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Questions tagged [free-lie-algebra]

A free Lie algebra, over a given field $K$, is a Lie algebra generated by a set $X$, without any imposed relations other than the defining relations of alternating bilinearity and the Jacobi identity.

Let $X$ be a set and $i: X \to L$ a morphism of sets from $X$ into a Lie algebra $L$. The Lie algebra $L$ is called free on $X$ if for any Lie algebra $A$ with a morphism of sets $f: X \to A$, there is a unique Lie algebra morphism $g: L \to A $ such that $f = g \circ i$.

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Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): First, consider the tensor algebra $T(M)$. Take the…
Martin Brandenburg
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Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? In mathematics, a free Lie algebra, over a given field $K$, is a Lie algebra generated by a set $X$, without any…
Zuriel
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The universal enveloping algebra of free Lie algebra is the tensor algebra on the Free Abelian Group?

Let $A$ be a set, and let $M_A$ be the free abelian group generated by $A$. Let $L(A)$ be the free Lie algebra generated by $A$. I am reading On Injective Homomorphisms For Pure Braid Groups, And Associated Lie Algebras By F. R. Cohen And Stratos…
Zuriel
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Free Lie algebra over the set $X=\{x\}$

How can I describe more concretely the free lie algebra over the singleton $X=\{x\}$? Is there any intuition on how to visualize the free Lie algebra when $X$ is more arbitrary? By the definition I just know that the free Lie algebra over X is a…
user2345678
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A software for calculations in free Lie algebras

I am interested to expand the symmetrised tensor products of several elements of the Philip Hall basis of a free Lie algebra in tensor form. For example, if the algebra has two generators $x$ and $y$, then we have $\ell_1=x$, $\ell_2=y$,…
Anatoliy Malyarenko
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Gröbner–Shirshov basis for free Lie algebras

I have already studied several papers related to composition-diamond lemma completely and now I am familiar with theoretical tools. For instance I realized the concept of associative Lyndon–Shirshov words, non-associative Lyndon–Shirshov words and…
Nil
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Difference between Free Lie algebra and universal enveloping algebra

I have seen so many questions about free Lie algebra and universal enveloping algebra. As in this question (Universal envoloping algebra of a free Lie algebra.), a reader asked for the universal enveloping algebra of a free Lie algebra. I am…
MANI
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The elements $f_1,\cdots, f_n$ generate $\mathfrak{\tilde n_{-} }$ freely

I've started studying Kac-Moody algebras and free lie algebras is a really new thing for me. I am trying to understand the statement (b) of theorem 1.2 in the following book Theorem 1.2, statement (b), more specifically: from the fact that…
user2345678
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Universal envoloping algebra of a free Lie algebra.

In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X. It is said that by the Poincaré–Birkhoff–Witt theorem the universal enveloping algebra of a free Lie…
LJR
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Defining a filtration on free Lie algebra in terms of generators

Let $L$ be the free Lie algebra generated by the set $X = \{x_i\}_{i \in \mathbb{N}}$ over a field $k$. A Lie monomial is a bracketed word of elements of $X$ of finite length, and $L$ is spanned by the Lie monomials. I would like to introduce a…
user65970
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Intuition for Kozsul algebras/Lie algebras

What is the intuition behind Koszul graded-commutative algebras and Lie algebras? Why is it an interesting property to study in commutative algebra? I know why it's interesting in homotopy theory but what about in algebraic geometry or commutative…
joe
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Poincare Series of a graded algebra (revisited)

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the formal series $$P(A) = \sum_{i \geq 0}…
Intuition
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Jacobi identity, free nilpotent Lie algebra

Is there a general formula for the Jacobi identity on the free $\nu$-nilpotent Lie algebra $\mathfrak{FL}(\nu,n)$ on $n$ generators? In math overflow I found a list of the Hall bases. Looking at Hall's paper, though, the proof only shows that the…
bliipbluup
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Canonical isomorphism between Polynomial algebra and Symmetric algebra.

I am studying "Introduction to Lie algebra" written by "J.E. Humphreys". In chapter 10, when he is giving the concept of universal enveloping algebra, he introduces the notion of the symmetric algebra and says that it is isomorphic to the polynomial…
MANI
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About embedding associative algebras

In this paper https://link.springer.com/content/pdf/10.1007/BF01877233.pdf, there is a corollary about embedding of countable associate algebras in a simple associative algebra with three generators. (Corollary #1). May you help me to sort out this…
Nil
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