Integrable derivations and Hochschild cohomology of block algebras of finite groups
Lead Research Organisation:
City, University of London
Department Name: Sch of Engineering and Mathematical Sci
Abstract
The product rule of the all familiar operation of taking derivatives of real valued functions has a plethora
of generalisations and applications in algebra. It leads to the notion of derivations of algebras - these
are linear endomorphisms of an algebra satifying the product rule. They represent the classes of
the first Hochschild cohomology of an algebra. The first Hochschild cohomology of an algebra
turns out to be a Lie algebra, and more precisely, a restricted Lie algebra if the underlying
base ring is a field of positive characteristic. The (restricted) Lie algebra structure extends to
all positive degrees in Hochschild cohomology - this goes back to pioneering work of Gerstenhaber
on defornations of algebras.
Modular representation theory of finite groups seeks to understand the connections between
the structure of finite groups and the associated group algebras. Many of the conjectures that drive
this area are - to date mysterious - numerical coincidences relating invariants of finite
group algebras to invariants of the underlying groups. The sophisticated cohomological
technology hinted at in the previous paragraph is expected to yield some insight regarding these
coincidences, and the present proposal puts the focus on some precise and unexplored
invariance properties of certain groups of integrable derivations under Morita, derived, or stable
equivalences between indecomposable algebra factors of finite group algebras, their character theory,
their automorphism groups, and the local structure of finite groups.
of generalisations and applications in algebra. It leads to the notion of derivations of algebras - these
are linear endomorphisms of an algebra satifying the product rule. They represent the classes of
the first Hochschild cohomology of an algebra. The first Hochschild cohomology of an algebra
turns out to be a Lie algebra, and more precisely, a restricted Lie algebra if the underlying
base ring is a field of positive characteristic. The (restricted) Lie algebra structure extends to
all positive degrees in Hochschild cohomology - this goes back to pioneering work of Gerstenhaber
on defornations of algebras.
Modular representation theory of finite groups seeks to understand the connections between
the structure of finite groups and the associated group algebras. Many of the conjectures that drive
this area are - to date mysterious - numerical coincidences relating invariants of finite
group algebras to invariants of the underlying groups. The sophisticated cohomological
technology hinted at in the previous paragraph is expected to yield some insight regarding these
coincidences, and the present proposal puts the focus on some precise and unexplored
invariance properties of certain groups of integrable derivations under Morita, derived, or stable
equivalences between indecomposable algebra factors of finite group algebras, their character theory,
their automorphism groups, and the local structure of finite groups.
Planned Impact
The impact of pure mathematics is omnipresent in modern technology.
The impact of pure mathematics is typically unplanned. Consistent
with this, the impact of the present project in the short term is expected to be within
the academic sphere.
Assuming positive outcomes of the research that is proposed to be undertaken,
this will have a substantial impact on where the area will move from there, and
by bringing together aspects of different areas of mathematics, influence the
methodology for some of these areas. The project will contribute to a vibrant
research environment, key to attracting the world's leading researchers.
The impact of pure mathematics is typically unplanned. Consistent
with this, the impact of the present project in the short term is expected to be within
the academic sphere.
Assuming positive outcomes of the research that is proposed to be undertaken,
this will have a substantial impact on where the area will move from there, and
by bringing together aspects of different areas of mathematics, influence the
methodology for some of these areas. The project will contribute to a vibrant
research environment, key to attracting the world's leading researchers.
Organisations
People |
ORCID iD |
| Markus Linckelmann (Principal Investigator) |
Publications
Benson D
(2021)
On the BV structure of the Hochschild cohomology of finite group algebras
in Pacific Journal of Mathematics
Benson D
(2019)
BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p' INERTIAL QUOTIENT
in The Quarterly Journal of Mathematics
Boltje R
(2020)
On Picard groups of blocks of finite groups
in Journal of Algebra
Eisele F
(2020)
The Picard Group of an Order and Külshammer Reduction
in Algebras and Representation Theory
Eisele F
(2018)
A counterexample to the first Zassenhaus conjecture
in Advances in Mathematics
Eisele F
(2018)
On Tate duality and a projective scalar property for symmetric algebras
in Pacific Journal of Mathematics
Eisele F
(2018)
A reduction theorem for $$\tau $$ t -rigid modules
in Mathematische Zeitschrift
Kessar R
(2018)
Weight conjectures for fusion systems
| Description | Two key events have been directly or indirectly made possible thanks to this grant. First, the hire of the postdoctoral researcher Florian Eisele, who did some outstanding work during his time at City, University of London. This includes a counter example to the last open Zassenhaus conjecture on integral group rings (in joint work with Leo Margolis), as well as finding an algebraic group structure on Picard groups of orders over complete discrete valuation rings, with applications to Donovan's conjecture in joint work with C. W. Eaton and M. Livesey. Second, the backing of this grant may well have tipped the balance for City University to grant a sabbatical semester at the MSRI, Berkeley, USA, during a special semester on representation theory, to which I was invited. This had the side effect that I met basically everyone I had mentioned in the grant application for extended periods in Berkeley. As a consequence, there has been significant output in the context of this grant. Here is the list (which does not contain a certain number of other papers which were not part of this project). The papers below make progress regarding some of the fundamental conjectures in block theory. Blocks with normal abelian defect and abelian p' inertial quotient. Submitted. (With D. J. Benson and R. Kessar.) The strong Frobenius numbers for cyclic defect blocks are equal to one. Submitted. On Picard groups of blocks of finite groups. To appear in a special volume in the Journal of Algebra in honour of M. Broue (2019). Dade's ordinary conjecture implies the Alperin-McKay conjecture. Arch. Math. (Basel) 112 (2019) (with R. Kessar) On automorphisms and focal subgroups of blocks. Geometric and topological aspects of the representation theory of finite groups, 235-249, Springer Proc. Math. Stat., 242, Springer, Cham, 2018. Descent of equivalences and character bijections. Geometric and topological aspects of the representation theory of finite groups, 181-212, Springer Proc. Math. Stat., 242, Springer, Cham, 2018 (with R. Kessar) Integrable derivations and stable equivalences of Morita type. Proc. Edinb. Math. Soc. (2) 61 (2018), no. 2, 343-362. On Morita and derived equivalences for cohomological Mackey algebras. Math. Z. 289 (2018), no. 1-2, 39-50. (With B. Rognerud.) On Tate duality and a projective scalar property for symmetric algebras. Pacific J. Math. 293 (2018), no. 2, 277-300. (with F. Eisele, M. Geline, and R. Kessar.) The grant administration was not entirely smooth at the end. Since Eisele had started one month later than planned, his 36 month appointment went one month beyond the official end date of the grant. Neither EPSRC nor City University nor I picked up on just how complicated that would be. The end result, which left me quite unimpressed, was that City asked me to pay for Eisele's last month from my personal account I have at City. Some less remote mechanisms for communication processes might have been helpful. |
| Exploitation Route | The structural properties of Picard groups of blocks in the joint paper with R. Boltje and R. Kessar are already being used towards new cases of Donovan's conjecure in work of Eaton, Eisele, and Livesey. |
| Sectors | Other |