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^\...
Marcos Gonzalez's user avatar
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 ...
Sergei Ivanov's user avatar
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 $R$},\\ ...
Oliver Nash's user avatar
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4 votes
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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 ...
user118746's user avatar
3 votes
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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 ...
cs89's user avatar
  • 971
2 votes
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83 views

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 ...
guido giuliani's user avatar
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 ...
user118746's user avatar
2 votes
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498 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 ...
GA316's user avatar
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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 $\...
GA316's user avatar
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What is the order of these binary trees?

In the book "Free Lie algebras" by the author Christophe Reutenauer, Example 4.2 (in subsection 4.1) gives the trees of degree $\le 5$ of a Hall set in magma $M(A)$, where $A=\{a,b\}$ as the ...
Chiên BUI's user avatar