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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.

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I recommend you the following reference book on Free Lie algebras.

C. Reutenauer, Christophe, Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press Oxford University Press (1993), ISBN 978-0-19-853679-6

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 $X$.

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