Editing Universal enveloping algebra (section)

==Informal construction==
The idea of the universal enveloping algebra is to embed a Lie algebra <math>\mathfrak{g}</math> into an associative algebra <math>\mathcal{A}</math> with identity in such a way that the abstract bracket operation in <math>\mathfrak{g}</math> corresponds to the commutator <math>xy-yx</math> in <math>\mathcal{A}</math> and the algebra <math>\mathcal{A}</math> is generated by the elements of <math>\mathfrak{g}</math>. There may be many ways to make such an embedding, but there is a unique "largest" such <math>\mathcal{A}</math>, called the universal enveloping algebra of <math>\mathfrak{g}</math>.

===Generators and relations===
Let <math>\mathfrak{g}</math> be a Lie algebra, assumed finite-dimensional for simplicity, with basis <math>X_1,\ldots X_n</math>. Let <math>c_{ijk}</math> be the [[structure constants]] for this basis, so that
:<math>[X_i,X_j]=\sum_{k=1}^n c_{ijk}X_k.</math>
Then the universal enveloping algebra is the associative algebra (with identity) generated by elements <math>x_1,\ldots x_n</math> subject to the relations
:<math>x_i x_j -                          x_j x_i=\sum_{k=1}^n c_{ijk}x_k</math>
and ''no other relations''. Below we will make this "generators and relations" construction more precise by constructing the universal enveloping algebra as a quotient of the tensor algebra over <math>\mathfrak g</math>.

Consider, for example, the Lie algebra  [[SL(2,C)|sl(2,C)]], spanned by the matrices
:<math display="block"> E = \begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}
\qquad
F = \begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix}
\qquad
H = \begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix}  ~,</math>
which satisfy the commutation relations <math>[H,E]=2E</math>, <math>[H,F]=-2F</math>, and <math>[E,F]=H</math>. The universal enveloping algebra of sl(2,C) is then the algebra generated by three elements <math>e,f,h</math> subject to the relations
:<math>he-eh=2e,\quad hf-fh=-2f,\quad ef-fe=h,</math>
and no other relations. We emphasize that the universal enveloping algebra ''is not'' the same as (or contained in) the algebra of <math>2\times 2</math> matrices. For example, the <math>2\times 2</math> matrix <math>E</math> satisfies <math>E^2=0</math>, as is easily verified. But in the universal enveloping algebra, the element <math>e</math> does not satisfy <math>e^2=0</math> because we do not impose this relation in the construction of the enveloping algebra. Indeed, it follows from the Poincaré–Birkhoff–Witt theorem (discussed [[#Poincaré–Birkhoff–Witt theorem|§ below]]) that the elements <math>1,e,e^2,e^3,\ldots</math> are all linearly independent in the universal enveloping algebra.

===Finding a basis===
In general, elements of the universal enveloping algebra are linear combinations of products of the generators in all possible orders. Using the defining relations of the universal enveloping algebra, we can always re-order those products in a particular order, say with all the factors of <math>x_1</math> first, then factors of <math>x_2</math>, etc. For example, whenever we have a term that contains <math>x_2 x_1</math> (in the "wrong" order), we can use the relations to rewrite this as <math>x_1 x_2</math> plus a [[linear combination]] of the <math>x_j</math>'s. Doing this sort of thing repeatedly eventually converts any element into a linear combination of terms in ascending order. Thus, elements of the form
:<math>x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}</math>
with the <math>k_j</math>'s being non-negative integers, span the enveloping algebra. (We allow <math>k_j=0</math>, meaning that we allow terms in which no factors of <math>x_j</math> occur.) The [[Poincaré–Birkhoff–Witt theorem]], discussed below, asserts that these elements are linearly independent and thus form a basis for the universal enveloping algebra. In particular, the universal enveloping algebra is always infinite dimensional.

The Poincaré–Birkhoff–Witt theorem implies, in particular, that the elements <math>x_1,\ldots, x_n</math> themselves are linearly independent. It is therefore common—if potentially confusing—to identify the <math>x_j</math>'s with the generators <math>X_j</math> of the original Lie algebra. That is to say, we identify the original Lie algebra as the subspace of its universal enveloping algebra spanned by the generators. Although <math>\mathfrak{g}</math> may be an algebra of <math>n\times n</math> matrices, the universal enveloping of <math>\mathfrak{g}</math> does not consist of (finite-dimensional) matrices. In particular, there is no finite-dimensional algebra that contains the universal enveloping of <math>\mathfrak{g}</math>; the universal enveloping algebra is always infinite dimensional. Thus, in the case of sl(2,C), if we identify our Lie algebra as a subspace of its universal enveloping algebra, we must not interpret <math>E</math>, <math>F</math> and <math>H</math> as <math>2\times 2</math> matrices, but rather as symbols with no further properties (other than the commutation relations).

===Formalities===
The formal construction of the universal enveloping algebra takes the above ideas, and wraps them in notation and terminology that makes it more convenient to work with. The most important difference is that the free associative algebra used in the above is narrowed to the [[tensor algebra]], so that the product of symbols is understood to be the [[tensor product]]. The commutation relations are imposed by constructing a [[Quotient space (linear algebra)|quotient space]] of the tensor algebra quotiented by the ''smallest'' [[two-sided ideal]] containing elements of the form <math>x_i x_j -x_j x_i-\Sigma c_{ijk}x_k</math>. The universal enveloping algebra is the "largest" [[unital associative algebra]] generated by elements of <math>\mathfrak g</math> with a [[Lie bracket]] compatible with the original Lie algebra.