The homotopy theory of strong homotopy algebras and bialgebras

The universal Hall bialgebra of a double 2-Segal space

Hall algebras and related constructions have had diverse applications in mathematics and physics, ranging from representation theory and quantum groups to Donaldson-Thomas theory and the algebra of BPS states. The theory of $2$-Segal spaces was introduced independently by Dyckerhoff-Kapranov and Gálvez-Carrillo-Kock-Tonks as a unifying framework for Hall algebras: every $2$-Space defines an algebra in the $\infty$-category of spans, and different Hall algebras correspond to different linearisations of this universal Hall algebra. A recurring theme is that Hall algebras can often be equipped wi…

Limits of Coalgebras, Bialgebras and Hopf Algebras

We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family of coalgebras (resp. bialgebras, Hopf algebras). The equalizers of two morphisms of coalgebras (resp. bialgebras, Hopf algebras) are also described explicitly. As a consequence the categories of coalgebras, bialgebras and Hopf algebras are shown to be complete and a explicit description for limits in the above categories is given.

Split extensions and actions of bialgebras and Hopf algebras

We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, an…

Twisted automorphisms of Hopf algebras

Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the first into Drinfeld twistings of the second. They possess a composition operation extending composition of bialgebra homomorphisms. Gauge transformations of twists, compatible with adjacent homomorphisms, give rise to gauge transformation of twisted homomorphisms, which behave nicely with respect to compositions. Here we study (gauge classes of) twisted automorphisms of cocommutative Hopf algebras. After revising well-known relations between twists, twisted forms of bialgebras and $R$-matrices (for commutative bialgebras)…

代数的广义分裂及其在由平均Lie双代数诱导的Leibniz代数双代数结构中的应用

Generalized splitting of algebras with application to a bialgebra structure of Leibniz algebras induced from averaging Lie bialgebras

本文扩展了operad的经典分裂概念，提出了由系数矩阵参数化的type-M pre-结构族，并将其应用于确定由平均Lie双代数诱导的Leibniz代数双代数结构。

Lie 2-bialgebras

In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras and symplectic Lie algebras.