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J. M. Pérez-Izquierdo

Publications and source records attributed to J. M. Pérez-Izquierdo.

7 recordsLinked to original sources
arXiv2026

Non-associative Solomon's descent algebras

Motivated by the observation that universal enveloping algebras of relatively free Sabinin algebras admit natural associative internal products, we revisit the classical relationship between Solomon's descent algebra, noncommutative symmetric functions, and Patras' descent algebras of graded bialgebras from a non-associative perspective. Using the language of algebras over cocommutative connected Hopf operads as a unifying framework, we describe a setting in which these connections extend beyond the associative case. This viewpoint allows us to interpret the free non-associative algebra as a non-associative analogue of Solomon's descent algebra inside a suitably defined non-associative Malvenuto-Reutenauer-type Hopf algebra. We further provide a detailed classification of this object as an algebra, as a coalgebra, and as an associative algebra with respect to its internal product.

math.RA↗
arXiv2017

Hyporeductive and Pseudoreductive Hopf algebras

In his generalization of reductive homogeneous spaces, Lev Sabinin showed that Lie's fundamental theorems hold for local analytic hyporeductive and pseudoreductive loops. We derive Sabinin's results in an algebraic context in terms of non-associative Hopf algebras that satisfy the analog of the hyporeductive and pseudoreductive identities for loops.

math.RA↗
arXiv2013

Hopf algebras in non-associative Lie theory

We review the developments in the Lie theory for non-associative products from 2000 to date and describe the current understanding of the subject in view of the recent works, many of which use non-associative Hopf algebras as the main tool.

math.RA↗
arXiv2009

Formal multiplications, bialgebras of distributions and non-associative Lie theory

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.

math.RA↗
arXiv2005

Ideals in non-associative universal enveloping algebras of Lie triple systems

The notion of a non-associative universal enveloping algebra for a Lie triple system arises when Lie triple systems are considered as Bol algebras (more generally, Sabinin algebras). In this paper a new construction for these universal enveloping algebras is given, and their properties are studied. It is shown that universal enveloping algebras of Lie triple systems have surprisingly few ideals. It is conjectured, and the conjecture is verified on several examples, that the only proper ideal of the universal enveloping algebra of a simple Lie triple system is the augmentation ideal.

math.RA↗
arXiv2004

Dimension filtration on loops

We show that the graded group associated to the dimension filtration on a loop acquires the structure of a Sabinin algebra after being tensored with a field of characteristic zero. The key to the proof is the interpretation of the primitive operations of Umirbaev and Shestakov in terms of the operations on a loop that measure the failure of the associator to be a homomorphism.

math.GR↗