# Fundamental Questions in Theory of Algebraic and Kac-Moody Groups

Lead Research Organisation: University of Warwick
Department Name: Mathematics

### Abstract

The research will lie on the interface of three research areas: Algebra, Geometry and Topology, Number Theory. The general aim is to study fundamental questions about reductive algebraic groups, occasionally expanding the results and the methods into other areas, such as Kac-Moody groups or 2-groups. The first question is integration of modules from the Lie algebra Lie(G) to G in positive characteristic. A number of methods will be deployed, hoping to shed light on some old conjectures by Humphreys and Verma. Another direction is the study of arithmetic differential operators on these groups. This include their geometry, their structure and their representation theory. The final direction is the study of locally symmetric spaces of the form H\G/K where G is a locally compact group, K is its compact subgroup, H is a lattice in G. One particular direction is to study lattices in G=PU(2,1) in the light of geometric properties H\B^2 where B^2=G/K is the unit ball in C^2. The following questions are of interest. What are commensurability classes of lattices that produce a smooth surface H\B^2 with c2=3 (or 6), c1^2=9 (or 18)? Are they always arithmetic? Can we classify all arithmetic lattices such that the surface H\B^2 has c2=6, c1^2=18? All these questions have serious potential applications, although only in fundamental science. The benefits to society and economy are not likely in short or medium terms.

### ORCID iD

Inna Capdeboscq (Primary Supervisor)
Matthew Westaway (Student)

### Publications

10 25 50
Rumynin D (2019) Integration of modules I: stability in Pacific Journal of Mathematics

Rumynin D (2019) Covering groups of nonconnected topological groups and 2-groups in Communications in Algebra

### Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509796/1 01/10/2016 30/09/2021
1789943 Studentship EP/N509796/1 03/10/2016 31/03/2020 Matthew Westaway

Description An algebraic group is a mathematical object which has both geometric and group theoretic structure. To understand complicated mathematical objects, like algebraic groups, we frequently try to represent them linearly, as linear objects are often easier to understand and compute with. Furthermore, geometric objects can be considered linearly by looking at their tangent spaces - for an algebraic group, this tangent space is a Lie algebra. Thus, to better understand algebraic groups, we seek to understand their representation theory and their Lie algebras, and the connections between the two. We work over fields of positive characteristic, an area where there are still many open questions.

The work funded through this award answers a question of Parshall and Friedlander from 1990, which addresses whether their work on the representation theory for Lie algebras in positive characteristic can be generalised for higher Frobenius kernels - certain objects whose representation theory somehow lies between that of the Lie algebra and of the algebraic group. We find that it can, constructing a 'higher universal enveloping algebra' and obtaining from it a family of algebras which deform the higher Frobenius kernels in an analogous way to the well-known results in the standard case. We obtain a number of results on the structure of these algebras, including finding Poincare-Birkhoff-Witt bases for them, and we furthermore are able to give significant results on their representation theory. Specifically, we find that irreducible modules over these algebras can be decomposed into irreducible modules over more well-studied algebras. These algebras furthermore appear to be the appropriate algebras for a localisation theorem, like that of Beilinson and Bernstein, for the arithmetic differential operators of Berthelot.

The work also explores approaches to a conjecture of Humphreys and Verma. Specifically, we wanted to find new methods through which to integrate restricted representations of Lie algebras in positive characteristic to representations over the corresponding algebraic groups. Two such techniques are developed through this work. The first is an algorithm through which the integration of modules can be achieved if certain 2-cocycles vanish in certain cohomology groups. This allows us to construct a test for the existence and uniqueness of integrated modules. The second uses exponentials and provides a condition, which we call being over-restricted, under which such integration happens. We also conjecture a generalisation for higher Frobenius kernels, which would simplify the Humphreys-Verma conjecture to a question about these higher Frobenius kernels rather than the whole algebraic group.
Exploitation Route The work on the question of Parshall and Friedlander can be taken forward in several ways. The algebras constructed through this work are new objects, about which many questions remain. In particular, it would be interesting to compute their centres and to develop a theory for these algebras analogous to the theory of finite W-algebras in the more well-known case. Furthermore, it would be nice to develop and prove a localisation theorem connecting the representation theory of these algebras with Berthelot's arithmetic differential operators.

The work related to the Humphreys-Verma conjecture also leads to further questions. One can try to compute the relevant cohomology groups and 2-cocycles for certain semisimple algebraic groups and projective indecomposable modules, hence improving our understanding of the Humphreys-Verma conjecture. As well as this, one could aim to prove results generalising our work on exponentials to higher Frobenius kernels. This would be particularly significant if one could give criteria for a module over the first Frobenius kernel to be able to be integrated to a module over the higher Frobenius kernels.
Sectors Other
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