Anti-dendriform algebras, new splitting of operations and Novikov-type algebras

We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti-O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}$$\end{document}-operators and anti-Rota-Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of q-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations.