# 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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### homogeneous subalgebra of a Lie superalgebra

In the case of Lie superalgebras which are defined as $L=L_{\bar{0}} \oplus L_{\bar{1}}$, I am a bit confused about the term "homogeneous subalgebra". Does it mean that the subalgebra which ...
• 1,316
1 vote
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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 ...
• 157
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### Is every solvable algebras are nilpotent?

An algebra $A$ is called nilpotent if $A^n=0$ for some positive integer $n$. Also, we call algebra $A$ is solvable if $A^{(n)}=0$, with solvable index $n$, where $A^{(n)} = A^{(n-1)}A^{(n-1)}$. Can we ...
• 19
1 vote
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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 ...
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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 $y$, ...
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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'...
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