The Poincaré–Birkhoff–Witt Theorem

Derives Lemma A30.3. Induct on \(n\). For \(n = 0\) there is nothing to prove. Let \(n \ge 1\) and assume the claim for \(n-1\). It suffices to treat a single word \(w = T_{a_{1}}T_{a_{2}}\cdots T_{a_{n}}\) of length exactly \(n\), and we induct on its number of inversions, the number of pairs \(i < j\) with \(a_{i} > a_{j}\). If \(w\) has no inversion its indices are already nondecreasing and \(w\) is an ordered monomial. If it has at least one, some adjacent pair is out of order, \(a_{i} > a_{i+1}\); otherwise the sequence would be nondecreasing. Apply Equation (18.12) to that pair alone:

The first word on the right has length \(n\) and exactly one inversion fewer, so it is handled by the inner induction; the second has length \(n-1\) and lies in \(U_{n-1}\), handled by the outer induction. Hence \(w\) is a combination of ordered monomials of degree at most \(n\).

Derives Proposition A30.5. The maps are constructed on \(P_{n}\) by induction on \(n\). Write \((A_{n})\), \((B_{n})\), \((C_{n})\) for the three conditions restricted as follows: \((A_{n})\) and \((B_{n})\) are Equations (A30.6) and (A30.7) for \(\abs{A} \le n\), and \((C_{n})\) is Equation (A30.8) applied to \(x_{A}\) with \(\abs{A} \le n-1\) — the largest range in which both sides are defined once \(\theta\) is known on \(P_{n}\), since \(\theta_{b}x_{A} \in P_{\abs{A}+1} \subseteq P_{n}\) by \((B_{n-1})\).

Base. \(P_{0} = \mathbb{K}\); set \(\theta_{a}1 = x_{a}\). Then \((A_{0})\) holds because \(a \le A\) is vacuous for the empty multi-index, \((B_{0})\) holds because the difference is \(0\), and \((C_{0})\) is empty.

Extension to \(P_{n}\). Assume \(\theta\) defined on \(P_{n-1}\) with \((A_{n-1})\), \((B_{n-1})\), \((C_{n-1})\). Let \(\abs{A} = n\).

1. If \(a \le A\), set \(\theta_{a}x_{A} = x_{a}x_{A}\), as Equation (A30.6) demands.

2. If not, let \(b\) be the smallest index occurring in \(A\), so that \(b < a\), and write \(A = (b,B)\) with \(b \le B\) and \(\abs{B} = n-1\). Then \(x_{A} = x_{b}x_{B} = \theta_{b}x_{B}\) by \((A_{n-1})\). Using \((B_{n-1})\) write

   \begin{equation}\tag{A30.9} \theta_{a}x_{B} = x_{a}x_{B} + w\ec\qquad w \in P_{n-1}\ec \end{equation}

   and define

   \begin{equation}\tag{A30.10} \theta_{a}x_{A} := x_{a}x_{A} + \theta_{b}w + C^{c}{}_{ab}\,\theta_{c}x_{B}\ep \end{equation}

   Every term on the right is already defined: \(w\) and \(x_{B}\) lie in \(P_{n-1}\).

Condition \((A_{n})\) holds by construction. For \((B_{n})\): in case 1 the difference is \(0\); in case 2 it is \(\theta_{b}w + C^{c}{}_{ab}\theta_{c}x_{B}\), and \((B_{n-1})\) gives \(\theta_{b}w \in P_{n}\) and \(\theta_{c}x_{B} \in P_{n}\).

Before proving \((C_{n})\), record what Equation (A30.10) says. With \(b < a\) and \(b \le B\), the monomial \(x_{a}x_{B}\) is \(x_{A'}\) for the multi-index \(A'\) obtained by inserting \(a\) into \(B\); its smallest index is still \(b\), so \(b \le A'\) and \(\abs{A'} = n\), and case 1 gives \(\theta_{b}(x_{a}x_{B}) = x_{b}x_{a}x_{B} = x_{a}x_{A}\), the variables commuting. Hence, by Equation (A30.9), \(\theta_{b}\theta_{a}x_{B} = x_{a}x_{A} + \theta_{b}w\), and Equation (A30.10) reads

using \(\theta_{b}x_{B} = x_{A}\) once more. So the commutation relation holds by fiat in the ordered case; the work is to show it then holds in general.

Proof of \((C_{n})\). Let \(\abs{A} = n-1\); the cases \(\abs{A} < n-1\) are \((C_{n-1})\). The claim is antisymmetric in \(a,b\) and trivial for \(a = b\), since \(C^{c}{}_{aa} = 0\), so assume \(a > b\).

Case 1: \(b \le A\). This is Equation (A30.11) with \(B = A\).

Case 2: the relation \(b \le A\) fails. Let \(c_{0}\) be the smallest index occurring in \(A\), so \(c_{0} < b < a\), and write \(A = (c_{0},B)\) with \(c_{0} \le B\) and \(\abs{B} = n-2\). We first establish the auxiliary relation

Indeed, by \((B_{n-1})\) write \(y = x_{b}x_{B} + w'\) with \(w' \in P_{n-2}\). The monomial \(x_{b}x_{B}\) is \(x_{A''}\) for the multi-index \(A''\) obtained by inserting \(b\) into \(B\); its smallest index is \(c_{0}\), because \(c_{0} \le B\) and \(c_{0} < b\), and \(\abs{A''} = n-1\). So Case 1 — just proved, for multi-indices of length \(n-1\) — applies to \(x_{A''}\) with the pair \((a,c_{0})\), and \((C_{n-1})\) applies to \(w' \in P_{n-2}\); adding the two gives Equation (A30.12).

Now compute, using \(x_{A} = \theta_{c_{0}}x_{B}\):

the second line by \((C_{n-1})\) applied to \(x_{B}\), \(\abs{B} = n-2\), and the third by Equation (A30.12) applied to \(y = \theta_{b}x_{B}\). Exchange \(a\) and \(b\) in Equation (A30.13) and subtract:

where the four mixed terms have been paired so that each parenthesis is a commutator. Every commutator in Equation (A30.14) acts on \(x_{B}\) with \(\abs{B} = n-2\), so \((C_{n-1})\) evaluates all three:

The quantity to be matched is, again by \((C_{n-1})\) on \(x_{B}\),

The first terms of Equations (A30.15) and (A30.16) agree, so \((C_{n})\) holds if and only if

after renaming the summation index in Equation (A30.16). Written with brackets, Equation (A30.17) is

which is the Jacobi identity: expanding the right-hand side, \(\comm{\comm{T_{a}}{T_{b}}}{T_{c_{0}}} = \comm{T_{a}}{\comm{T_{b}}{T_{c_{0}}}} - \comm{T_{b}}{\comm{T_{a}}{T_{c_{0}}}}\), and the second term equals \(+\comm{\comm{T_{a}}{T_{c_{0}}}}{T_{b}}\) by antisymmetry. This completes the induction, and with it the construction of \(\theta\) on all of \(P = \bigcup_{n}P_{n}\).

Proof of Theorem A30.1. Derives Theorem A30.1. By Proposition A30.5 the map \(T_{a} \mapsto \theta_{a}\) is a representation of \(\mathfrak{g}\) on the vector space \(P\), and by Remark 18.6 it extends uniquely to a homomorphism of associative algebras with unit

(The extension of Remark 18.6 is stated there for a representation on a vector space; that \(P\) is infinite-dimensional plays no part, since no topology or trace is involved.)

Evaluate \(\Phi\) of an ordered monomial on the constant polynomial \(1\). Reading Equation (A30.2) from right to left, the letter \(T_{M}\) acts \(n_{M}\) times on multi-indices all of whose entries are \(M\), then \(T_{M-1}\) acts on multi-indices whose entries are all \(\ge M-1\), and so on: at every step the acting index is \(\le\) every index already present, so Equation (A30.6) applies at every step and

Distinct ordered monomials of \(U(\mathfrak{g})\) thus have distinct monomials of \(P\) as images, and the monomials of a polynomial algebra are linearly independent. Hence a vanishing linear combination of ordered monomials of \(U(\mathfrak{g})\) maps to a vanishing linear combination of distinct monomials of \(P\), forcing every coefficient to vanish: the ordered monomials are linearly independent. With Lemma A30.3 they are a basis.

The two consequences follow at once. The monomials of degree one are the \(T_{a}\) themselves, part of a basis, so they are linearly independent in \(U(\mathfrak{g})\) and the map \(\mathfrak{g} \to U(\mathfrak{g})\) is injective. And if some element of the free associative algebra on the symbols \(T_{a}\) were annihilated in \(U(\mathfrak{g})\) beyond what Equation (18.12) forces, its reduction to ordered form — which uses those relations only — would be a nontrivial vanishing combination of basis elements.

Derives Proposition A30.9. (1) Each word on the right of Equation (A30.22) is a product of \(n\) elements of \(\mathfrak{g}\), hence lies in \(U_{n}\). Take \(p = T_{1}^{n_{1}}\cdots T_{M}^{n_{M}}\) of degree \(n\). Every one of the \(n!\) words in Equation (A30.22) is a rearrangement of the same letters, so by Equation (A30.4) each equals the ordered monomial \(T_{1}^{n_{1}}\cdots T_{M}^{n_{M}}\) of \(U(\mathfrak{g})\) modulo \(U_{n-1}\); averaging, \(\omega(p) - T_{1}^{n_{1}}\cdots T_{M}^{n_{M}} \in U_{n-1}\), which is the assertion about \(\sigma_{n}\). Thus \(\omega\) maps a basis of \(S(\mathfrak{g})\) to a family of elements that is “triangular” with respect to the filtration, with the ordered monomials as leading terms. Such a family is a basis: a nontrivial vanishing combination would have a highest degree \(n\) occurring in it, and its image under \(\sigma_{n}\) would be a nontrivial vanishing combination of distinct monomials of \(S^{n}(\mathfrak{g})\). Surjectivity follows because the leading terms exhaust the basis of Theorem A30.1.

(2) On \(U(\mathfrak{g})\) the map \(\ad_{X}\) is a derivation of the associative product, \(\ad_{X}(uv) = (\ad_{X}u)v + u(\ad_{X}v)\), as one checks by expanding \(X(uv) - (uv)X\) and inserting \(\pm uXv\). Hence

using \(\ad_{X}Y = \comm{X}{Y}\) for \(Y \in \mathfrak{g}\). Average Equation (A30.23) over \(\pi\) and group the resulting \(n!\,n\) words by the value \(i = \pi(j)\) of the slot that was hit: for fixed \(i\), the pairs \((\pi,j)\) with \(\pi(j) = i\) put \(\comm{X}{X_{i}}\) in position \(j\) and let the remaining letters occupy the remaining positions in every possible order, each exactly once — which is the definition of \(\omega\left(X_{1}\cdots\comm{X}{X_{i}}\cdots X_{n}\right)\) up to the common factor \(1/n!\). Therefore

the last equality because \(\ad_{X}\) acts on \(S(\mathfrak{g})\) as the derivation extending \(\ad_{X}\), which is exactly the Leibniz sum on the right.

Derives Corollary A30.10. Write \(p = k^{a_{1}\cdots a_{r}}T_{a_{1}}\cdots T_{a_{r}}\) for the corresponding element of \(S^{r}(\mathfrak{g})\), the same expression read in commuting symbols. Then \(C_{r} = \omega(p)\): by linearity \(\omega(p) = k^{a_{1}\cdots a_{r}}\,\frac{1}{r!}\sum_{\pi} T_{a_{\pi(1)}}\cdots T_{a_{\pi(r)}}\), and renaming the summation indices in each of the \(r!\) terms turns it into \(k^{a_{1}\cdots a_{r}}T_{a_{1}}\cdots T_{a_{r}}\), because \(k^{a_{1}\cdots a_{r}}\) is totally symmetric. Now \(p \neq 0\): the coefficient of the commuting monomial \(T_{1}^{n_{1}}\cdots T_{M}^{n_{M}}\) in \(p\) is the multinomial coefficient \(r!/(n_{1}!\cdots n_{M}!)\) times the corresponding component of \(k^{a_{1}\cdots a_{r}}\), and in characteristic zero that factor cannot vanish. Since \(\omega\) is injective by Proposition A30.9, \(C_{r} = \omega(p) \neq 0\). For Equation (18.20) the array is \(\kappa^{ab}\), which is invertible, hence nonzero.