# Dr Jan Grabowski SFHEA

Senior Lecturer in Algebra and Geometry## 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.

## Web Links

Personal pages: http://www.maths.lancaster.ac.uk/~grabowsj

## Research Grants

*October 2014*: EPSRC First Grant EP/M001148/1 (£98,663), 16-month project entitled "Reconstructing broken symmetry".

*January 2014*: Faculty Travel Grant (£282), to fund attendance at meeting "Cluster algebras and combinatorics", February 2014, Münster, Germany.

*February 2012:* London Mathematical Society Scheme 1 Conference Grant (£2,567.80), to part-fund conference "Future directions for quantum groups" held in Lancaster in September 2012.

*February 2012:* Faculty Research Grant (£1,000), to part-fund conference "Future directions for quantum groups" held in Lancaster in September 2012.

*November 2011:* Faculty Travel Grant (£348), to fund attendance at British Mathematical Colloquium, April 2012, Canterbury, Kent.

*October 2011:* Departmental Research Incentivisation Grant (£550), to fund research visit of Sira Gratz to Lancaster University.

## Current Teaching

MATH225 Abstract Algebra

## Qualifications

*December 2013:* Postgraduate Certificate in Academic Practice (PGCert), Lancaster University.

*April 2006:* PhD in Mathematics, Queen Mary, University of London, under the supervision of Prof. Shahn Majid.

*July 2002:* Master of Mathematics with Honours (MMath(Hons)), First class, University of Warwick.

## Career Details

* August 2019-present: *Senior Lecturer in Pure Mathematics, Department of Mathematics and Statistics, Lancaster University.

*September 2011-August 2019:* Lecturer in Pure Mathematics, Department of Mathematics and Statistics, Lancaster University.

*October 2009-September 2011: *Fixed-Term Fellow by Special Election in Mathematics, Keble College, Oxford.

* October 2008-October 2009: *Stipendiary Lecturer in Mathematics, Keble College, Oxford.

* January-March 2008-2010: *Lecturer and class tutor, Mathematical Institute, Oxford.

* October 2005-September 2008: *Research Fellow and Tutor in Mathematics, Keble College, Oxford.

## Professional Role

Part I Director of Studies (joint) (September 2016-present)

Chair of University Web Steering Group (September 2017-present)

Marketing and Publicity Officer, Department of Mathematics and Statistics (September 2012-present)

Outreach Officer, Department of Mathematics and Statistics (September 2011-June 2016)

Deputy Admissions Tutor, Department of Mathematics and Statistics (September 2011-August 2014)

## External Roles

Subject Editor (Algebra), Glasgow Mathematical Journal

Member of EPSRC Peer Review College

Carnegie Research Assessor, Carnegie Trust for the Universities of Scotland

## PhD Supervisions Completed

Thomas Booker-Price (October 2013-December 2017). Thesis: *Applications of Graded Methods to Cluster Variables in Arbitrary Types*

## PhD Supervision Interests

I am currently offering PhD projects in three areas: 1. Quantum groups, including producing novel quantum groups from the double bosonisation construction. 2. Cluster algebras, their quantizations and representation theory. 3. Relationships with mathematical physics, including Verlinde algebras, quantum cohomology and integrable systems and their relationships to quantum cluster algebras.

Selected Publications

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. 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

All Publications

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

Non-associative algebras

Participation in conference

Quantum homogeneous spaces

Participation in conference

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

EPSRC (External organisation)

Membership of committee

Journal of the London Mathematical Society (Journal)

Publication peer-review

STEMNET (Science, Technology, Engineering and Mathematics Network)

School Engagement

EPSRC (External organisation)

Membership of committee

Journal of Algebra (Journal)

Publication peer-review

Netherlands Organisation for Scientific Research - NWO (External organisation)

Membership of committee

EPSRC (External organisation)

Membership of committee

Future directions for quantum groups

Participation in conference

XV International Conference on Representations of Algebras (ICRA XV)

Participation in conference

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

- Algebra and Geometry