Editing Universal enveloping algebra (section)

==Examples in particular cases==
If <math>\mathfrak{g} = \mathfrak{sl}_2</math>, then it has a basis of matrices<blockquote><math>H = \begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}, \text{ }
E = \begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix}, \text{ }
F = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}</math></blockquote>which satisfy the following identities under the standard bracket:<blockquote><math>[H,E] = -2E</math>, <math>[H,F] = 2F</math>, and <math>[E,F] = - H </math></blockquote>this shows us that the universal enveloping algebra has the presentation<blockquote><math>U(\mathfrak{sl}_2) = \frac{\mathbb{C}\langle x,y,z\rangle}{(xy - yx + 2y, xz - zx - 2z, yz - zy + x)}</math></blockquote>as a non-commutative ring.

If <math>\mathfrak{g}</math> is ''abelian'' (that is, the bracket is always {{math|0}}), then <math>U(\mathfrak{g})</math> is commutative; and if a [[basis (linear algebra)|basis]] of the [[vector space]] <math>\mathfrak{g}</math>  has been chosen, then <math>U(\mathfrak{g})</math> can be identified with the [[polynomial]] algebra over {{math|''K''}}, with one variable per basis element.

If <math>\mathfrak{g}</math> is the Lie algebra corresponding to the [[Lie group]] {{math|''G''}}, then <math>U(\mathfrak{g})</math> can be identified with the algebra of left-invariant [[differential operator]]s (of all orders) on {{math|''G''}}; with <math>\mathfrak{g}</math> lying inside it as the left-invariant [[vector field]]s as first-order differential operators.

To relate the above two cases: if <math>\mathfrak{g}</math> is a vector space {{math|''V''}} as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the [[partial derivative]]s of first order.

The center <math>Z(\mathfrak{g})</math> consists of the left- and right- invariant differential operators; this, in the case of {{math|''G''}} not commutative, is often not generated by first-order operators (see for example [[Casimir operator]] of a semi-simple Lie algebra).

Another characterization in Lie group theory is of <math>U(\mathfrak{g})</math> as the [[convolution]] algebra of [[Distribution (mathematics)|distribution]]s [[Support (mathematics)#Support of a distribution|support]]ed only at the [[identity element]] {{math|''e''}} of {{math|''G''}}.

The algebra of differential operators in {{math|''n''}} variables with polynomial coefficients may be obtained starting with the Lie algebra of the [[Heisenberg group]]. See [[Weyl algebra]] for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.

The universal enveloping algebra of a finite-dimensional Lie algebra is a filtered [[quadratic algebra]].