I work in noncommutative algebra both in pure mathematics and in mathematical physics, including quantum gravity. Among my results, I introduced one of the two main classes quantum groups or Hopf algebras, and have also developed noncommutative differential geometry or `quantum geometry', braided algebra, a Lie theory for finite groups, and a new way of thinking about nonassociative algebra such as the octonions. Some of my best recent papers are probably:
0. `Lie theory and coverings of finite groups', in press J. Algebra, 389 (2013) 137-150 (with K. Rietsch)
We introduce the notion of a locally skew IP quandle as a refinement of the notion of a `Lie algebra' of a finite group. We prove that this applies to many types of groups including Coxeter reflection groups and we show that when group a group is generated by such a `Lie algebra' is noncommutative de Rham cohomology has trivial classical part.
1. `Almost commutative Riemannian Geometry: wave operators', Commun. Math. Phys. 310 (2012) 569-609
The paper `quantizes' a general class of Riemannian geometries with static metric, including the famous Schwarzschild `black hole'. I show that as a quantum gravity effect the usual infinite time-dilation at the black hole event horizon is now made finite within this approach to noncommutative geometry.
2. `Hopf quasigroups and the algebraic 7-sphere', J. Algebra, 323 (2010) 3067-3110 (with J. Klim)
We show how Hopf-algebra methods can be adapted to develop a theory of algebraic quasigroups. This includes the 7-sphere which is well-known not be a group (unlike the circle and the 3-sphere) and we describe this using our algebraic methods, including its differential geometry. We obtain a new description of the structure constants of the Lie algebra g_2 directly in terms of the structure constants of the octonions.
3. `*-Compatible connections in noncommutative Riemannian geometry', J. Geom. Phys. 25 (2011) 95-124 (with E.J. Beggs)
Using bar categories, we develop the right notion of *-structure in noncommutative geometry and solve it to find natural bimodule connections on quantum groups and quantum spheres. This includes a unique `Levi-Civita' torsion free, metric compatible and *-compatible connection on the quantum group C_q(SU_2).
4. `Bar categories and star operations', Alg. and Representation Theory 12 (2009) 103-152 (with E.J. Beggs)
We show how to extend the innocent notion of `complex conjugation' to braided and other very general settings based on monoidal categories. An application is to the geometry of quantum groups at roots of unity and their braided algebra versions.
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