L-quadri-algebras

李代数上的dg Loday-Pirashvili模

Dg Loday-Pirashvili modules over Lie algebras

本文研究李代数上dg Loday-Pirashvili模的推广与等价刻画，证明其可消解任意经典Loday-Pirashvili模，并通过dg导子与Kapranov函子构造Leibniz∞[1]代数结构，揭示其与扭曲Atiyah类的联系。

${\mathcal O}$-operators of Loday algebras and analogues of the classical Yang-Baxter equation

We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and necessary condition on the existence of the $2^{n+1}$ operations on an algebra with the $2^{n}$ operations in an associative cluster. The analogues of the classical Yang-Baxter equation in these algebras can be understood as the $\mathcal O$-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerat…

Loday-type Algebras and the Rota-Baxter Relation

In this brief note we would like to report on an observation concerning the relation between Rota-Baxter operators and Loday-type algebras, i.e. dendriform di- and trialgebras. It is shown that associative algebras equipped with a Rota-Baxter operator of arbitrary weight always give such dendriform structures.

Homotopy Loday Algebras and Symplectic $2$-Manifolds

Using the technique of higher derived brackets developed by Voronov, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic $2$-manifold. The algebra we obtain has a particularly nice structure, in that it accommodates the Dorfman bracket of a Courant algebroid as the binary operation in the hierarchy of operations, and the defect in the symmetry of each operation is measurable in a certain precise sense. We move to call such an algebra a homotopy Dorfman algebra, or a $D_\infty$-algebra, which leads to the construction of a homotopy Courant algebro…

Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees

This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …

Loday-Ronco Hopf 代数的一种量子化

A Quantization of the Loday-Ronco Hopf Algebra

本文基于 Eynard-Orantin 拓扑递归构造 Loday-Ronco Hopf 代数的量子化 $k[Y^\infty]_h$，证明其 Hopf 代数性质并刻画拓扑递归解空间的商子代数结构。