Octo-bialgebras (Q2883151)

The Loday algebras are referred to the series of algebras with a common property of ``splitting associativity'', naturally appearing and with applications in many parts of mathematics and physics, as algebraic \(K\)-theory, operads, homology, Hopf, Lie, and Leibniz algebras, combinatorics, arithmetic, quantum field theory, etc. On the other hand, for an algebra, a bialgebra structure means that there is an algebra structure on the dual space satisfying certain compatible conditions. In the paper under review the authors construct a bialgebra structure for octo-algebras (Loday algebras with 8 binary operations introduced by \textit{Philippe Leroux}, On some remarkable operads constructed from Baxter operators. \url{arXiv:math/0311214}). The main idea is to follow the Drinfeld construction of Lie bialgebras. The construction is equivalent to a double construction of a quadri-algebra with a nondegenerate 2-cocycle or a double construction of an octo-algebra with a nondegenerate invariant bilinear form. Some properties of octo-bialgebras are given, including the study of the coboundary cases which leads to a construction from an analogue of the classical Yang-Baxter equation in an octo-algebra.