Duality for Generalised Differentials on Quantum Groups and Hopf quivers

TL;DR

This paper develops a unified theory of generalized differential and codifferential structures on algebras and coalgebras, especially Hopf algebras, using duality, cocycles, and braided algebra constructions, with applications to quantum groups and Hopf quivers.

Contribution

It introduces and classifies generalized differential structures on algebras and coalgebras, including dual notions and their constructions, extending the theory to quantum groups and Hopf quivers.

Findings

Classification of generalized differential structures via cocycles.

Construction methods using braided-antisymmetrizers and braided tensor algebras.

Application to quantum groups and finite groups with dual pairings.

Abstract

We study generalised differential structures $\Omega^1,d$ on an algebra $A$, where $A\tens A\to \Omega^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs $(\Lambda^1,\omega)$ where $\Lambda^1$ is a right module and $\omega$ a right module map, and the Hopf algebra bicovariant case corresponds to morphisms $\omega:A^+\to \Lambda^1$ in the category of right crossed (or Drinfeld-Radford-Yetter) modules over $A$. When $A=U(g)$ the generalised left-covariant differential structures are classified by cocycles $\omega\in Z^1(g,\Lambda^1)$. We then introduce and study the dual notion of a codifferential structure $(\Omega^1,i)$ on a coalgebra and for Hopf algebras the self-dual notion of a strongly bicovariant differential graded algebra $(\Omega,d)$ augmented by a codifferential $i$ of degree -1. Here $\Omega$ is a graded super-Hopf algebra extending the Hopf algebra $\Omega^0=A$ and, where applicable, the dual super-Hopf algebra gives the same structure on the dual Hopf algebra. We show how to construct such objects from first order data, with both a minimal construction using braided-antisymmetrizes and a maximal one using braided tensor algebras and with dual given via braided-shuffle algebras. The theory is applied to quantum groups with $\Omega^1(C_q(G))$ dually paired to $\Omega^1(U_q(g))$, and to finite groups in relation to (super) Hopf quivers.