Editing Universal enveloping algebra (section)

===Using basis elements===
One way is to suppose that the Lie algebra can be given a [[totally ordered]] basis, that is, it is the [[free vector space]] of a totally ordered set. Recall that a free vector space is defined as the space of all finitely supported functions from a set {{math|''X''}} to the field {{math|''K''}} (finitely supported means that only finitely many values are non-zero); it can be given a basis <math>e_a:X\to K</math> such that <math>e_a(b) = \delta_{ab}</math> is the [[indicator function]] for <math>a,b\in X</math>. Let <math>h:\mathfrak{g}\to T(\mathfrak{g})</math> be the injection into the tensor algebra; this is used to give the tensor algebra a basis as well. This is done by lifting: given some arbitrary sequence of <math>e_a</math>, one defines the extension of <math>h</math> to be

:<math>h(e_a\otimes e_b \otimes\cdots \otimes e_c) = h(e_a) \otimes h(e_b) \otimes\cdots \otimes h(e_c)</math>

The Poincaré–Birkhoff–Witt theorem then states that one can obtain a basis for <math>U(\mathfrak{g})</math> from the above, by enforcing the total order of {{math|''X''}} onto the algebra. That is, <math>U(\mathfrak{g})</math> has a basis

:<math>e_a\otimes e_b \otimes\cdots \otimes e_c</math>

where <math>a\le b \le \cdots \le c</math>, the ordering being that of total order on the set {{math|''X''}}.<ref>{{harvnb|Hall|2015}} Theorem 9.10</ref> The proof of the theorem involves noting that, if one starts with out-of-order basis elements, these can always be swapped by using the commutator (together with the [[structure constants]]). The hard part of the proof is establishing that the final result is unique and independent of the order in which the swaps were performed.

This basis should be easily recognized as the basis of a [[symmetric algebra]]. That is, the underlying vector spaces of <math>U(\mathfrak{g})</math> and the symmetric algebra are isomorphic, and it is the PBW theorem that shows that this is so.  See, however, the section on the algebra of symbols, below, for a more precise statement of the nature of the isomorphism.

It is useful, perhaps, to split the process into two steps. In the first step, one constructs the [[free Lie algebra]]: this is what one gets, if one mods out by all commutators, without specifying what the values of the commutators are. The second step is to apply the specific commutation relations from <math>\mathfrak{g}.</math> The first step is universal, and does not depend on the specific <math>\mathfrak{g}.</math> It can also be precisely defined: the basis elements are given by [[Hall word]]s, a special case of which are the [[Lyndon word]]s; these are explicitly constructed to behave appropriately as commutators.