What is the meaning of "Word Problem" in free algebras

Question

I am looking for a clear definition of Word Problem in free Lie algebras. May you please describe that?

Answer 1

The word problem is, in general, the problem of deciding whether two words in the generators of a given Lie algebra actually represent the same element.

There does exist a wikipedia page about the word problem for groups, which seems to give a pretty good description. The problem is the same for Lie algebras; these are algebraic objects which can also be concisely described using generators and relations.

There appears to have been some work done on this problem, for example here, although I do not have access to this document.

For an example (I'll stick to group theory since the idea is the same), look at for example the dihedral group. A presentation of $D_n$ is

$$\langle r,s | r^n = s^2 = (rs)^2 = 1 \rangle.$$

Suppose that $n=4$. Then we have the dihedral group of order $8$ on our hands. But what if I wrote down two words in the generators... say, $r^3$ and $srs$. These look different.

However, I know that $r^3 r = r^4 = 1 \implies r^3 = r^{-1}$

and $rsrs = (rs)^2 = 1 \implies srs = r^{-1}.$

The two words which I wrote down actually represent the same element. See? Tricky right?