# How to get to the lyndon-basis in free Lie algebras for matrices

Assume that I have a matrix algebra where the underlying vector space is $\mathbb{C}^{n \times n}$ for a fixed $n \in \mathbb{N}$.

As matrix-multiplication is an associative binary operation, we are dealing with an associative algebra. As for associate algebra, one can define a Lie-bracket as the ordinary commutator, namely $[A,B] = AB - BA$.

I have problems understanding the free associative algebra and especially the free Lie-algebra. I don't see how to connect them.

As for associative algebras, I think that the free associative algebra over $X$ with $X = \{ x_1 , \ldots, x_k \}$ is the algebra containing all words over $X$, call it $A(X)$.

Questions here: Are the $X$ here introduced additional symbols so that $A(X)$ is infinite dimensional, almost like the (noncommutative) polynomial ring? Does the substitution homomorphism play any role at this point? With other words: Is this construction still something finite dimensional as the $n\times n$ matrices have $e_{ij} = \text{matrix with entries } \delta_{ij}$ as a basis?

Now, how is the free Lie-algebra constructed?

Is it a correct approach to take the free vector space that is generated by $A(X)$, call it $V(A(X))$, and factoring out the relations that define a Lie-algebra (antireflexivity, jacoby identity)?

Any help and literature reference is greatly appreciated! Please note that I do like to understand the matrix situation in particular, so I would not like to dip too deep into algebra.

About your first question. Let $R$ be a commutative ring. Then the free $R$-algebra on a set $X$ is exactly the algebra of polynomials with coefficients in $R$ and noncommuting indeterminates taken from $X$.