Professor Jan Grabowski CMath FIMA SFHEA

Professor of Algebra

Research Overview

My research is in the area of quantum algebra. Here "algebra" refers to the major branch of mathematics that studies the structure of mathematical objects and their symmetries. The adjective "quantum" highlights a relationship with the quantum theory of physics. In the mathematics of that theory, many of the variables do not commute - that is, a times b need not equal b times a. Many of the structures in quantum algebra have a multiplication that is not commutative. The ones I study are often not far from being commutative, though. One can often deform a commutative product of two elements a and b by putting in an extra parameter, q say, and changing the product so that now ab=qba. Then ba=(1/q)ab and if q does not equal 1, the product is not commutative. We can think of being allowed to vary q and talk of q=1 as the "classical limit", just as classical physics can be recovered from quantum physics. The study of these deformations of well-understood classical algebraic and geometric objects is what makes up the field of quantum algebra.

More specifically, I work with Lie algebras, quantized enveloping algebras, quantized coordinate rings, partial flag varieties and quantum cluster algebras.

Selected Publications

Segre products of cluster algebras
Grabowski, J., Hindmarch, L. 5/12/2024 In: Bulletin of the London Mathematical Society. 56, 12, p. 3777-3785. 10 p.
Journal article

Deformed cluster maps of type A_2N
Grabowski, J., Hone, A., Kim, W. 28/02/2024 In: arxiv.org. 40 p.
Journal article

Slices of groupoids are group-like
Cooney, N., Grabowski, J. 7/02/2020 In: arxiv.org. 7 p.
Journal article

Automorphism groupoids in noncommutative projective geometry
Cooney, N., Grabowski, J.E. 15/08/2022 In: Journal of Algebra. 604, p. 296-323. 28 p.
Journal article

Graded Frobenius cluster categories
Grabowski, J.E., Pressland, M. 1/01/2018 In: Documenta Mathematica. 23, p. 49-76. 23 p.
Journal article

Graded quantum cluster algebras of infinite rank as colimits
Grabowski, J.E., Gratz, S. 11/2018 In: Journal of Pure and Applied Algebra. 222, 11, p. 3395-3413. 18 p.
Journal article

Graded cluster algebras
Grabowski, J. 12/2015 In: Journal of Algebraic Combinatorics. 42, 4, p. 1111-1134. 24 p.
Journal article

Graded quantum cluster algebras and an application to quantum Grassmannians
Grabowski, J., Launois, S. 23/05/2014 In: Proceedings of the London Mathematical Society. 109, 3, p. 697-732. 36 p.
Journal article

Cluster algebras of infinite rank
Grabowski, J., Gratz, S., Groechenig, M. 04/2014 In: Journal of the London Mathematical Society. 89, 2, p. 337-363. 27 p.
Journal article

A quantum analogue of the dihedral action on Grassmannians
Allman, J.M., Grabowski, J. 06/2012 In: Journal of Algebra. 359, 1, p. 49-68. 20 p.
Journal article

Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases
Grabowski, J., Launois, S. 2011 In: International Mathematics Research Notices. 2011, 10, p. 2230-2262. 33 p.
Journal article

Examples of quantum cluster algebras associated to partial flag varieties
Grabowski, J. 2011 In: Journal of Pure and Applied Algebra. 215, 7, p. 1582-1595. 14 p.
Journal article

Braided enveloping algebras associated to quantum parabolic subalgebras
Grabowski, J. 14/10/2011 In: Communications in Algebra. 39, 10, p. 3491-3514. 24 p.
Journal article

Braided Lie bialgebras associated to Kac-Moody algebras
Grabowski, J. 2008 In: Journal of Lie Theory. 18, 1, p. 125-140. 16 p.
Journal article

Inductive constructions for Lie bialgebras and Hopf algebras
Grabowski, J. 2006 University of London.
Doctoral Thesis

On Lie induction and the exceptional series
Grabowski, J. 2005 In: Journal of Algebra and Its Applications. 4, 6, p. 707-737. 30 p.
Journal article

A triple construction for Lie bialgebras
Grabowski, J. 2005 In: Pacific Journal of Mathematics. 221, 2, p. 281-301. 22 p.
Journal article

Schubert Calculus via Cluster Categories
04/01/2022 → 31/12/2022
Research

Combinatorial and Geometric Methods in Representation Theory
20/01/2020 → 19/01/2022
Research

Supplementary Issue of Glasgow Mathematical Journal on Non-Associative Algebras
01/12/2018 → 30/11/2020
Research

Reconstructing Broken Symmetry
28/11/2014 → 27/03/2016
Research

St Bernadette's Catholic Primary School
School Engagement

Noncommutative Geometry and Topology Seminar, Prague
Invited talk

Charles University
Visiting an external academic institution

CLAN (Clusters and their Applications in the North) Network Meeting
Invited talk

Combinatorial Representation Theory in Leeds
Participation in conference - Academic

UKRI Talent Peer Review College (External organisation)
Membership of committee

Aarhus University
Visiting an external academic institution

Geometric and Homological Methods in Representation Theory
Participation in conference -Mixed Audience

Sira Gratz
Hosting an academic visitor

Non-associative algebras
Participation in conference -Mixed Audience

Quantum homogeneous spaces
Participation in conference -Mixed Audience

MAXIMALS seminar
Invited talk

University of York (External organisation)
Membership of committee

Algebra seminar
Invited talk

Carnegie Trust for the Universities of Scotland (External organisation)
Member of an organisation

Proceedings of the London Mathematical Society (Journal)
Publication peer-review

Journal of Algebra (Journal)
Publication peer-review

Charlie Beil
Hosting an academic visitor

Algebra and Geometry seminar
Invited talk

Algebras and Representation Theory (Journal)
Publication peer-review

Algebra seminar
Invited talk

Algebra seminar
Invited talk

Advances in Mathematics (Journal)
Publication peer-review

Algebra seminar
Invited talk

London Algebra Colloquium
Invited talk

London Mathematical Society Lecture Note Series (Journal)
Publication peer-review

Journal of Algebra (Journal)
Publication peer-review

Advances in Applied Mathematics (Journal)
Publication peer-review

Sian Fryer
Hosting an academic visitor

Algebra-Geometry seminar
Invited talk

Algebras and Representation Theory (Journal)
Publication peer-review

Matthew Pressland
Hosting an academic visitor

Invited talk at Special Session on "Combinatorics and Geometry of Quantum Algebras".
Invited talk

Sian Fryer
Hosting an academic visitor

Invited talk at LMS Workshop on Cluster Algebras and Preprojective Algebras, Cardiff University
Invited talk

4 lectures at instructional course "Kent Algebra Days PGR"
Invited talk

Oberseminar zur Algebra und Algebraischen Kombinatorik
Invited talk

Pure Mathematics Colloquium
Invited talk

London Algebra Colloquium
Invited talk

Invited talk at conference "Cluster algebras and combinatorics", University of Münster
Invited talk

Glasgow Mathematical Journal (Journal)
Editorial activity

Algebra-Geometry seminar, Newcastle University
Invited talk

Launois Stéphane
Hosting an academic visitor

Pure Mathematics seminar, University of Leicester
Invited talk

Engineering and Physical Sciences Research Council (External organisation)
Membership of committee

Journal of the London Mathematical Society (Journal)
Publication peer-review

STEMNET (Science, Technology, Engineering and Mathematics Network)
School Engagement

Engineering and Physical Sciences Research Council (External organisation)
Membership of committee

Journal of Algebra (Journal)
Publication peer-review

Netherlands Organisation for Scientific Research - NWO (External organisation)
Membership of committee

Engineering and Physical Sciences Research Council (External organisation)
Membership of committee

Future directions for quantum groups
Participation in conference -Mixed Audience

XV International Conference on Representations of Algebras (ICRA XV)
Participation in conference -Mixed Audience

Algebras and Representation Theory (Journal)
Publication peer-review

Algebra seminar, University of Glasgow
Participation in workshop, seminar, course

Pure Mathematics seminar, Lancaster University
Participation in workshop, seminar, course

Sira Gratz
Hosting an academic visitor

29th meeting of ARTIN (Algebra and Representation Theory in the North)
Participation in workshop, seminar, course

Kent Algebra Days 2011
Participation in conference -Mixed Audience

  • Algebra and Geometry
  • Mathematical Physics (Maths and Stats)