BIALGEBRA STRUCTURES OF 2-ASSOCIATIVE ALGEBRAS

This work is devoted to the study of 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 any associative algebra and the classification of these structures in low dimensions.