MY ResEARCH RESULTS

MY ResEARCH RESULTS

Research outline

I'm interested in almost all aspects of pure mathematics and mathematical structures needed to address fundamental problems in theoretical physics. A fuller narrative account is below but here as some of my favourite results, some of them with collaborators.

1980s An explanation of the confinement of quarks using infinitesimal holonomy and the background field method

Among the first to introduce the use of loop variables as functionally Fourier dual to gauge fields.

Introduced quantum Born reciprocity (or position-momentum/observer-observed interchange) as a key input for quantum gravity

Pioneered the modern theory of quantum groups introducing one of the two main types (the bicrossproducts associated to Lie group factorisations)

Introduced the centre of a monoidal category (as part of my more general duality construction) and an algebraic formulation of Drinfelds double as a `double cross product’

1990s Introduced a new kind of braided algebra done with knots and tangles (the systematic theory of hopf algebras in braided categories with main theorems such as reconstruction and bosonisation)

Introduced the first successful model of quantum spacetime with quantum group symmetry, a model which was later shown to predict variable speed of light (this is currently tested by a satellite in orbit)

Introduced cogravity: the idea of curvature in momentum space (as equivalent to quantum spacetime via quantum Fourier transform)

Pioneered the `quantum groups approach to noncommutative geometry’ including first to introduce quantum group gauge theory and frame bundles

Constructed Drinfeld quantum groups by an inductive construction on braided categories. Part of this is a canonical braided-Hopf algebras associated to an object in a braided category

Classified differential calculi on all main classes of quantum groups

Showed that the famous Octonions are in fact associative when viewed in a certain monoidal category

Introduced a theory of braided-Lie algebras generating Drinfeld quantum groups

Introduced cocycle twists of module algebras (extending Drinfeld twist of quantum groups)

2000s No-go theorem forcing invariant differential calculi on the main quantum groups to have extra dimensions or else be nonassociative

Found the quantum Riemannian and complex structure on the quantum q-sphere

Found the quantum Riemanian structure on finite group S3 (its Ricci curvature is constant)

Harmonic analysis on quantum spacetime to find variable speed of light prediction (previously had been speculated)

Found braided-Hopf algebra structure on the Fomin-Kirillov algebra relating noncommutative geometry of Sn to classical cohomology of the flag variety

Gave a construction for the Fock space of anyons using braided categories

Related braided-Lie algebras to differential calculi on quantum groups

Introduced notion of complex conjugation (`bar’) for monoidal categories

2010s Advanced a previous bimodule-connection approach to noncommutative differential geometry with first solved model on quantum spacetime

First actual construction of noncommutative-geometric or quantum black hole

Introduced Lie theory of finite groups and applied it to Killing form and coverings

Introduced duality for differential structures on quantum groups

New point of view of Riemannian geometry as cocycle extension data for the exterior algebra on the manifold

Fuller research statement

My earliest works were in high-energy particle physics at Harvard. I introduced ideas such as `infinite spin’ regularization ([29-30 ] in my list of publications) as a way to handle the infinities in quantum field theory, an infinitesimal proof of quark confinement using the background field method [31], and a Fourier duality between loops and gauge fields on a manifold [36]. I also worked on Yang-Mills gauge theory [32-33].

Starting with my 1988 PhD thesis, I helped pioneer the modern theory of ‘quantum groups’ or Hopf algebras, introducing one of the two main classes of these objects, namely the ‘bicrossproduct’ quantum groups C[M] U(g) associated to Lie group factorisations X = GM [34-38]. These have become increasingly important, most recently in applications to quantum spacetime where my model [127,149] based on them could provide the first experimental tests of quantum gravity, see expositions [12-13,23-24]. I also worked on the other class of ‘quasitriangular’ quantum groups Uq(g) of Drinfeld and Jimbo q-deforming the enveloping algebras of simple Lie algebras g. I have results on the Drinfeld quantum double construction, including its formulation[36] as a double cross product H H which is now more or less standard in algebra circles. At the end of the 1980’s I was among those developing the connection between quantum groups and braided categories. Results included the interpretation of q-dimensions as the categorical rank[43], relations of that with the theory of modular functions[53,54], an important functor [50] from modules of a quasitriangular Hopf algebra to modules of its double (so-called `crossed modules’), and generalised Tannaka-Krein reconstruction theorems for quasi-quantum groups and ‘braided groups’ or Hopf algebras in braided categories [47-48,57,63]. Another important result was a duality operation for monoidal categories [39,46,75] a special case of which was later called the `centre’ by V.G. Drinfeld and connected with the quantum double (in a famous letter to me after reading the preprint version of [46]). A later application was the quantum double of a quasi-quantum group found in [139].

A large body of my subsequent work introduced and studied these braided groups and ‘braided geometry’ as the deeper structure behind q-deformations [58-107]. I showed that every quasitriangular quantum group has a braided version by `transmutation’[63-64] and that every braided group of a certain type has an associated ordinary quantum group by `bosonisation’[65], both constructions are now standard in algebra circles. Among results was a new inductive construction [102] of all quantum groups Uq(g) for g simple by repeatedly adjoining additive braided groups such as the n-dimensional additive quantum plane Aq . For example Uq(sln+1) = Aq Uq(sln) Aq . Iterating this provides a new ‘block decomposition’ of any Uq(g) into a product of braided spaces and a Cartan part. The construction included Lusztig’s construction of Uq(g), and also gave a fresh approach to the structure theory of semisimple Lie algebras themselves. Other braided groups related to ‘noncommutative exterior algebras’ of finite groups [159] and are conjecturally related to Lusztig canonical bases as well as to geometry in characteristic p. Braided groups were also shown to underly the theory of nonrelativistic anyons [114] and to provide a generalisation of cyclic cohomology appropriate to q-deformation examples [115]. Braided group computations are best done by a new kind of `braided algebra’ consisting of algebra operations `wired up’ like the wiring in a computer except that there is a nontrivial braiding operation each time one `wire’ jumps over the other. I was the first to seriously use such methods for algebra [57,64-65], which represent a new kind of `braided logic’ which might also provide the right semantics for quantum computers.

Also using tensor category ideas, in 1999 I introduced[108] a now well-known approach to the nonassociative Octonions algebra, as associative in a nontrivial monoidal category of modules over a certain quasi-Hopf algebra structure on (Z_2)^3. It then becomes by a theorem of Mac Lane ‘as good as’ associative in the sense that all linear algebra constructions can be done as usual with bracketing inserted afterwards via an associator. I recently[117-118] applied this to a theory of algebraic Moufang loops (such as the unit octonions) and obtained a new description of the structure constants of the Lie algebra g2.

The other large body of my work since the 1990s is the pioneering of what I call the ‘quantum groups approach to noncommutative differential geometry’ [121-191]. In fact, while the Uq (g) quantum groups arose from quantum integrable systems and statistical physics, the bicrossproduct ones arose specifically from the search for a generalisation of geometry to situations where both gravitational effects (curvature) and quantum effects are present, the so-called ‘Planck scale’. From a mathematical point of view this amounts to extending differential geometry to allow a noncommutative ‘ring of functions’. The main problem is that usual ideas of topological spaces, sheaf theory and local trivialisations do not work well when the ‘coordinate rings’ are noncommutative. A solution to this problem appeared in 1992 in work [126]: we developed purely algebraic replacements for these ideas sufficient to define principal bundles and connections (with quantum group fibre) on noncommutative algebras, and sufficient also to include nontrivial examples such as a q-monopole over a noncommutative q-sphere. In key works [145,154,] I extended this to noncommutative Riemannian geometry on potentially any unital algebra using a quantum frame bundle approach with quantum group fibre. In more recent work on ‘formalism’ I have looked at the notion of ∗-structure needed to define real forms [177] and at Riemannian geometry based on ∗-compatible connections [181]. This extends the frame bundle theory to a slightly more general theory based on vector bundles (as projective modules) and hence closer towards Alain Connes’ approach to noncommutative geometry modelled on the Dirac operator without reference to quantum frame bundles. There are also links between Connes’ approach and bicrossproduct quantum groups [172]. My early results include classification theorems for bicovariant differential calculi both on q-deformation quantum groups in 1998 [137], quantum doubles [155] and bicrossproduct ones [144,160]. By now the geometry of quantum groups has its own Mathematics Subject Classification code 58B32 and featured as an integral part of a 6-month programme on Noncommutative Geometry that I organised in 2006 at the Newton Institute along with Alain Connes and Albert Schwarz.

A couple of important things have emerged from this quantum groups approach to noncommutative geometry. The first is applications to the noncommutative geometry of finite groups even where the ring of functions is commutative (for example I found [154] that S3 is ‘Einstein’ while A4 is Ricci flat [160]). This theory is expected to have deep connections with canonical bases as an extension of Schur-Weyl duality [159] and in the meantime the notion of Killing form and `Lie algebra’ provides a new tool in the theory of finite simple groups. In [184] it is proved that this Killing form is non-degenerate in certain cases. The theory also applies finite-dimensional Hopf algebras[155, 161] such as quantum groups at roots of unity (which could be viewed as quantizations of finite groups of Lie type) and potentially also to finite sets[173-174,179]. The other is a fundamental anomaly or obstruction to the construction of associative differential calculi of classical dimensions and appropriate symmetry. This was confirmed in a semiclassical analysis [170] for all standard quantum groups Cq(G) and [180] for all U(g) regarded as quantizations of g∗, for g simple. These two theorems imply that in such highly noncommutative contexts one must either resolve the anomaly with extra cotangent directions [170] or live with a forced nonassociative differential geometry [116,180,182]. The first approach can be interpreted as an intrinsic ‘time evolution’ induced by the anomaly and may (or may not) be connected to the modular automorphism induced in the operator algebras approach. It is what I have called `the algebraic origin of time’ [171]. The second approach is more radical but we show in [170,180,182] how `cochain twists’ may be used to non-associatively quantize any classical structure under the action of a symmetry. The use of Drinfeld twists by cocycles and cochains as a method of transforming or `quantizing’ algebras had been introduced by me in the early 1990s and used notably in [92,108,148]. Some expositions were in [127, 164] and a recent application was in [175]. More recent work [169,176] also hints at a deep theory of `noncommutative complex structures’. The first of these included among other things a noncommutative Dolbeault complex for the q-sphere and proved (for generic q) a q-Borel Weil Bott theorem whereby the irreducible representations of Uq(su2) are obtained as holomorphic sections of quantum line bundles (monopoles) associated to representations of the U(1) fibre in the q-Hopf fibration. Noncommutative CP^n and more nontrivial Grassmannians were obtained in [176] as an approach to noncommutative twistor theory, recovering in the process θ-deformed S^4 that had been introduced by Connes and Landi, as well as noncommutative versions of several of the key bundles in twistor theory.

Reflecting a maturing of the field, such constructions are not only of mathematical interest. Other ‘fuzzy’ spheres as quotients of U(su2) as a coadjoint quantisation arise in 3D quantum gravity without cosmological constant [165], while for 3D quantum gravity with cosmological constant we found [178] q-fuzzy spheres similarly based on Uq(su2) viewed as a coordinate algebra. In [184] I used twisting and braided category methods to introduce noncommutative differential calculi on Uq(su2) and q-fuzzy spheres, opening the way to the Riemannian and complex geometry underlying the model spacetime of full 3D quantum gravity with cosmological constant and point sources. Meanwhile in [183,185] I have achieved a long-cherished goal of constructing a noncommutative Schwarzschild black hole, turning the idea of `extra dimensions’ in the calculus around to define the wave operator.

Many mathematicians have explicitly picked up on my work. These include traditional ‘pure Hopf algebraists’ such as D. Radford, S. Montgomery, M. Cohen, M. Takeuchi, category theorists such as B. Pareigis, D. Yetter, V. Turaev, R. Street, and operator algebraists such as S. Baaj, G. Skandalis, T. Yamanouchi and more recently A. Connes. Also geometers such as J-H. Lu and A. Weinstein and experts in q-special functions such as T. Koornwinder. In general, my work includes both abstract theorems about Hopf algebras, the construction of nontrivial examples and links with mathematical physics. On the mathematics side, I have invited contributions to the Notices of the AMS and to the Princeton Companion of Mathematics [25-26].

In 1993 I was awarded an international prize for my work, the `Bleuler medal’. In 1998 I was featured in Faces of Maths, a photographic study of leading UK mathematicians. In 1995, I published a 600+ page textbook [1] explaining some of the foundations of the subject and which remains a standard reference today. In 1998 I gave a part III course `Quantum groups' in the pure maths tripos at Cambridge University; its lecture notes were published by the LMS[2]. Also, since 1992 I have organised a weekly seminar series on quantum groups and quantum geometry, held some terms in DPMMS and some terms in DAMTP when I was in Cambridge and now held in Queen Mary, University of London since I moved here. My latest book [4] addresses the true nature of space and time.