Related papers: Hopf heap modules, Rota-Baxter operators, and thei…
This is a survey on spherical Hopf algebras. We give criteria to decide when a Hopf algebra is spherical and collect examples. We discuss tilting modules as a mean to obtain a fusion subcategory of the non-degenerate quotient of the…
A relative Rota-Baxter operator on Lie 2-groups is introduced as a pair of relative Rota-Baxter operators on the underlying Lie groups which is also a Lie groupoid morphism. Such an operator induces a factorization theorem for Lie 2-groups…
The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain…
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We…
In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the…
Covariant Hom-bimodules are introduced and the structure theory of them in the Hom-setting is studied in a detailed way. The category of bicovariant Hom-bimodules is proved to be a (pre)braided monoidal category and its structure theory is…
In this article we discuss the Hopf algebras spanned by the adjacency matrices in detail. We show that there two Hopf algebraic structures concerning the adjacency matrices, one is the copy of Connes-Kreimer Hopf algebra, another one is the…
We endow the set of isomorphic classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform…
Motivated by Quantum Mechanics considerations, we expose some cross product constructions on a groupoid structure. Furthermore, critical remarks are made on some basic formal aspects of the Hopf algebra structure.
We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant…
The theme of this article is the algebraic combinatorics of leaf-labeled rooted binary trees and forests of such trees. The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees.…
We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of…
We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's…
This paper is devoted to the study of Hopf braces projections in a monoidal setting. Given a cocommutative Hopf brace ${\mathbb H}$ in a strict symmetric monoidal category ${\sf C}$, we define the braided monoidal category of left…
In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general…
Transformation of operator algebras under Hopf algebra twist is studied. It is shown that that adjoint module algebras are stable under the twist. Applications to vector fields on non-commutative space-time are considered.
We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle…
Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of…
The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique…