Nilpotency and Dimension Series for Loops

We take a step towards the development of a nilpotency theory for loops based on the commutatorassociator filtration instead of the lower central series. This nilpotency theory shares many essential features with the associative case. In particular, we show that the isolator of the nth commutator-associator subloop coincides with the nth dimension subloop over a field of characteristic zero. The lower central series for groups can be defined in two essentially different ways. Namely, the lower central series of a group G is a descending filtration of G by normal subgroups G = G1 ⊇ G2 ⊇ . . . defined inductively by setting either: Gi = [G,Gi−1]where [G,H ] is the largest of all subgroupsK of G with the property that K/H is contained in the centre of G/H ; or, Gi to be generated by all commutators [x, y] with x ∈ Gp and y ∈ Gq with p+ q ≥ i. These two definitions are equivalent for groups. However, in the non-associative case they give rise to rather different objects. The first definition, with “groups” replaced by “loops”, produces Bruck’s lower central series, see [1]. An analog of the second definiton for loops was introduced in [6] under the name of “commutator-associator filtration”. The terms of the commutator-associator filtration contain, but do not necessarily coincide with the corresponding terms of the lower central series. The main advantage of the commutator-associator filtration is the existence of a rich algebraic structure on the associated graded abelian group, consisting of an infinite number of multilinear operations. It can be seen that two of the operations, namely those induced by the loop commutator and the loop associator, satisfy the Akivis identity. However, the complete identification of this algebraic structure is a non-trivial problem. In this paper we set up a nilpotency theory for loops based on the commutator-associator filtration. In this theory the standard techniques of the theory of nilpotent groups can be applied and various results valid for groups can be extended to loops. In particular, we shall prove that for an arbitrary loop the isolators of the terms of the commutator-associator filtration coincide with the dimension series. As a corollary, we identify the algebraic structure on the graded Q-vector space associated to the commutator-associator filtration: it turns out to be a Sabinin algebra. Throughout the text we make the emphasis on the similarities, rather than differences, between nilpotency theories for groups and for general loops. This should not leave the impression that extending the nilpotency theory from groups to loops is a straightforward task. In particular, the residual nilpotency of the free loop, established for the lower central series by Higman [3], remains an open question for the commutatorassociator filtration. We did not strive for completeness; many relevant topics (such as applications to particular classes of loops, relation to the nilpotency of the multiplication group of the loop et cetera) have remained outside the scope of this paper. Acknowledgments. I would like to thank Liudmila Sabinina and José Maŕıa Pérez Izquierdo for discussions. This work was supported by the CONACyT grant CO2-44100. 2000 Mathematics Subject Classification. 20N05,17D99. 1