# Questions tagged [free-lie-algebras]

The tag has no usage guidance.

19 questions
Filter by
Sorted by
Tagged with
35 views

130 views

### Lyndon basis of free Lie algebras

Let $A = \{a,b,c,d\}$ be a set of totally ordered alphabets, a Lyndon word over $A$ is a word $w$ in $A^*$ such that if $w=uv$ is a factorization of $w$ into non-empty subwords, then $u<v$ in ...
148 views

### Free Lie algebra and nilpotent groups in Rothschild and Stein's paper

In Rothschild, Linda Preiss; Stein, Elias M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(1976), 247-320 (1977). ZBL0346.35030. PDF at archive.ymsc.tsinghua.edu.cn ...
53 views

### Lie subalgebra as the intersection of the subalgebra and the Lie subalgebra

Let $X = (X_1, \dots X_q)$ be indeterminates. Let $A(X)$ be the free algebra over $X$. Let $L(X) \subset A(X)$ be the free Lie algebra over $X$. I consider some finite set $Y \subset L(X)$ and ...
99 views

### Criterion to construct a $\mathbb{Z}$-basis of a free $\mathbb{Z}$-Lie algebra

Let $L(\mathbb{Z},n)$ (resp. $L(\mathbb{Q},n)$) be the free Lie algebra over $\mathbb{Z}$ (resp. over $\mathbb{Q}$) with generating set $\{x_1,\dots,x_n\}$. Let $\mathcal B$ be a $\mathbb{Q}$-basis ...
63 views

### Lyndon words and free groups [closed]

It is well known that Lyndon words form a basis for free Lie algebras. Is there any analog result for free groups? What is the connection between Lyndon words and free groups? Since groups and Lie ...
117 views

### super Lyndon words

Lyndon words form a basis for Free Lie algebras. In this direction, I need the reference for the super Lyndon words for free Lie superalgebras. Given the definition of super Lyndon words, how to ...
81 views

### Lyndon basis of free Lie superalgebras

Lyndon basis for Free Lie algebras is well known in the literature. My question is, what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of ...
269 views

### Lyndon words and Hall basis

I am looking for an algorithm to produce Hall basis from Lyndon words. First I will recall the definition of the Hall set following Serre's presentation. Let $X$ be a finite set and let $M(X)$ be ...
155 views

### Structure constants of Lyndon-Shirshov basis of the free Lie ring

Let $X$ be an alphabet, ${\sf Lyn}$ be the set of Lyndon words on $X$ and $L$ be the free Lie ring on $X.$ For $w\in {\sf Lyn}$ we denote by $[w]$ the corresponding element of the Lyndon-Shirshov ...
112 views

### How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly?

Let $V$ be a vector space with a basis $v_1, v_2, \ldots, v_n$. Let $T(V)$ be the tensor algebra of $V$. Let $S(Lie(V))$ be the symmetric algebra of the free Lie algebra of $V$. I think that $T(V)$ is ...
349 views

### Free groups and free restricted Lie algebras

If $G$ is any group and $\gamma_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows \mathcal{L}(G) = \bigoplus_{k=1}^{\infty} \gamma_k(G) / \...
134 views

### Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
541 views

### Bracket of lyndon words?

Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the ...
Recently I did some explicit computations that involved the BCH series, $\log(e^x e^y)$. Here $x$ and $y$ are non-commuting variables, and the BCH series lives in the graded completion $FL(x,y)$ of ...