Generalised Noncommutative Geometry on Finite Groups and Hopf

J. Noncommut. Geom. 13 (2019), 1055–1116 Journal of Noncommutative Geometry DOI 10.4171/JNCG/345 © European Mathematical Society

Generalised noncommutative geometry on ﬁnite groups and Hopf quivers

Shahn Majid and Wen-Qing Tao

Abstract. We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of S3. We show how quiver geometries arise naturally in the context of quantum principal bundles. We provide a formulation of bimodule Riemannian geometry for quantum metrics on a quiver, with a fully worked example on 2 points; in the quiver case, metric data assigns matrices not real numbers to the edges of a graph. The paper builds on the general theory in our previous work [19].

Mathematics Subject Classiﬁcation (2010). 81R50, 58B32, 16G20. Keywords. Hopf algebra, nonsurjective calculus, quiver, duality, ﬁnite group, bimodule connection.

1. Introduction

Noncommutative differential geometry is an extension of geometry to the case where the coordinate algebra A may be noncommutative or “quantum.” The starting point is a “differential structure” on A defined as a pair .1; d/ where the space of “1-forms” 1 is an A-A-bimodule (so one can multiply by the algebra from the left or the right) and d A 1 obeys the Leibniz rule. One usually requires that W ! the map A A 1 given by .a b/ adb, a; b A is surjective. Recently W ˝ ! ˝ D 2 in [19] we began a study of differentials in noncommutative geometry where this condition is dropped. One still has a standard differential calculus 1 .A A/ x D ˝ given by the image of this map, but there are many situations where this image is not easy to describe and where a larger 1 is the much more natural object. This also links in to the wider use of differential graded algebras in other contexts where

The second author was funded by 2016YXMS006 and NSFC11601167. PACIFIC JOURNAL OF MATHEMATICS Vol. 284, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.284.213

NONCOMMUTATIVE DIFFERENTIALS ON POISSON–LIE GROUPS AND PRE-LIE ALGEBRAS

SHAHN MAJID AND WEN-QING TAO

We show that the quantisation of a connected simply connected Poisson–Lie group admits a left-covariant noncommutative differential structure at lowest deformation order if and only if the dual of its Lie algebra admits a pre- Lie algebra structure. As an example, we ﬁnd a pre-Lie algebra structure underlying the standard 3-dimensional differential structure on ރq [SU2]. At the noncommutative geometry level we show that the enveloping algebra U(m) of a Lie algebra m, viewed as quantisation of m∗, admits a connected differential exterior algebra of classical dimension if and only if m admits a pre-Lie algebra structure. We give an example where m is solvable and we extend the construction to tangent and cotangent spaces of Poisson–Lie groups by using bicross-sum and bosonisation of Lie bialgebras. As an example, we obtain a 6-dimensional left-covariant differential structure on the G ∗ bicrossproduct quantum group ރ[SU2]I Uλ(su2).

1. Introduction

It is well-known following[Drinfeld 1987] that the semiclassical objects underlying quantum groups are Poisson–Lie groups. This means a Lie group together with a Poisson bracket such that the group product is a Poisson map. The inﬁnitesimal notion of a Poisson–Lie group is a Lie bialgebra, meaning a Lie algebra g equipped with a “Lie cobracket” δ : g → g ⊗ g forming a Lie 1-cocycle and such that its adjoint is a Lie bracket on g∗. Of the many ways of thinking about quantum groups, this is a “deformation” point of view in which the coordinate algebra on a group is made noncommutative, with commutator controlled at lowest order by the Poisson bracket. In recent years, the examples initially provided by quantum groups have led to a signiﬁcant “quantum groups approach” to noncommutative differential geometry in

Majid was on leave at the Mathematical Institute, Oxford, during 2014 when this work was completed. Tao was supported by the China Scholarship Council. MSC2010: 17D25, 58B32, 81R50. Keywords: noncommutative geometry, quantum group, left-covariant, differential calculus, bicovariant, deformation, Poisson–Lie group, pre-Lie algebra, (co)tangent bundle, bicrossproduct, bosonisation.

213 PHYSICAL REVIEW D 91, 124028 (2015) Cosmological constant from quantum spacetime

Shahn Majid* and Wen-Qing Tao Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, United Kingdom (Received 12 December 2014; published 9 June 2015) i i We show that a hypothesis that spacetime is quantum with coordinate algebra ½x ;t¼λPx , and spherical symmetry under rotations of the xi, essentially requires in the classical limit that the spacetime metric is the Bertotti-Robinson metric, i.e., a solution of Einstein’s equations with a cosmological constant and a non-null electromagnetic field. Our arguments do not give the value of the cosmological constant or the Maxwell field strength, but they cannot both be zero. We also describe the quantum geometry and the full moduli space of metrics that can emerge as classical limits from this algebra.

DOI: 10.1103/PhysRevD.91.124028 PACS numbers: 04.60.-m, 02.20.Uw, 02.40.Gh

I. INTRODUCTION calculus first appeared in Ref. [6] as did those for another, which we call the “β family” and which generalizes the Recently in [1], a new phenomenon was uncovered standard one. In our case, we come to these same differ- whereby the constraints of noncommutative algebra force ential calculi out of a systematic classification theory [7] the form of quantum metric and hence of its classical limit. based on pre-Lie algebras. Remarkably. we then find for the Put another way, if a spacetime is quantized, as is by now α family, in Sec. III, that this time there is a moduli of widely accepted as a plausible model of quantum gravity quantum metrics, and in Sec. IV we consider their classical effects, then this would be visible classically as quantiz- limits and show that in the spherically symmetric case they ability conditions [2] on the classical spacetime metric so n−2 n−2 S dS2 S 2 as to extend to the quantum algebra. Thus the quantum are all locally of the form × or ×AdS spacetime hypothesis potentially has strong and observable depending on the sign of one of the two curvature-scale δ δ¯ consequences for classical general relativity (GR). parameters ; . This means that they are the Levi-Bertotti- – Specifically, Ref. [1] looked at the most popular quantum Robinson metric [8 11], which has been of interest in a ’ spacetime algebra, the bicrossproduct or Majid-Ruegg number of contexts in GR and is known to solve Einstein s model [3] with generators xi;t, i ¼ 1; …;n− 1, and equation with a cosmological constant and Maxwell field. relations We can write the value of the cosmological constant here as ð − 2Þð − 3Þ ½xi;xj¼0; ½xi;t¼λxi; ð1Þ Λ ¼ n n δ − 2 2 q GN; λ ¼ ıλ λ 1 where P and P is a real quantization parameter, q2G ¼ ððn − 3Þδ − δ¯Þ; usually assumed in this context to be the Planck time. Here N 2 n ¼ 4 but we will consider other dimensions also. The paper [1] showed that in the 2D case the quantizability where q is the Maxwell field coupling in suitable units. In constraints force a strong gravitational source or an our context δ > 0, so that for small q we are forced to expanding universe depending on a sign degree of freedom. Λ > 0. Moreover, the arguments that force us to this form This provided a toy model, but in 4D the constraints were of metric depend on the structure of the differential algebra so strong that there was no fully invertible quantum metric when spacetime is noncommutative, which is believed to be at all. The analysis depended on the differential structure on a quantum gravity effect. In 2D, there is no Sn−2 factor and the algebra, and we used the standard one as in Refs. [4,5]. being the limit of a quantum metric in the α family in 2D In the present paper, we will now consider the same forces the metric to be de Sitter or anti–de Sitter for some phenomenon for another natural choice of differential scale δ¯. structure on (1), which we call the “α family” and which The further noncommutative Riemannian geometry for we show, in Sec. II, is the only good alternative that treats our quantum metrics in the α family is obtained by the same the xi equally in the sense of rotationally invariant and methods as in Ref. [1], and a brief outline of this is included works in all dimensions. The relations for this differential in Sec. V. We work in this paper with one particular algebra (1) assumed to be some local model of quantum spacetime. λ *On leave at the Mathematical Institute, Oxford, United The general analysis at lowest order in , i.e., at the level of Kingdom. a general Poisson structure on spacetime and the constraints [email protected] on the classical metric arising from being quantizable along

Contents lists available at ScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

Duality for generalised diﬀerentials on quantum groups

Shahn Majid, Wen-Qing Tao 1

Queen Mary University of London, School of Mathematical Sciences, Mile End Rd, London E1 4NS, UK a r t i c l e i n f o a b s t r a c t

Article history: We study generalised differential structures (Ω1, d) on an Received 12 September 2014 algebra A, where A ⊗ A → Ω1 given by a ⊗ b → adb need Available online 26 May 2015 not be surjective. The finite set case corresponds to quivers Communicated by J.T. Stafford with embedded digraphs, the Hopf algebra left covariant 1 1 MSC: case to pairs (Λ , ω)where Λ is a right module and ω a primary 81R50, 58B32, 20D05 right module map, and the Hopf algebra bicovariant case corresponds to morphisms ω : A+ → Λ1 in the category of Keywords: right crossed (or Drinfeld–Radford–Yetter) modules over A. Noncommutative geometry When A = U(g)the generalised left covariant differential Quantum group structures are classified by cocycles ω ∈ Z1(g, Λ1). We Hopf algebra then introduce and study the dual notion of a codifferential Differential calculus structure (Ω1, i)on a coalgebra and for Hopf algebras the Bicovariant self-dual notion of a strongly bicovariant differential graded Quiver algebra (Ω, d) augmented by a codifferential i of degree −1. Shuffle algebra Crossed module Here Ωis a graded super-Hopf algebra extending the Hopf 0 Duality algebra Ω = A and, where applicable, the dual super- Hopf 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 (Λ1, ω)on a Hopf algebra A extends canonically to a strongly bicovariant differential graded algebra via the

E-mail addresses: [email protected] (S. Majid), [email protected] (W.-Q. Tao). 1 The second author was supported by the China Scholarship Council. http://dx.doi.org/10.1016/j.jalgebra.2015.03.023 0021-8693/© 2015 Elsevier Inc. All rights reserved. April 20, 2015 7:55 WSPC/S0219-4988 171-JAA 1550135

Journal of Algebra and Its Applications Vol. 14, No. 8 (2015) 1550135 (24 pages) c World Scientiﬁc Publishing Company DOI: 10.1142/S0219498815501352

Coquasitriangular structures on Hopf quivers

Hua-Lin Huang∗ and Wen-Qing Tao†,‡ School of Mathematics Shandong University, Jinan 250100, P. R. China ∗[email protected] †[email protected]

Received 1 August 2013 Accepted 12 November 2014 Published 24 April 2015

Communicated by T. Lenagan

The complete list of the coquasitriangular structures of the graded Hopf algebra over a connected Hopf quiver is given.

Keywords: Hopf algebra; coquasitriangular structure; Hopf quiver.

Mathematics Subject Classiﬁcation: 16T20, 16G20

1. Introduction Quasitriangular Hopf algebras were introduced and deeply studied by Drinfeld [5, 6]. The notion of coquasitriangular (CQT) Hopf algebras is dual to that of the quasitriangular Hopf algebras, and therefore CQT Hopf algebras have properties dual to by Miss Wenqing Tao on 05/06/15. For personal use only. quasitriangular ones. Due to their fascinating connections with many branches of

J. Algebra Appl. 2015.14. Downloaded from www.worldscientific.com mathematics and physics, (co)quasitriangular Hopf algebras have attracted inten- sive attention since their appearance. CQT Hopf algebras are essentially noncommutative algebras but with noncommutativity controlled by universal R-matrices, i.e. CQT structures, so they are also of particular interest in noncommutative geometry [7]. The very fundamental problems in the theory of (co)quasitriangular Hopf algebras are to determine the conditions for a Hopf algebra to admit (co)quasitriangular structures and to construct the complete list of the (co)quasitriangular structures. The techniques of Hopf quivers were introduced in [4, 8, 9, 22]andturnout to be useful in studying some interesting classes of Hopf algebras and their repre- sentations. A recent work [11] shows that the quiver techniques can also help to understand the (co)quasitriangularity of Hopf algebras. More precisely, it is proved

‡Corresponding author. Current address: School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK.

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