^\psi\ & A}$")
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