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

TL;DR

This paper develops a non-associative Lie theory connecting formal loops, Sabinin algebras, and bialgebras, revealing new identities and extending classical results to non-associative contexts.

Contribution

It introduces a non-associative framework linking formal multiplications, Sabinin algebras, and bialgebras, and explores identities and Ado's theorem in this setting.

Findings

Constructed a non-associative bialgebra from formal loops.

Compared functors from formal loops to Sabinin algebras, highlighting differences.

Discovered a new identity involving Bernoulli numbers.

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.