Duality for generalised differentials on quantum groups

JOURNAL OF ALGEBRA

卷:439

Article

Majid, Shahn¹  Tao, Wen-Qing¹ 

[1] Queen Mary Univ London, Sch Math Sci, London EL 4NS, England

关键词: Noncommutative geometry;    Quantum group;    Hopf algebra;    Differential calculus;    Bicovariant;    Quiver;    Shuffle algebra;    Crossed module;    Duality;   

DOI  :  10.1016/j.jalgebra.2015.03.023

来源: Elsevier

【 摘 要 】

We study generalised differential structures (Omega(1), d) on an algebra A, where A circle times A -> Omega(1) given by a circle times b -> adb need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs (Lambda(1), w) where Lambda(1) is a right module and w a right module map, and the Hopf algebra bicovariant case corresponds to morphisms w : A(+) -> Lambda(1) in the category of right crossed (or Drinfeld-Radford-Yetter) modules over A. When A = U(theta) the generalised left covariant differential structures are classified by cocycles w is an element of Z(1) (theta, 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 superHopf algebra gives the same structure on the dual Hopf algebra. Accordingly, group 1-cocycles correspond precisely to codifferential structures on algebraic groups and function algebras. Among general constructions, we show that first order data (Lambda(1), w) on a Hopf algebra A extends canonically to a strongly bicovariant differential graded algebra via the braided super-shuffle algebra. The theory is also applied to quantum groups with Omega(1)(C-q,(G)) dually paired to Omega(1) (U-q(theta)). (C) 2015 Elsevier Inc. All rights reserved.

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