Editing Universal enveloping algebra (section)

===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.