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The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are:

  1. $a^b= b^{-1}ab$, taking the conjugate in $F$.
  2. $[a,b]= aba^{-1}b^{-1}$, taking the commutator of two elements in $F$.

And that's all. (If I were allowing only conjugation then this structure would be a conjugation quandle, but I'm also allowing to take commutators, but not to take products- the group product is not part of the structure). Are such structures at all studied or known?

Question: Is the full set of relations in this structure known (in the sense of universal algebra)? Is there a proof in the literature?
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Have you looked up the "Lie ring method" in group theory? It seems related. You have an identity something like $[a,b^{-1},c]^{b}[b,c^{-1},a]^{c}[c,a^{-1},b]^{a}] = 1$, which is reminiscent of the Jacobi identity ( I may have got the formula wrong from memory, but it can be found in group theory texts, and may be due to P. Hall). Taking the direct sum of the quotients of successive terms of the lower central series gives a Lie ring like structure which has been looked at in the literature. – Geoff Robinson Dec 4 '12 at 8:09
IIRC that identity is due to Witt (or both?) (You get from it to Jacobi's by setting $a=\exp(At)$ and so on and computing the third derivative of the left hand side of he Witt identity) – Mariano Suárez-Alvarez Dec 4 '12 at 12:36
I don't think I have Witt-Hall because I can't multiply. I can only conjugate and take commutators. – Daniel Moskovich Dec 4 '12 at 13:58
You're asking for relations in the sense of universal algebra? If so, what kind of quantification are you using? – Ian Agol Dec 4 '12 at 16:29
It's probably not quite what you are asking, but I'll mention it anyway: A lot of the relations among commutators (and with conjugation, and especially the interaction of commutators and powers) are considered in the study of "commutator collection" and "basic commutators". The big work on them is Ward's "Basic Commutators", Philos. Trans. Roy. Soc. London Series A, vol 264 (1969), 343-412, MR 0251148 – Arturo Magidin Dec 5 '12 at 6:51
up vote 8 down vote accepted

There is a notion of multiplicative Lie algebras introduced here: Ellis, Graham J. On five well-known commutator identities. J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 1–19. The signature there does include multiplication, though. The problem of finding axioms was solved there (I think somebody finally proved that the five standard commutator identities suffice). If the product operation is removed from the signature, the correct first question would be if the class is first order axiomatizable. It is not clear. One can write a bunch of axioms which certainly hold, but this list is not complete:

  1. $[x,x]=[y,y]$ (call this element 1)

  2. $[x,1]=[1,x]=1$

  3. $[[x,y], [y,x]]=1$

  4. $[x,y]=1, [x,z]=1 \to [x,[y,z]]=1$

  5. $[x,y]^z=[x^z,y^z]$

  6. $x^z=x \leftrightarrow [x,z]=1$

  7. $\exists x,y, [x,y]\ne 1 \to \exists x \exists a, [x,a]=1, a\ne 1, a\ne x$

(The last axiom follows from the fact that if all elements of a group are of order 2, then the group is Abelian.)

I think it is clear that the class cannot be axiomatized only by universal formulas (because it is not closed under taking subalgebras). It is closed under taking ultraproducts, which is a good news.

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Great to see you active again! I was already quite worried you left MO. – user9072 Dec 18 '12 at 12:44
me too !!! :) :) – Alexander Chervov Dec 18 '12 at 13:06

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