Questions tagged [free-lie-algebras]

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Associativity of the Campbell-Baker-Hausdorff operation on a Banach-Lie algebra

Let $(\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}, \Vert \cdot \Vert_\mathfrak{g})$ be an infinite-dimensional Banach-Lie algebra, and let us define for any $a,b \in \mathfrak{g}$ the series $$~ Z^\...
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4 votes
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Is the natural map from the free Lie algebra to the free associative algebra injective?

$\newcommand{\im}{\operatorname{im}}$Given a set $X$ and non-zero unital commutative ring $R$, let: \begin{align} A &= \mbox{free unital, associative algebra on $X$ with coefficients in $R$},\\ ...
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Elements of the Hall basis described via permutations

Good morning, Suppose that $\mathfrak{g}$ is a free graded Lie algebra generated by the elements $1,\dots, n$, i.e. assume that $\mathfrak{g}_1=\mathrm{span}\{1,\dots, n\}$. Let us focus on the Hall ...
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2 votes
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How can we interpret Groebner basis in a special case?

Let consider a free Lie algebra generated by $X$ with a set of relations $S$ such that the degree of leading monomial of relations in $S$ are greater than or equal to $2$. Let assume that we compute ...
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1 vote
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is this the correct universal property of free Lie superalgebras?

Consider a $\Bbb Z_2$ graded set $A$. Universal property of free Lie superalgebra $FLS(A)$: Let $\mathfrak g$ be a Lie superalgebra and let $\Phi: A \to \mathfrak g$ be a set map which preserves the $\...
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1 answer
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Ideal of the free Lie algebra L(x,y) generated by x

Let $L=L(x,y)$ be the free Lie algebra generated by letters $x,y.$ For a vector subspace $V\leq L$ we denote by $[V,L]$ the vector space spanned by brackets $[v,l],v\in V,l\in L.$ A vector subspace $V\...
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2 votes
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282 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 ...
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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 ...
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  • 409
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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 ...
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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 ...
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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 ...
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2 votes
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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 ...
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2 votes
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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 ...
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3 votes
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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 ...
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7 votes
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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 ...
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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 ...
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13 votes
2 answers
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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) / \...
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4 votes
0 answers
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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 ...
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11 votes
2 answers
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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 ...
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24 votes
1 answer
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The BCH series in terms of Lyndon words

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 ...
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4 votes
1 answer
300 views

About the term "tangential derivation" on a free Lie algebra.

Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, ...
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8 votes
4 answers
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list of Hall basis

Anyone know a place where the standard Hall basis is listed up to at least 5-fold brackets? And for graded Lie algebras? The rules are clear but I'd rather not turn the crank myself. Google search ...
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1 vote
2 answers
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Invariant space of lifted Chevalley automorphisms of the tensor algebra

Question. Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form $\left[a,\left\lbrace ...
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