Nilpotent Sabinin algebras

Abstract

In this paper we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of the Ado theorem for loops whose commutator-associator filtration is of finite length.

Hopf algebras in non-associative Lie theory

“…This is not the standard definition of nilpotency in loops; however, there are strong indications that it is the correct one [18,34,35,39]. Since the commutator-associator filtration is defined in terms of words (namely, commutators, associators and associator deviations and their compositions) nilpotent loops of class n form a variety.…”

“…, X n ) for all n ≥ 2 satisfying the relations ( 9)- (11). It can be shown that if l is a flat Sabinin algebra in this sense, there exists a decomposition (8) which gives rise to the operations on l, [39].…”

“…If applied to a Sabinin algebra l, it gives a formal loop whose tangent algebra has the same brackets as l but whose multioperator is zero. In particular, it provides formal integration for flat Sabinin algebras [39]. If l is not flat, one defines the Baker-Campbell-Hausdorff series to be L(x, Φ(x, y)),…”

“…Then a Sabinin algebra is nilpotent of class n if all the brackets of weight greater than n vanish in it. For a flat Sabinin algebra, nilpotency is the same as the nilpotency of the Lie algebra in which it is a summand according to the construction of Section 3.2, see [39].…”