Algebras, hyperalgebras, nonassociative bialgebras and loops

Pérez-Izquierdo, José M.

doi:10.1016/j.aim.2006.04.001

Cited by 86 publications

(45 citation statements)

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“…These structures, now known as Sabinin algebras, can be integrated under some convergence conditions to local loops: essentially, they are the analog of Lie algebras in the non-associative setting. Shestakov and Umirbaev later showed [SU02] that the set of primitive elements in any bialgebra has the structure of a Sabinin algebra, and it was proved by the second author of the present paper [PI07], that each Sabinin algebra arises in this way. The main purpose of the present paper is to show how the Lie theory for nonassociative formal multiplications can be constructed by first passing from a formal multiplication to the corresponding bialgebra of distributions, and then to the Sabinin algebra of the primitive elements of the latter.…”

Section: Introductionmentioning

confidence: 99%

“…Operations x\y and x/y induce the corresponding operations on distributions; these operations were first considered in [PI07]. We shall simply write µ\ν and µ/ν to denote these operations.…”

Section: Formal Multiplications and Bialgebras Of Distributionsmentioning

confidence: 99%

“…An affirmative answer (with a modified version of the operations p( ; ; ) and, hence, of the functor UX) was given in [PI07]. Given a Sabinin algebra (V, ; , , Φ ′ ) the corresponding bialgebra is denoted by U (V ) and has the following universal property: any homomorphism of Sabinin algebras from V to a unital algebra A extends to a unique homomorphism of unital algebras U (V ) → A.…”

Section: Bialgebras Of Distributions and Sabinin Algebrasmentioning

confidence: 99%

“…In [PI07] w ′ was called the linearization of w. It was proved that certain bialgebras constructed from Malcev algebras satisfy the identity (3) above. That was surprising since the construction of those bialgebras [PIS04] had no relation with Moufang loops.…”

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confidence: 99%

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Formal multiplications, bialgebras of distributions and nonassociative Lie theory

Mostovoy

Pérez-Izquierdo

2010

Transformation Groups

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Abstract. We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras.Starting with a formal multiplication we construct a non-associative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how identities in loops produce identities in bialgebras. While associativity in loops translates into associativity in algebras, other loop identities (such as the Moufang identity) produce new algebra identities. Finally, we define a class of unital formal multiplications for which Ado's theorem holds and give examples of formal loops outside this class.A by-product of the constructions of this paper is a new identity on Bernoulli numbers. We give two proofs: one coming from the formula for the non-associative logarithm, and the other (due to D. Zagier) using generating functions.

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Section: Introductionmentioning

confidence: 99%

Section: Formal Multiplications and Bialgebras Of Distributionsmentioning

confidence: 99%

Section: Bialgebras Of Distributions and Sabinin Algebrasmentioning

confidence: 99%

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confidence: 99%

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Formal multiplications, bialgebras of distributions and nonassociative Lie theory

Mostovoy

Pérez-Izquierdo

2010

Transformation Groups

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“…They were initially introduced by Mikheev and Sabinin as tangent structures to general affine connections, see [11,5]. Later, it was proved that primitive elements in a non-associative bialgebra form a Sabinin algebra [12], and that every Sabinin algebra arises this way [10].…”

Section: The Jennings Theoremmentioning

confidence: 99%

Nilpotency and Dimension Series for Loops

Mostovoy

2008

Communications in Algebra

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Abstract. 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 subgroupsdefined inductively by setting either:is the largest of all subgroups K of G with the property that K/H is contained in the centre of G/H; or, G i to be generated by all commutators [x, y] with x ∈ G p and y ∈ G q 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 Liudm...

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Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

Concepción

Makhlouf

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Algebras, hyperalgebras, nonassociative bialgebras and loops

Cited by 86 publications

References 14 publications

Formal multiplications, bialgebras of distributions and nonassociative Lie theory

Formal multiplications, bialgebras of distributions and nonassociative Lie theory

Nilpotency and Dimension Series for Loops

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

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