Editing Universal enveloping algebra (section)

==Poincaré–Birkhoff–Witt theorem ==
{{main article|Poincaré–Birkhoff–Witt theorem}}
The Poincaré–Birkhoff–Witt theorem gives a precise description of <math>U(\mathfrak{g})</math>. This can be done in either one of two different ways: either by reference to an explicit [[vector basis]] on the Lie algebra, or in a [[coordinate-free]] fashion.

===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.

===Coordinate-free===
One can also state the theorem in a coordinate-free fashion, avoiding the use of total orders and basis elements. This is convenient when there are difficulties in defining the basis vectors, as there can be for infinite-dimensional Lie algebras. It also gives a more natural form that is more easily extended to other kinds of algebras.  This is accomplished by constructing a [[filtration (mathematics)|filtration]] <math>U_m \mathfrak{g}</math> whose limit is the universal enveloping algebra <math>U(\mathfrak{g}).</math>

First, a notation is needed for an ascending sequence of subspaces of the tensor algebra. Let
:<math>T_m\mathfrak{g} = K\oplus \mathfrak{g}\oplus T^2\mathfrak{g} \oplus \cdots \oplus T^m\mathfrak{g}</math>
where
:<math>T^m\mathfrak{g} = T^{\otimes m} \mathfrak{g} = \mathfrak{g}\otimes \cdots \otimes \mathfrak{g}</math>

is the {{math|''m''}}-times tensor product of <math>\mathfrak{g}.</math> The <math>T_m\mathfrak{g}</math> form a [[filtration (mathematics)|filtration]]:
:<math>K\subset \mathfrak{g}\subset T_2\mathfrak{g} \subset \cdots \subset T_m\mathfrak{g} \subset\cdots</math>

More precisely, this is a [[filtered algebra]], since the filtration preserves the algebraic properties of the subspaces. Note that the [[limit (category theory)|limit]] of this filtration is the tensor algebra <math>T(\mathfrak{g}).</math>

It was already established, above, that quotienting by the ideal is a [[natural transformation]] that takes one from <math>T(\mathfrak{g})</math> to <math>U(\mathfrak{g}).</math> This also works naturally on the subspaces, and so one obtains a filtration <math>U_m \mathfrak{g}</math> whose limit is the universal enveloping algebra <math>U(\mathfrak{g}).</math>

Next, define the space
:<math>G_m\mathfrak{g} = U_m \mathfrak{g}/U_{m-1} \mathfrak{g}</math>
This is the space <math>U_m \mathfrak{g}</math> modulo all of the subspaces <math>U_n \mathfrak{g}</math> of strictly smaller filtration degree. Note that <math>G_m\mathfrak{g}</math> is ''not at all'' the same as the leading term <math>U^m\mathfrak{g}</math> of the filtration, as one might naively surmise. It is not constructed through a set subtraction mechanism associated with the filtration.

Quotienting <math>U_m \mathfrak{g}</math> by <math>U_{m-1} \mathfrak{g}</math> has the effect of setting all Lie commutators defined in <math>U_m \mathfrak{g}</math> to zero. One can see this by observing that the commutator of a pair of elements whose products lie in <math>U_{m} \mathfrak{g}</math> actually gives an element in <math>U_{m-1} \mathfrak{g}</math>. This is perhaps not immediately obvious: to get this result, one must repeatedly apply the commutation relations, and turn the crank. The essence of the Poincaré–Birkhoff–Witt theorem is that it is always possible to do this, and that the result is unique.

Since commutators of elements whose products are defined in <math>U_{m} \mathfrak{g}</math> lie in <math>U_{m-1} \mathfrak{g}</math>, the quotienting that defines <math>G_m\mathfrak{g}</math> has the effect of setting all commutators to zero. What PBW states is that the commutator of elements in <math>G_m\mathfrak{g}</math> is necessarily zero. What is left are the elements that are not expressible as commutators.

In this way, one is lead immediately to the [[symmetric algebra]].  This is the algebra where all commutators vanish. It can be defined as a filtration <math>S_m \mathfrak{g}</math> of symmetric tensor products <math>\operatorname{Sym}^m \mathfrak{g}</math>. Its limit is the symmetric algebra <math>S(\mathfrak{g})</math>. It is constructed by appeal to the same notion of naturality as before. One starts with the same tensor algebra, and just uses a different ideal, the ideal that makes all elements commute:

:<math>S(\mathfrak{g}) = T(\mathfrak{g}) / (a\otimes b - b\otimes a)</math>

Thus, one can view the Poincaré–Birkhoff–Witt theorem as stating that <math>G(\mathfrak{g})</math> is isomorphic to the symmetric algebra <math>S(\mathfrak{g})</math>, both as a vector space ''and'' as a commutative algebra.

The <math>G_m\mathfrak{g}</math> also form a filtered algebra; its limit is <math>G(\mathfrak{g}).</math>  This is the [[associated graded algebra]] of the filtration.

The construction above, due to its use of quotienting, implies that the limit of <math>G(\mathfrak{g})</math> is isomorphic to <math>U(\mathfrak{g}).</math>  In more general settings, with loosened conditions, one finds that <math>S(\mathfrak{g})\to G(\mathfrak{g})</math> is a projection, and one then gets PBW-type theorems for the associated graded algebra of a [[filtered algebra]]. To emphasize this, the notation <math>\operatorname{gr}U(\mathfrak{g})</math> is sometimes used for <math>G(\mathfrak{g}),</math> serving to remind that it is the filtered algebra.

===Other algebras===
The theorem, applied to [[Jordan algebra]]s, yields the [[exterior algebra]], rather than the symmetric algebra. In essence, the construction zeros out the anti-commutators. The resulting algebra is ''an'' enveloping algebra, but is not universal. As mentioned above, it fails to envelop the exceptional Jordan algebras.