Some results on L-dendriform algebras

Abstract

We introduce a notion of L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the $\mathcal{O}$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce a notion of $\mathcal{O}$-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra.

15 pages

References in corpus (3)

• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators

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• Special identities for the pre-Jordan product in the free dendriform algebra
• Replicating of binary operads, Koszul duality, Manin products and average operators
• Splitting of operads and Rota-Baxter operators on operads
• Rota-Baxter Operators on pre-Lie superalgebras and beyond
• Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
• The Cayley-Dickson process for dialgebras
• A trick to compute certain Manin products of operads
• Polarization and deformations of generalized dendriform algebras
• Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer-Cartan equations
• Domain transfer convolutional attribute embedding
• Free associative algebras, noncommutative Grobner bases, and universal associative envelopes for nonassociative structures
• Splitting of operations for alternative and Malcev structures