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.

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About subalgebras of free algebras

Let $F$ be any free algebra in an homogeneous variety with the set $X$ of free generators and let $R$ be a subalgebra of $F$ generated by a set $Y$ which is linearly independent modulo $F_2$ ($F_2$ is ...
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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 ...
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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 ...
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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. (...
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About finitely presented algebras and free product

In fact, the free product of two finitely presented Lie algebras is also a finitely presented Lie algebra. Let consider the definition of dialgebras: A diassociative algebra is a $K$-linear space, ...
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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 ...
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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)...