07.06.13 · representation-theory / lie-algebraic

Free Lie algebras, the Hall basis, and Magnus's theorem

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Anchor (Master): Magnus 1937 *Über Beziehungen zwischen höheren Kommutatoren* (Crelle 177); Witt 1937 *Treue Darstellung Liescher Ringe* (Crelle 177); Serre *Lie Algebras and Lie Groups* Part I Ch. IV; Reutenauer *Free Lie Algebras* (Oxford 1993); Lothaire *Combinatorics on Words* Ch. 5 (Lyndon words)

Intuition Beginner

Start with a handful of letters, say $x$ and $y$ . A Lie algebra lets you form one new operation, the bracket $[a, b]$ , which measures how much two things fail to commute. The free Lie algebra is what you get when you bracket these letters together in every possible way and impose only the two rules every Lie algebra must obey, with nothing extra. It is the most unconstrained algebra of brackets you can build from a set of letters.

A clean way to picture it: write words in the letters, like $x y$ or $x y x$ , where juxtaposition means "do one after another". This word algebra is the home of everything. Inside it, the bracket means $[a, b] = ab - ba$ . The free Lie algebra is the part of the word algebra you can reach using brackets alone, never plain multiplication.

The surprise is that this bracket-reachable part can be described exactly, given an explicit basis, and even counted. A short formula tells you how many independent brackets of each length exist.

Visual Beginner

Picture a tree. At the leaves sit the letters $x$ and $y$ . Each internal node is a bracket of its two children, so a binary tree with letters at the leaves names a nested bracket like $[[x, y], x]$ . The free Lie algebra is the space spanned by all such trees, after you quietly identify trees that the Lie rules force to be equal.

Around this tree-world sits a larger box: the world of plain words, where letters are simply strung together. Every tree casts a shadow into the box by expanding each bracket as $ab - ba$ . The picture to hold is two nested regions: a big box of all words, and inside it a curved region of exactly the "bracket-expressible" words.

The diagram captures the central mechanism: brackets live inside words, and the free Lie algebra is the precise bracket-shaped slice of the word algebra.

Worked example Beginner

Take two letters $x$ and $y$ and look at brackets of length two and three.

Length one gives the letters themselves: $x$ and $y$ . Two independent elements.

Length two gives a single new element, $[x, y]$ . Why only one? Because the bracket flips sign when you swap its inputs, so $[y, x] = - [x, y]$ , and $[x, x] = 0$ . So the length-two part is one-dimensional, spanned by $[x, y]$ .

Length three is more interesting. The candidates are $[[x, y], x]$ and $[[x, y], y]$ . You might also write $[x, [x, y]]$ , but that equals $- [[x, y], x]$ , so it is not new. The Jacobi rule handles every other rearrangement. The result: the length-three part is two-dimensional.

Now expand $[x, y]$ as a word: $[x, y] = x y - y x$ . And $[[x, y], x] = (x y - y x) x - x (x y - y x) = x y x - y xx - xx y + x y x$ , which simplifies to $2 x y x - y xx - xx y$ . The bracket has become a specific combination of length-three words.

Check your understanding Beginner

Formal definition Intermediate+

Let $X$ be a set and $k$ a commutative ring (take $k = Q$ or $Z$ unless stated otherwise).

Definition (free Lie algebra by universal property). The free Lie algebra on $X$ is a Lie $k$ -algebra $L (X)$ together with a map $ι : X \to L (X)$ such that for every Lie $k$ -algebra $g$ and every map $f : X \to g$ there is a unique Lie-algebra homomorphism $\tilde{f} : L (X) \to g$ with $\tilde{f} \circ ι = f$ . This universal property determines $L (X)$ up to unique isomorphism.

Construction inside the tensor algebra. Let $T (X) = ⨁_{n \geq 0} X^{\otimes n}$ be the free associative algebra on $X$ (the algebra of words in the letters of $X$ ), with multiplication the concatenation of words. Equip $T (X)$ with the commutator bracket $[a, b] = ab - ba$ . Let $L (X) \subseteq T (X)$ be the smallest Lie subalgebra containing $X$ . The inclusion $X ↪ L (X)$ satisfies the universal property above, so this subalgebra is the free Lie algebra.

Grading. Both $T (X)$ and $L (X)$ are graded by word length: $T (X) = ⨁_{n} T_{n}$ , $L (X) = ⨁_{n} L_{n}$ , with $L_{n} = L (X) \cap T_{n}$ . The degree-one piece $L_{1}$ is the span of $X$ .

The coproduct. Give $T (X)$ the cocommutative coproduct $Δ : T (X) \to T (X) \otimes T (X)$ determined by declaring each generator primitive: $Δ (x) = x \otimes 1 + 1 \otimes x$ for $x \in X$ , extended as an algebra map. An element $w$ is primitive if $Δ (w) = w \otimes 1 + 1 \otimes w$ . This is the Hopf-algebra apparatus of unit 03.12.49.

Counterexamples to common slips

Not every element of $T_{n}$ is a Lie element. In degree two, $x y$ is not in $L (X)$ ; only the antisymmetric combination $x y - y x = [x, y]$ is. The Lie elements form a proper subspace cut out by the primitivity condition.
The free Lie algebra is not the free associative algebra. $L (X) ⊊ T (X)$ as soon as $∣ X ∣ \geq 1$ and $n \geq 2$ . The relation $U (L (X)) = T (X)$ at the level of enveloping algebras (Magnus's theorem) does not say $L (X) = T (X)$ .
"Free" is relative to the ring. Over a field of characteristic $p$ , the alternating law $[a, a] = 0$ is strictly stronger than antisymmetry $[a, b] = - [b, a]$ , and the restricted (p-)structure enters. The clean dimension count below is the characteristic-zero statement.

Key theorem with proof Intermediate+

Theorem (Dynkin-Specht-Wever / Friedrichs' criterion). Work over a field $k$ of characteristic zero. For a homogeneous element $w \in T_{n} (X)$ with $n \geq 1$ , the following are equivalent:

(i) $w \in L_{n} (X)$ , i.e. $w$ is a Lie element (a $k$ -combination of nested brackets of letters);

(ii) $w$ is primitive: $Δ (w) = w \otimes 1 + 1 \otimes w$ ;

(iii) $D (w) = n w$ , where $D$ is the Dynkin map sending a word $x_{1} x_{2} \dots x_{n}$ to the left-normed bracket $[[\dots [x_{1}, x_{2}], x_{3}], \dots, x_{n}]$ .

Proof. (i) $\Rightarrow$ (ii). The coproduct $Δ$ is an algebra homomorphism, and the primitive elements $P (T (X)) = {w : Δ w = w \otimes 1 + 1 \otimes w}$ form a Lie subalgebra under the commutator, because the commutator of two primitives is primitive (a direct computation with $Δ (ab) = Δ (a) Δ (b)$ ). The generators $x \in X$ are primitive by definition, so the Lie subalgebra they generate, namely $L (X)$ , lands in $P (T (X))$ .

(ii) $\Rightarrow$ (iii). The Dynkin map factors as $D = π \circ ℓ$ where $ℓ$ is left bracketing. One computes $D$ on a word by the recursion $D (x_{1} \dots x_{n}) = [D (x_{1} \dots x_{n - 1}), x_{n}]$ . A bookkeeping argument with the coproduct shows that on a primitive element of degree $n$ the map $D$ acts as multiplication by $n$ ; the key point is that $D$ kills any decomposable (product) contribution, and primitivity removes exactly those contributions.

(iii) $\Rightarrow$ (i). If $D (w) = n w$ then $w = \frac{1}{n} D (w)$ , and $D (w)$ is by construction a $k$ -combination of left-normed brackets, hence a Lie element. (Division by $n$ uses characteristic zero.) $□$

The operator $e_{n} = \frac{1}{n} D$ restricted to $T_{n}$ is therefore an idempotent, the Dynkin idempotent, projecting $T_{n}$ onto $L_{n}$ . Its image is the degree- $n$ free Lie component.

Bridge. This criterion builds toward the Campbell-Baker-Hausdorff formula, which appears again in 07.06.15: the series $lo g (exp X exp Y)$ is manifestly primitive because $exp$ and $lo g$ pass between group-like and primitive elements, so the Dynkin-Specht-Wever theorem is exactly what guarantees it is a Lie series. The foundational reason the free Lie algebra is so rigid is that it is cut out of the tensor algebra by a single linear condition, primitivity, and this is dual to the Milnor-Moore description of $U (g)$ in 03.12.49 where primitives recover the Lie algebra. Putting these together, the Dynkin idempotent is the central insight that turns an existence statement (Lie elements exist) into an algorithm (project any word onto its Lie part), and this generalises to the higher Eulerian idempotents that refine the decomposition of $T (X)$ .

Exercises Intermediate+

Exercise 2 (easy, multiple choice).

Magnus's theorem states that the universal enveloping algebra of the free Lie algebra $L (X)$ is:

A. the symmetric algebra $S (X)$ B. the free associative (tensor) algebra $T (X)$ C. the group algebra of the free group on $X$ D. the exterior algebra $Λ (X)$

Hint

The enveloping algebra is the smallest associative algebra receiving $L (X)$ and respecting brackets as commutators. Since $L (X)$ already sits inside $T (X)$ generating it, ask what $T (X)$ is.

Answer

B. Magnus's theorem gives $U (L (X)) ≅ T (X)$ . Feedback-correct: the free associative algebra is the enveloping algebra of the free Lie algebra, which is why $L (X)$ embeds in $T (X)$ . Feedback-wrong: $S (X)$ (A) is the enveloping algebra of the abelian Lie algebra on $X$ ; the group algebra (C) and exterior algebra (D) are different constructions entirely.

Exercise 5 (medium, short-answer).

A Lyndon word is a word strictly smaller in lexicographic order than all of its proper cyclic rotations. With alphabet ${x < y}$ , list all Lyndon words of length $\leq 3$ and check that their count in each degree equals $dim L_{n}$ from Exercise 3.

Hint

Length-one words are Lyndon. For length two and three, test each word against its rotations; e.g. $x y$ versus $y x$ , and $xx y, x y y$ versus their rotations.

Answer

Length $1$ : $x, y$ (two words, matching $dim L_{1} = 2$ ). Length $2$ : $x y$ only, since $y x$ rotates to the smaller $x y$ (one word, matching $dim L_{2} = 1$ ). Length $3$ : $xx y$ and $x y y$ are Lyndon; $x y x, y xx, y x y, y y x, xxx, y y y$ fail (repeats or a smaller rotation exists). That is two words, matching $dim L_{3} = 2$ . The number of Lyndon words of length $n$ equals the Witt number, and the standard bracketing of Lyndon words gives a basis of $L (X)$ .

Exercise 6 (hard, short-answer).

Assume Magnus's theorem $U (L (X)) = T (X)$ and the Poincaré-Birkhoff-Witt theorem 07.06.02. Deduce the Witt dimension formula $dim L_{n} = \frac{1}{n} \sum_{d ∣ n} μ (d) q^{n / d}$ for $q = ∣ X ∣$ by comparing graded Hilbert series.

Hint

By PBW, $T (X) = U (L (X))$ has graded Hilbert series $\prod_{n \geq 1} (1 - t^{n})^{- d i m L_{n}}$ . The left side is $\frac{1}{1 - q t}$ . Take logarithms and Möbius-invert.

Answer

PBW gives a basis of $T (X) = U (L (X))$ from ordered monomials in a homogeneous basis of $L (X)$ , so the Hilbert series factor: $\sum_{n} q^{n} t^{n} = \frac{1}{1 - q t} = \prod_{n \geq 1} (1 - t^{n})^{- d i m L_{n}}$ . Taking $- lo g$ of both sides, $- lo g (1 - q t) = \sum_{n \geq 1} dim L_{n} (- lo g (1 - t^{n}))$ . Expanding $- lo g (1 - u) = \sum_{m \geq 1} u^{m} / m$ and matching the coefficient of $t^{N}$ : $\frac{q ^{N}}{N} = \sum_{d ∣ N} \frac{d i m L _{d} \cdot d}{N}$ , i.e. $q^{N} = \sum_{d ∣ N} d dim L_{d}$ . Möbius inversion yields $N dim L_{N} = \sum_{d ∣ N} μ (N / d) q^{d} = \sum_{d ∣ N} μ (d) q^{N / d}$ , the Witt formula.

Advanced results Master

Theorem 1 (Magnus's theorem). The natural map $U (L (X)) \to T (X)$ induced by the inclusion $L (X) ↪ T (X)$ is an isomorphism of associative algebras. Consequently $L (X)$ is the Lie algebra of primitive elements of the Hopf algebra $T (X)$ , and $L (X) ↪ T (X)$ is an embedding. Over $Z$ the free Lie ring $L_{Z} (X)$ is a free $Z$ -module, hence torsion-free, with $T_{Z} (X)$ as its enveloping algebra.

Theorem 2 (Hall basis). Fix a Hall set $H$ : a totally ordered set of bracketed monomials defined recursively so that $X \subseteq H$ , and a bracket $[u, v]$ lies in $H$ iff $u < v$ , $u, v \in H$ , and (if $v = [v_{1}, v_{2}]$ ) $u \geq v_{1}$ . Then the images in $L (X)$ of the elements of $H$ of degree $n$ form a $k$ -basis of $L_{n} (X)$ . The Lyndon basis is the special Hall set obtained from the lexicographic order via standard factorisation of Lyndon words.

Theorem 3 (Witt dimension formula). For $∣ X ∣ = q < \infty$ and characteristic zero, $dim_{k} L_{n} (X) = \frac{1}{n} \sum_{d ∣ n} μ (d) q^{n / d}$ , where $μ$ is the number-theoretic Möbius function. Equivalently the formal character of $L (X)$ as a $GL (X)$ -representation is given by Witt's necklace-counting identity, and $dim L_{n}$ equals the number of Lyndon words of length $n$ over a $q$ -letter alphabet.

Theorem 4 (Magnus embedding of the free group). Let $F$ be the free group on $X$ and $T (X)$ the completion of $T (X)$ by powers of the augmentation ideal. The map $μ : F \to T (X)^{\times}$ , $x \mapsto 1 + x$ , $x^{- 1} \mapsto 1 - x + x^{2} - \dots$ , is an injective group homomorphism (the Magnus embedding). It identifies the lower central series of $F$ with the filtration by bracket-degree: $γ_{n} (F) / γ_{n + 1} (F) ≅ L_{n} (X)$ as abelian groups, so the associated graded of the free group is the free Lie ring.

Synthesis. The foundational reason these results cohere is that the tensor algebra $T (X)$ is a single object wearing three hats: as an associative algebra it is the free word algebra, as a Hopf algebra its primitives are $L (X)$ , and as a filtered algebra it receives the free group through the Magnus embedding. Putting these together, Magnus's theorem $U (L (X)) = T (X)$ is exactly the statement that the enveloping-algebra construction, applied to the freest Lie input, returns the freest associative output, and this is dual to the primitive-element description that recovers $L (X)$ from $T (X)$ . The central insight is that one combinatorial invariant, the Witt number, simultaneously counts a Lie-algebra dimension, a set of Lyndon words, and a lower-central-series rank; this generalises the elementary necklace-counting of aperiodic words. The bridge is that the Hall and Lyndon bases turn the abstract freeness into an explicit algorithm, so that every later structure built on the free Lie algebra, the Campbell-Baker-Hausdorff series of 07.06.15 above all, becomes computable term by term.

Full proof set Master

Proposition 1 (Primitives form the free Lie algebra over a field of characteristic zero). Let $k$ have characteristic zero. The space $P (T (X))$ of primitive elements of $T (X)$ equals the Lie subalgebra $L (X)$ generated by $X$ .

Proof. The inclusion $L (X) \subseteq P (T (X))$ is step (i) $\Rightarrow$ (ii) of the key theorem: $X$ consists of primitives, and primitives are closed under the commutator (Exercise 4), so the generated Lie subalgebra lies in $P (T (X))$ . For the reverse inclusion, the Dynkin idempotent $e_{n} = \frac{1}{n} D$ on $T_{n}$ projects onto $L_{n}$ , and the key theorem shows $e_{n}$ fixes every primitive of degree $n$ . Hence each homogeneous primitive lies in the image of $e_{n}$ , which is $L_{n}$ . Summing over $n$ , $P (T (X)) \subseteq L (X)$ . The two inclusions give equality. $□$

Proposition 2 (Magnus: $U (L (X)) ≅ T (X)$ ). The associative-algebra map $φ : U (L (X)) \to T (X)$ extending $L (X) ↪ T (X)$ is an isomorphism.

Proof. Surjectivity is immediate: $T (X)$ is generated as an algebra by $X \subseteq L (X)$ , and $φ$ hits every generator. For injectivity, use the universal property of $T (X)$ as the free associative algebra: the map $X \to U (L (X))$ , $x \mapsto x$ , extends to an algebra homomorphism $ψ : T (X) \to U (L (X))$ . By construction $φ \circ ψ$ fixes the generators $X$ , so $φ \circ ψ = id_{T (X)}$ . Then $φ$ is surjective with a right inverse, and $ψ$ is surjective because $U (L (X))$ is generated by the image of $L (X)$ , which $ψ$ reaches via the Lie-bracket-as-commutator relations holding in $T (X)$ . A right-invertible surjection between these graded connected algebras of equal finite graded dimensions in each degree (forced by PBW for $U (L (X))$ matched against the word count of $T (X)$ ) is an isomorphism. Hence $φ$ is bijective. $□$

Proposition 3 (Freeness of the integral free Lie ring). The underlying $Z$ -module of $L_{Z} (X)$ is free; in particular $L_{Z} (X)$ is torsion-free.

Proof. By Magnus's theorem over $Z$ , $L_{Z} (X)$ is the Lie subring of primitives of the tensor ring $T_{Z} (X)$ , which is a free $Z$ -module on the words in $X$ . A Hall set $H$ (Theorem 2) provides explicit elements whose images in each degree- $n$ component are $Z$ -linearly independent and span $L_{n}$ , because the leading word of the Hall bracket of degree $n$ (its lexicographically greatest monomial) is distinct for distinct Hall elements and appears with coefficient $\pm 1$ . This unitriangular relationship to a subset of the free $Z$ -basis of $T_{n}$ exhibits each $L_{n}$ as a free $Z$ -module on the degree- $n$ Hall elements. A graded module that is free in each degree is free, and a free module is torsion-free. $□$

Connections Master

Universal enveloping algebra and PBW 07.06.02. Magnus's theorem is the extreme case of the enveloping-algebra construction: $U (L (X)) = T (X)$ says the enveloping algebra of the freest Lie input is the free associative algebra. The Witt dimension formula is derived precisely by feeding the PBW basis of 07.06.02 into a Hilbert-series comparison, so the counting of free Lie generators rests on PBW.
Bialgebra, Hopf algebra, Milnor-Moore 03.12.49. The primitive-element characterisation of $L (X)$ uses the cocommutative coproduct $Δ (x) = x \otimes 1 + 1 \otimes x$ supplied by 03.12.49. Milnor-Moore identifies the primitives of $U (g)$ with $g$ ; for the free case this specialises to $P (T (X)) = L (X)$ , the content of Proposition 1, and the Dynkin idempotent is the projection onto these primitives.
Campbell-Baker-Hausdorff 07.06.15. The BCH series $lo g (exp X exp Y)$ lives in the completed free Lie algebra: it is a primitive element of the completed tensor algebra, hence a Lie series by the Dynkin-Specht-Wever theorem proved here. The free Lie algebra is therefore the natural home in which BCH is stated and its Lie-ness is guaranteed, and Dynkin's explicit coefficient formula is the Dynkin idempotent applied to $lo g (exp X exp Y)$ .
Lie algebra 03.04.01. The free Lie algebra is the left adjoint to the forgetful functor from Lie algebras to sets: every Lie algebra in the sense of 03.04.01 generated by a set $X$ is a quotient of $L (X)$ , so $L (X)$ is the universal source of bracket identities and the place where relations among iterated brackets are tested.

Historical & philosophical context Master

The free Lie algebra entered mathematics through two independent 1937 papers in the same volume of Crelle's journal. Wilhelm Magnus, studying the lower central series of free groups, introduced the embedding $x \mapsto 1 + x$ of the free group into the units of a power-series ring and showed that the associated graded recovers a free Lie ring ^[Magnus1937]. In the companion paper, Ernst Witt proved that the free Lie ring is a free abelian group and computed its graded ranks by Möbius inversion, giving the formula now bearing his name ^[Witt1937]. Friedrichs later isolated the criterion that an element of the tensor algebra is a Lie element exactly when it is primitive, a result independently due to Specht and Wever and now standardly attributed to all three.

Jean-Pierre Serre's Lie Algebras and Lie Groups placed the free Lie algebra at the centre of Part I, using it to give a clean, coordinate-free proof that the Campbell-Baker-Hausdorff series is a Lie series ^[Serre1965]. Philosophically, the free Lie algebra exemplifies the categorical idea of a free object: it is the universal solution to imposing only the unavoidable axioms, so that every theorem proved about $L (X)$ transfers to every Lie algebra by the universal property. The combinatorial miracle, that a structure defined by a universal property turns out to have an explicit basis indexed by Lyndon words and a closed-form dimension count, is a recurring theme linking algebra to the combinatorics of words ^{[Reutenauer1993]}.

Bibliography Master

@article{Magnus1937,
  author  = {Magnus, Wilhelm},
  title   = {{\"U}ber Beziehungen zwischen h{\"o}heren Kommutatoren},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {177},
  year    = {1937},
  pages   = {105--115}
}

@article{Witt1937,
  author  = {Witt, Ernst},
  title   = {Treue Darstellung Liescher Ringe},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {177},
  year    = {1937},
  pages   = {152--160}
}

@book{Serre1965,
  author    = {Serre, Jean-Pierre},
  title     = {Lie Algebras and Lie Groups},
  publisher = {W. A. Benjamin (orig.) / Springer Lecture Notes in Mathematics 1500 (reprint)},
  year      = {1965},
  note      = {1964 lectures at Harvard; Part I treats free Lie algebras and the Campbell-Hausdorff formula}
}

@book{Reutenauer1993,
  author    = {Reutenauer, Christophe},
  title     = {Free Lie Algebras},
  publisher = {Oxford University Press},
  series    = {London Mathematical Society Monographs, New Series},
  volume    = {7},
  year      = {1993}
}

@book{BourbakiLie,
  author    = {Bourbaki, Nicolas},
  title     = {Groupes et alg{\`e}bres de {L}ie, Chapitres 1--3},
  publisher = {Hermann (orig.) / Springer (reprint)},
  year      = {1972},
  note      = {Ch. II: free Lie algebras and their enveloping algebras}
}

@book{Lothaire1983,
  author    = {Lothaire, M.},
  title     = {Combinatorics on Words},
  publisher = {Cambridge University Press},
  series    = {Encyclopedia of Mathematics and its Applications},
  volume    = {17},
  year      = {1983},
  note      = {Ch. 5: Lyndon words and the factorisation theorem}
}

Prerequisites

07.06.02
03.04.01
03.12.49

Tier anchors

beginner: Reutenauer *Free Lie Algebras* (Oxford 1993) Ch. 0 informal account of words and brackets; Stillwell *Naive Lie Theory* Ch. 1 (the bracket as a measure of non-commutativity)
intermediate: Serre *Lie Algebras and Lie Groups* (Benjamin 1965 / Springer LNM 1500) Part I Ch. IV §4–5; Bourbaki *Groupes et algèbres de Lie* Ch. II §2–3; Reutenauer *Free Lie Algebras* Ch. 0–5
master: Magnus 1937 *Über Beziehungen zwischen höheren Kommutatoren* (Crelle 177); Witt 1937 *Treue Darstellung Liescher Ringe* (Crelle 177); Serre *Lie Algebras and Lie Groups* Part I Ch. IV; Reutenauer *Free Lie Algebras* (Oxford 1993); Lothaire *Combinatorics on Words* Ch. 5 (Lyndon words)

References

Serre — Lie Algebras and Lie Groups · Part I Ch. IV: free Lie algebras, the Friedrichs criterion, the Campbell-Hausdorff formula; Magnus's theorem U(L(X)) = T(X)
Magnus — Über Beziehungen zwischen höheren Kommutatoren · Journal für die reine und angewandte Mathematik (Crelle) 177 (1937), 105–115; the Magnus embedding of the free group into the units of the formal power-series ring
Witt — Treue Darstellung Liescher Ringe · Journal für die reine und angewandte Mathematik (Crelle) 177 (1937), 152–160; the dimension formula via Möbius inversion and the realisation of L(X) inside T(X)
Reutenauer — Free Lie Algebras · London Mathematical Society Monographs, New Series 7, Oxford University Press 1993; Hall and Lyndon bases, the Dynkin idempotent, the shuffle Hopf algebra
Bourbaki — Groupes et algèbres de Lie · Ch. II §2 Free Lie algebras; §3 The enveloping algebra of a free Lie algebra; the Poincaré-Birkhoff-Witt basis
Lothaire — Combinatorics on Words · Cambridge University Press 1983, Ch. 5: Lyndon words, the factorisation theorem, and the standard bracketing

Estimated time

beginner: 18m
intermediate: 45m
master: 85m