Editing Universal enveloping algebra (section)

==Universal property==
The universal enveloping algebra, or rather the universal enveloping algebra together with the canonical map  <math>h\colon\mathfrak{g}\to U(\mathfrak{g})</math>, possesses a [[universal property]].<ref>{{harvnb|Hall|2015}} Theorem 9.7</ref> Suppose we have any Lie algebra map
:<math>\varphi: \mathfrak{g} \to A</math>
to a unital associative algebra {{math|''A''}} (with Lie bracket in {{math|''A''}} given by the commutator). More explicitly, this means that we assume
:<math>\varphi([X,Y])=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X)</math>
for all <math>X,Y\in\mathfrak{g}</math>. Then there exists a ''unique'' unital [[algebra homomorphism]]

:<math>\widehat\varphi\colon U(\mathfrak{g}) \to A</math>
such that
:<math>\varphi = \widehat \varphi \circ h </math>

where <math>h:\mathfrak{g}\to U(\mathfrak{g})</math> is the canonical map. (The map <math>h</math> is obtained by embedding <math>\mathfrak{g}</math> into its [[tensor algebra]] and then composing with the [[Quotient space (linear algebra)|quotient map]] to the universal enveloping algebra. This map is an embedding, by the Poincaré–Birkhoff–Witt theorem.)

To put it differently, if <math>\varphi\colon\mathfrak{g}\rightarrow A</math> is a linear map into a unital algebra <math>A</math> satisfying <math>\varphi([X,Y])=\varphi(X)\varphi(Y)-\varphi(Y)\varphi(X)</math>, then <math>\varphi</math> extends to an algebra homomorphism of <math>\widehat\varphi: U(\mathfrak{g}) \to A</math>. Since <math> U(\mathfrak{g})</math> is generated by elements of <math>\mathfrak{g}</math>, the map <math>\widehat{\varphi}</math> must be uniquely determined by the requirement that
:<math>\widehat{\varphi}(X_{i_1}\cdots X_{i_N})=\varphi(X_{i_1})\cdots \varphi(X_{i_N}),\quad X_{i_j}\in\mathfrak{g}</math>.
The point is that because there are no other relations in the universal enveloping algebra besides those coming from the commutation relations of <math>\mathfrak{g}</math>, the map <math>\widehat{\varphi}</math> is well defined, independent of how one writes a given element <math>x\in U(\mathfrak{g})</math> as a linear combination of products of Lie algebra elements.

The universal property of the enveloping algebra immediately implies that every representation of <math>\mathfrak{g}</math> acting on a vector space <math>V</math> extends uniquely to a representation of <math>U(\mathfrak{g})</math>. (Take <math>A=\mathrm{End}(V)</math>.) This observation is important because it allows (as discussed below) the Casimir elements to act on <math>V</math>. These operators (from the center of  <math>U(\mathfrak{g})</math>) act as scalars and provide important information about the representations. The [[Casimir element|quadratic Casimir element]] is of particular importance in this regard.

===Other algebras===
Although the canonical construction, given above, can be applied to other algebras, the result, in general, does not have the universal property. Thus, for example, when the construction is applied to [[Jordan algebra]]s, the resulting enveloping algebra contains the [[special Jordan algebra]]s, but not the exceptional ones: that is, it does not envelope the [[Albert algebra]]s. Likewise, the Poincaré–Birkhoff–Witt theorem, below, constructs a basis for an enveloping algebra; it just won't be universal. Similar remarks hold for the [[Lie superalgebra]]s.