Modeling free Lie algebras with matrix algebras

Question

I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that appears in such expressions. These are collectively referred to as the "Hall basis". [See here ]

I would like to find a shortcut in terms of matrix algebras. In particular I was wondering if there is a matrix algebra of $n$-dimensional matrices in which any member can be written as a sum of commutators of up to degree $d$. I think $n$ will grow exponentially with $d$.

I am trying to use such a method to streamline some calculations in the dynamics of open quantum systems were such expansions (e.g. Magnus expansion ) occur naturally but are hard to work with. I think the solution must have to do with triangular matrices but I might be missing something. Perhaps the same thing can be done with a little overhead by rephrasing in terms of products and nil-potency as opposed to commutators. Any pointers in helpful directions are much appreciated!

Answer 1

This is not supposed to be an answer to your question, just a bunch of casual remarks.

1. Free Lie algebras indeed may be approximated (in some strict sense) by, say, Lie algebras $sl_n(K)$ (observing that there is no nontrivial identity satisfied by $sl_n(K)$'s for all $n$), but this appears to be not very constructive, at least I cannot see how it can buy something for kind of computations you are interested in.

2. In arXiv:1008.2380 , Bremner and Hu study (in a very computational-oriented way) invariants of actions of low-dimensional matrix Lie algebras on a free Lie algebra. May be this is related to what you are looking for.

3. There is a lot of other works dedicated to various computations in free Lie algebras, sometimes on the borderline with computer science. (Some of them are referenced in the paper by Munthe-Kaas and Owren you link to, so you are probably aware of that). Usually these computatuions are performed not in terms of matrix algebras, but, again, may be it could be useful for the sort of calculus you are interested in.

4. In: M. Grayson and R. Grossman, Models for free nilpotent Lie algebras, J. Algebra 35 (1990), 177-191 http://pubs.rgrossman.com/dl/journal-005.pdf , they tackle a related problem: computations in free nilpotent Lie algebras. The latters admit a represemtation by (upper-triangular) matrices - unlike in the case of free algebras this is not approximation, but a true embedding - but the authors are unsatisfied with it and pursue another one, by vector fields.