Bialgebra structures of 2-associative algebras

TL;DR

This paper explores new bialgebra structures related to 2-associative algebras, including their construction from associative algebras and classification in low dimensions, expanding the algebraic framework of these structures.

Contribution

It introduces and classifies 2-associative bialgebras, 2-bialgebras, and 2-2-bialgebras, extending the understanding of algebraic structures with dual associative operations.

Findings

Constructed 2-associative bialgebras from associative algebras

Classified these structures in low dimensions

Extended the algebraic framework of 2-associative structures

Abstract

This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative algebra and the classification of these structures in low dimensions.