Dr Lewis Topley

Research Associate in Representation Theory

Working in the Algebra, Geometry and Topology group at the University of Kent, with Stephane Launois.

Research interests:

  • ordinary and modular representation theory of Lie algebras and algebraic groups;
  • finite and affine W-algebras;
  • Yangians in positive characteristics;
  • Poisson algebras and deformation theory.

A brief summary of my current research goals:

Many algebras arising in representation theory and Lie theory are finite extensions of their centres, and arise as specialisations of one parameter quantisations over some principal ideal domain. Classic examples include the enveloping algebras of restricted Lie algebras, as well as quantised enveloping algebras and their restricted Hopf duals specialised at a root of unity. The fact that these algebras occur as specialisations of a family of algebras means that the centre acquires a Poisson structure, and there are many examples where the representation theory of the algebra in question appears to be influenced by the Poisson geometry of the spectrum of the centre. For example, the dimensions of simple modules appear to be controlled by the dimensions of the symplectic leaves of the underlying central characters [6]. This fascinating relationship is only well understood in a small handful of cases and one thread of my current research - specifically my work with Launois [11] - aims to develop tools which may offer broader, more conceptual explanations of these kinds of phenomena. One case where the connections mentioned above are especially well developed is for enveloping algebras in positive characteristics, where finite W-algebras provide machinery for elucidating the relationship between representation theory and Poisson geometry. Another thread to my research aims to develop the theory of modular finite W-algebras and extrapolate consequences in the classical representation theory of Lie algebras [4, 8, 10]. In type A this approach will be especially effective thanks to the connections between shifted Yangians and modular finite W-algebras [12].

In future work I plan to explore modular affine W-algebras (vertex operator algebras) and their relationship with their finite counterparts via Zhu's functor. We expect to find that these algebras admit central reductions, analogous to reduced enveloping algebras, which are all C_2-cofinite.

Room 271, Sibson Building,

The University of Kent

Canterbury CT2 7FS

United Kingdom

Email: L.Topley@kent.ac.uk

Phone: 01227 816 387

Curriculum Vitae: download here

2 page version

Papers and preprints (download from the arxiv)

  1. Invariants of centralisers in positive characteristic, J. Algebra 399 (2014), pp. 1021--1050.
  2. Derived subalgebras of centralisers and finite W-algebras (joint with Alexander Premet) Compos. Math. 150 (2014), no. 9, pp. 1485--1548.
  3. Centralisers in Classical Lie Algebras PhD. Thesis, August 2014.
  4. A Morita theorem for modular finite W-algebras Math. Z. 285 ( 2017) 3-4, pp. 685--705.
  5. Harish-Chandra invariants and the centre of the reduced enveloping algebra J. Pure App. Alg. 221 (2017), pp. 490--498.
  6. A Non-restricted counterexample to the first Kac-Weisfeiler conjecture Proc. A.M.S. 45 (2016) 5, pp. 1937--1942.
  7. Transfer results for free Frobenius extensions (joint with Stephane Launois) accepted for publication in J. Algebra (2017).
  8. Modular finite W-algebras (joint with Simon M. Goodwin) accepted for publication in I. M. R. N. (2018).
  9. On the semicentre of a Poisson algebra (joint with Cesar Lecoutre) accepted for publication in Algebr. Represent. Theory (2018).
  10. Minimal dimensional representations of reduced enveloping algebras for gl_n (joint with Simon M. Goodwin) arxiv:1805.01327 (2018).
  11. The orbit method for Poisson orders (joint with Stephane Launois) arXiv:1711.05542 (2017).
  12. The p-centre of the Yangian and shifted Yangians (joint with Jonathan Brundan) accepted for publication in Mosc. Math. J. (2018).
  13. Smoothness of stabilisers in generic characteristic (joint with Ben Martin and David Stewart) arXiv:1810.12628 (2018).
  14. A proof of the first Kac-Weisfeiler conjecture in large characteristics (joint with Ben Martin and David Stewart) arXiv:1810.12632 (2018).
  15. Restricted shifted Yangians and restricted finite W-algebras (joint with Simon M. Goodwin), in preparation.

Teaching:

2016-2017: Groups and rings MA565

2017-2018: Graphs and Combinatorics MA595

Office hours: Thursday 15:00-16:00 & Friday 10:00-11:00

A selection of my professional activities from 2015 onwards: