Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0012073

Hopf group-coalgebras

Alexis Virelizier
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study algebraic properties of Hopf group-coalgebras, recently introduced by Turaev. We show the existence of integrals and traces for such coalgebras, and we generalize the main properties of quasitriangular and ribbon Hopf algebras to the setting of Hopf group-coalgebras.

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexis Virelizier. 2000-12-11. Hopf group-coalgebras. https://arxiv.org/abs/math/0012073

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Polylogarithmic characterizations of the reduced coaction Lie algebra and the double shuffle Lie algebra, and their relation to the Kashiwara--Vergne Lie algebra

The reduced coaction Lie algebra $\mathfrak{rc}_0$ is defined by a skew-symmetric condition together with an algebraic equation involving the reduced coaction, a non-cyclic refinement of the necklace cobracket. The double shuffle Lie algebra $\mathfrak{dmr}_0$ encodes the regularized double shuffle relations in the study of formal multiple zeta values, while the Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2$ arose in the study of the Kashiwara--Vergne conjecture. Our main result is the description of $\mathfrak{dmr}_0$ and $\mathfrak{rc}_0$ by vanishing conditions of different strengths on the same object: the defect of Drinfeld's pentagon equation; $\mathfrak{dmr}_0$ is characterised by the vanishing of the associated two-variable multiple polylogarithms, and $\mathfrak{rc}_0$ by the vanishing of those whose second index has depth $1$. We also establish explicit relations among these three Lie algebras. First, the skew-symmetric elements of $\mathfrak{dmr}_0$ embed into $\mathfrak{rc}_0$. Second, under an additional condition, we construct an injective Lie algebra morphism from $\mathfrak{rc}_0$ to $\mathfrak{krv}_2$.

math.QA↗
arXiv2026

Graded Necklace Lie Bialgebras and Batalin-Vilkovisky Formalism

An involutive Lie bialgebra induces a Batalin-Vilkovisky operator on its exterior algebra. We introduce a graded generalization of the necklace Lie bialgebra, which depends on a choice of a quiver $Q$. We relate the resulting Batalin-Vilkovisky structure to the Batalin-Vilkovisky structure coming from a degree $-1$ symplectic form on a suitably defined representation variety of the quiver $Q$. The morphism intertwining these Batalin-Vilkovisky algebras will be given by a twisted trace, recovering the usual (super)trace and the odd trace.

math.QA↗
arXiv2026

Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Hopf modules to be semisimple. We also show that $A$ can be embedded into a left $H$-comodule algebra of a specific form when $H$ and $A$ are $\mathbb{N}_0$-graded. As a consequence, we prove that if $H$ is finite-dimensional and $A^{\mathrm{co} H}=\Bbbk$, then $A$ is finite-dimensional and $\dim A$ divides $\dim H$.

math.QA↗