Lecturer in Algebra and Geometry

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.

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.

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

*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.

MATH112 Discrete Mathematics, MATH225 Abstract Algebra

*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.

*September 2011-present:* 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.

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)

Subject Editor (Algebra), Glasgow Mathematical Journal

Member of EPSRC Peer Review College

Carnegie Research Assessor, Carnegie Trust for the Universities of Scotland

## Graded Frobenius cluster categories

Grabowski, J. E. & Pressland, M. 2018 In : Documenta Mathematica. 23, p. 49-76 23 p.Research output: Contribution to journal › Journal article

- E-pub ahead of print
## Graded quantum cluster algebras of infinite rank as colimits

Grabowski, J. E. & Gratz, S. 23/12/2017 In : Journal of Pure and Applied Algebra. 222, 11, p. 3395-3413 18 p.Research output: Contribution to journal › Journal article

## Graded cluster algebras

Grabowski, J. 12/2015 In : Journal of Algebraic Combinatorics. 42, 4, p. 1111-1134 24 p.Research output: Contribution to journal › Journal article

## Graded quantum cluster algebras and an application to quantum Grassmannians

Grabowski, J. & Launois, S. 2014 In : Proceedings of the London Mathematical Society. 109, 3, p. 697-732 36 p.Research output: Contribution to journal › 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.Research output: Contribution to journal › Journal article

- Published
## 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.Research output: Contribution to journal › 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.Research output: Contribution to journal › 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.Research output: Contribution to journal › 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.Research output: Contribution to journal › 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.Research output: Contribution to journal › Journal article

- Unpublished
## Inductive constructions for Lie bialgebras and Hopf algebras

Grabowski, J. 2006 University of London.Research output: Thesis › 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.Research output: Contribution to journal › Journal article

## A triple construction for Lie bialgebras

Grabowski, J. 2005 In : Pacific Journal of Mathematics. 221, 2, p. 281-301 22 p.Research output: Contribution to journal › Journal article

## Advances in Mathematics (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Algebra seminar

Activity: Talk or presentation › Invited talk

## Algebras and Representation Theory (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Algebra and Geometry seminar

Activity: Talk or presentation › Invited talk

## Hosted academic visitor

Activity: Other › Types of External academic engagement - Hosting an academic visitor

## Journal of Algebra (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Proceedings of the London Mathematical Society (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Carnegie Trust for the Universities of Scotland (External organisation)

Activity: Membership › Member of an organisation

## Algebra seminar

Activity: Talk or presentation › Invited talk

## University of York (External organisation)

Activity: Membership › Membership of committee

## MAXIMALS seminar

Activity: Talk or presentation › Invited talk

## Quantum homogeneous spaces

Activity: Participating in or organising an event › Participation in conference

## Non-associative algebras

Activity: Participating in or organising an event › Participation in conference

## Algebra seminar

Activity: Talk or presentation › Invited talk

## Visit of Sian Fryer, University of Leeds

Activity: Other › Types of External academic engagement - Hosting an academic visitor

## Visit of Matthew Pressland, University of Bath

Activity: Other › Types of External academic engagement - Hosting an academic visitor

## Algebras and Representation Theory (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Algebra-Geometry seminar

Activity: Talk or presentation › Invited talk

## Visit of Sian Fryer, University of Leeds

Activity: Other › Types of External academic engagement - Hosting an academic visitor

## Journal of Algebra (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## Advances in Applied Mathematics (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## London Mathematical Society Lecture Note Series (Journal)

Activity: Publication peer-review and editorial work › Publication peer-review

## London Algebra Colloquium

Activity: Talk or presentation › Invited talk

## Algebra seminar

Activity: Talk or presentation › Invited talk

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

Activity: Talk or presentation › Invited talk

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

Activity: Talk or presentation › Invited talk

## 4 lectures at instructional course "Kent Algebra Days PGR"

Activity: Talk or presentation › Invited talk

## Oberseminar zur Algebra und Algebraischen Kombinatorik

Activity: Talk or presentation › Invited talk

## Pure Mathematics Colloquium

Activity: Talk or presentation › Invited talk

## London Algebra Colloquium

Activity: Talk or presentation › Invited talk

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

Activity: Talk or presentation › Invited talk

## Glasgow Mathematical Journal (Journal)

Activity: Publication peer-review and editorial work › Editorial activity

## Algebra-Geometry seminar, Newcastle University

Activity: Talk or presentation › Invited talk

## Visit of Dr Stephane Launois, University of Kent

Activity: Other › Types of External academic engagement - Hosting an academic visitor

## EPSRC (External organisation)

Activity: Membership › Membership of committee

## STEMNET (Science, Technology, Engineering and Mathematics Network)

Activity: Visiting an external institution › School Engagement

## Future directions for quantum groups

Activity: Participating in or organising an event › Participation in conference

## XV International Conference on Representations of Algebras (ICRA XV)

Activity: Participating in or organising an event › Participation in conference

## 29th meeting of ARTIN (Algebra and Representation Theory in the North)

Activity: Participating in or organising an event › Participation in workshop, seminar, course

## Kent Algebra Days 2011

Activity: Participating in or organising an event › Participation in conference

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