Standard Lyndon bases of Lie algebras and enveloping algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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.

Lyndon words and the free Lie algebra

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 $$\begin{array}{cc}b\left[a\right]=a,\text{for}\hspace{0.17em}a\in A,b\left[\ell \right]=[b\left[m\right],b\left[n\right]],& \text{(1.1)}\end{array}$$ 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 $𝔤$ is a Lie algebras and $B$ is an ordered basis of $𝔤,$ 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 $U𝔤$ of $𝔤\text{.}$ (1.8) ([Bou1989] I § 3 Prop.3) Let $𝔤$ be a Lie algebra and let $J$ be a Lie ideal of $𝔤\text{.}$ Let $U𝔤$ be the enveloping algebra of $𝔤$ and let $I$ be the ideal in $U𝔤$ generated by $J\text{.}$ Then the enveloping algebra of $\overline{𝔤}=𝔤/J$ is $U\overline{𝔤}=U𝔤/I\text{.}$ 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 ${\ell }_1\ge \cdots \ge {\ell }_k\text{.}$

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