# 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

### Papers and preprints (download from the arxiv)

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

### A selection of my professional activities from 2015 onwards:

- This year I am co-organising the seminar series “Algebra, Geometry and Topology” at the University of Kent, with Chris Bowman.
- In December 2017 I took part in the INdAM workshop in Rome on "Affine, vertex and W-algebras".
- In September 2017 I visited Tomoyuki Arakawa at Massachusetts Institute of Technology to start work on the theory of affine W-algebras and related vertex algebras in positive characteristics.
- In July 2017 I attended a summer school in Dijon "Current topics in the theory of algebraic groups".
- In January 2017 I took part in the winter school and subsequent conference on "Geometry and Representation theory" at the Erwin Schrödinger Institute in Vienna.
- In April 2016 I attended a conference in honour of my PhD supervisor Sasha Premet at the ICMS in Edinburgh "Representation theory and symplectic singularities". If you look carefully at the conference photos you can see the reflection of my spectacles floating like a spectre next to Gunter Malle's head!
- In July 2016 I attended the conference "Representation theory in Samos" at the University of the Aegean Sea, Samos.
- In May 2016 I visited Jon Brundan at the University of Oregon to start work on the theory of Yangians in positive characteristics.
- In 2016 I arranged a reading group in Padova on the subject of "Category O for symmetrisable Kac-Moody algebras" where we examined some combinatorial aspects of Kazhdan-Lusztig theory, mostly drawn from the work of Peter Fiebig.
- The above reading group was inspired by a series of lectures given by Prof Fiebig at a workshop attended in Marienheide, 2015, entitled "Representations of algebraic groups and Lie algebras in characteristic p".
- In August 2015 Simon Goodwin visited me in Padova and we started work on a series of papers on the theory of modular finite W-algebras and the representations of restricted Lie algebras.
- In January 2015 I gave a talk at the conference "Geometry and representations of Cherednik algebras, and category O" at Paris VII