Rhizaform algebras
Zafar Normatov
TL;DR
Rhizaform algebras introduce a two-operation splitting of anti-associativity, defining $x\ast y = x\succ y + x\prec y$ so that $(A,\ast)$ is anti-associative and $(A,L_\succ,R_\prec)$ forms a bimodule. The framework is connected to operator theory via $\mathcal{O}$-operators and Rota-Baxter operators to construct compatible rhizaform structures, and to Connes cocycles, which yield natural rhizaform realizations and double constructions. The paper develops an operator-theoretic and cohomological toolkit for rhizaforms, including invertible $\mathcal{O}$-operators and double constructions, culminating in a complete classification of 2-dimensional rhizaform algebras and a unified view of nilpotency notions. This advances the study of anti-associative algebras by providing a dendriform-like splitting, a rich interplay with cohomology, and concrete low-dimensional models for further exploration.
Abstract
Any anti-associative algebra gives rise to a Jacobi-Jordan algebra by [x, y] = xy + yx. This article aims to introduce the concept of "rhizaform algebras", which offer an approach to addressing anti-associativity. These algebras are defined by two operations whose sum is anti-associative, with the left and right multiplication operators forming bimodules of the sum of anti-associative algebras. This characterization parallels that of dendriform algebras, where the sum of operations preserves associativity. Additionally, the notions of O-operators and Rota-Baxter operators on anti-associative algebras are presented as tools to interpret rhizaform algebras. Notably, anti-associative algebras with nondegenerate Connes cocycles admit compatible rhizaform algebra structures.