(B1) is a basis by definition of $\mathbb{Q}\left[{A}^{*}\right]$ and $\text{(B2)}=\text{(B1)}$ by (1.4). (B3) is a basis by (1.3) and the Poincaré-Birkhoff-Witt Theorem.

$\square $

Last updated: 19 May 2015

This is an excerpt of the paper *Standard Lyndon bases of Lie algebras and enveloping algebras* by Pierre Lalonde Arun Ram.

The first author's research was supported by FCAR and NSERC grants. The second author's research was partially supported by a National
Science Foundation Postdoctoral Fellowship.

In this section we give a short summary of the facts about Lyndon words and the free Lie algebra which we shall use. All of the facts in this section are well known. A comprehensive treatment of free Lie algebras (and Lyndon words) appears in the book by C. Reutenauer [Reu1993u1993].

$A$ be an ordered alphabet and let ${A}^{*}$ be the set of all words in the alphabet $A$ (the free monoid generated by $A\text{).}$ Let $\mid u\mid $ denote the length of the word $u\in {A}^{*},$ and let $u<v$ denote that the word $u$ is lexicographically smaller than the word $v\text{.}$ A word $\ell \in {A}^{*}$ is a Lyndon word if it is lexicographically smaller than all its cyclic rearrangements. Let $\ell $ be a Lyndon word and let $m,n$ be words such that $\ell =mn$ and $n$ is the longest Lyndon word appearing as a proper right factor of $\ell \text{.}$ Then $m$ is also a Lyndon word ([Lot1983] Prop 5.1.3). The standard bracketing of a Lyndon word is given (inductively) by $$\begin{array}{cc}b\left[a\right]=a,\phantom{\rule{1em}{0ex}}\text{for}\hspace{0.17em}a\in A,\phantom{\rule{2em}{0ex}}b\left[\ell \right]=[b\left[m\right],b\left[n\right]],& \text{(1.1)}\end{array}$$ where $\ell =mn$ and $n$ is the longest Lyndon word appearing as a proper right factor of $\ell \text{.}$ We shall use the following facts: (1.2) ([Lot1983] Lemma 5.3.2 or [Reu1993] Thm. 5.1) For each Lyndon word $\ell \text{,}$ $$b\left[\ell \right]=\ell +\sum _{\underset{\mid v\mid =\mid \ell \mid}{v>\ell}}{a}_{v}v,$$ for some integers ${a}_{v}\text{.}$ (1.3) ([Lot1983] Theorem 5.3.1 or [Reu1993] Thm. 4.9) The elements $b\left[\ell \right],$ where $\ell $ is a Lyndon word, are a basis of $\text{Lie}\left(A\right)\text{.}$ (1.4) ([Lot1983] Thm. 5.1.5 or [Reu1993] Cor. 4.7) Every word $w$ has a unique factorization $w={\ell}_{1}\cdots {\ell}_{k},$ such that the ${\ell}_{i}$ are Lyndon words and ${\ell}_{1}\ge \cdots \ge {\ell}_{k}\text{.}$

For each $w\in {A}^{*}$ define $$b\left[w\right]=b\left[{\ell}_{1}\right]\cdots b\left[{\ell}_{k}\right],$$ where $w={\ell}_{1}\cdots {\ell}_{k},$ the factors ${\ell}_{i}$ are Lyndon words and ${\ell}_{1}\ge \cdots \ge {\ell}_{k}\text{.}$ The following result is essentially the same as Theorem 5.1 in [Reu1993]. The fact that the length of the words is preserved is clear from the proof given there. (1.5) ([Reu1993] Thm. 5.1) For each $w\in {A}^{*}$ $$b\left[w\right]=w+\sum _{\underset{\mid v\mid =\mid w\mid}{v>w}}{a}_{v}v,$$ for some integers ${a}_{v}\text{.}$

The free Lie algebra $\text{Lie}\left(A\right)$ with generating set $A$ can be viewed as the span of the letters in $A$ and all brackets of letters in $A\text{.}$ $\mathbb{Q}\left[{A}^{*}\right]$ is the associative algebra of $\mathbb{Q}\text{-linear}$ combinations of words in the alphabet $A$ where the product is juxtaposition. The algebra $\mathbb{Q}\left[{A}^{*}\right]$ is graded by the length of the words. We shall have need of the following: (1.6) ([Bou1989] II § 3 Thm. 1 or [Reu1993] Thm. 0.5) $\mathbb{Q}\left[{A}^{*}\right]$ is the enveloping algebra of $\text{Lie}\left(A\right)\text{.}$ (1.7) (Poincaré-Birkhoff-Witt theorem, [Bou1989] I § 3 Cor. 3 to Thm. 1) If $\U0001d524$ is a Lie algebras and $B$ is an ordered basis of $\U0001d524,$ then the set of products ${\ell}_{1}\cdots {\ell}_{k},$ ${\ell}_{i}\in B,$ ${\ell}_{1}\ge \cdots \ge {\ell}_{k},$ is a basis of the enveloping algebra $U\U0001d524$ of $\U0001d524\text{.}$ (1.8) ([Bou1989] I § 3 Prop.3) Let $\U0001d524$ be a Lie algebra and let $J$ be a Lie ideal of $\U0001d524\text{.}$ Let $U\U0001d524$ be the enveloping algebra of $\U0001d524$ and let $I$ be the ideal in $U\U0001d524$ generated by $J\text{.}$ Then the enveloping algebra of $\stackrel{\u203e}{\U0001d524}=\U0001d524/J$ is $U\stackrel{\u203e}{\U0001d524}=U\U0001d524/I\text{.}$ The following well known result follows easily from (1.1)-(1.5).

Each of the following is a basis of $\mathbb{Q}\left[{A}^{*}\right]\text{.}$ (B1) The set of words ${A}^{*}\text{.}$ (B2) The set of products ${\ell}_{1}\cdots {\ell}_{k},$ where the ${\ell}_{i}$ are Lyndon words and ${\ell}_{1}\ge \cdots \ge {\ell}_{k}\text{.}$ (B3) The set of products $b\left[{\ell}_{1}\right]\cdots b\left[{\ell}_{k}\right],$ where the ${\ell}_{i}$ are Lyndon words and ${\ell}_{1}\ge \cdots \ge {\ell}_{k}\text{.}$

(B1) is a basis by definition of $\mathbb{Q}\left[{A}^{*}\right]$ and $\text{(B2)}=\text{(B1)}$ by (1.4). (B3) is a basis by (1.3) and the Poincaré-Birkhoff-Witt Theorem.

$\square $