# 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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### 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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### 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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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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### 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)...
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 ...