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Details of Grant 

EPSRC Reference: EP/X035328/1
Title: The Lie algebra of derivations of a block of a finite group
Principal Investigator: Linckelmann, Professor M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Sch of Engineering and Mathematical Sci
Organisation: City, University of London
Scheme: Standard Research
Starts: 01 October 2023 Ends: 30 September 2026 Value (£): 358,933
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
28 Feb 2023 EPSRC Mathematical Sciences Prioritisation Panel February 2023 Announced
Summary on Grant Application Form
Lie groups and Lie algebras arise in Physics as symmetry groups of

physical systems and their tangent spaces, which may be regarded as

infinitesimal symmetry motions.

These notions have long been of interest in many areas of Mathematics.

Lie algebras arise, for instance, as operators on algebras respecting

Leibniz' product rule, called derivations.

The derivations on an algebra can be interpreted as representatives of

the first Hochschild cohomology of an algebra. The Lie algebra structure

on this space extends to a graded Lie algebra structure on Hochschild

cohomology - this goes back to pioneering work of Gerstenhaber, in the

context of the deformation theory of algebras.

The use of this technology in Physics tends to be over fields of

characteristic zero, but the underlying concepts have analogues over

fields of prime characteristic, and makes this technology available

for investigations in the modular representation theory of finite group

algebras over local rings and fields. In fact, there are `many more'

finite-dimensional Lie algebras over fields of prime characteristic than

over the complex numbers.

A particular feature of modular representation theory of finite groups

is that it is driven by a great number of conjectures, some of which

predict remarkable structural connection between various direct factors

of finite group algebras, and other simply predicting mysterious numerical

coincidences.

Hochschild cohomology in general has turned out to be useful for

reformulations and variations of those conjectures. Expectations are

high that investigating the (graded and restricted) Lie algebra structure

of Hochschild cohomology in the context of finite group algebras and their

direct factors should contribute to an understanding of some parts of those

conjectures.

The present proposal takes precisely these expectations as a starting

point, putting the focus on higher structural aspects of

Hochschild cohomology and their impact on invariants of finite group

algebras and their blocks. We set out describing this programme in a

sequence of nine conjectures, ranging from basic questions - such as

the non-vanishing of the first Hochschild cohomology of blocks - via

explicit calculations in certain classes of finite groups to

currently elusive conjectures on numerical and structural aspects of

finite groups and their blocks.

Key Findings
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Organisation Website: http://www.city.ac.uk