Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras

Abstract

This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra, defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively. We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. Free totally compatible dialgebras are constructed. We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra, generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.

17 pages

References in corpus (5)

• Loday-type Algebras and the Rota-Baxter Relation
• Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
• Character formulas for the operad of two compatible brackets and for the bihamiltonian operad
• Operads of compatible structures and weighted partitions
• Integrable matrix equations related to pairs of compatible associative algebras

Cited by in corpus (4)

• Combinatorial Hopf algebra for interconnected nonlinear systems
• Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras