# Dr Jan Grabowski

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.

Selected Publications Show all 14 publications

Grabowski, J.E., Pressland, M. 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

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

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

Algebra and Geometry seminar
Invited talk

Algebras and Representation Theory (Journal)
Publication peer-review

Algebra seminar
Invited talk

Algebra seminar
Invited talk

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

Publication peer-review

Visit of Sian Fryer, University of Leeds

Algebra-Geometry seminar
Invited talk

Algebras and Representation Theory (Journal)
Publication peer-review

Visit of Matthew Pressland, University of Bath

Visit of Sian Fryer, University of Leeds

Pure Mathematics Colloquium
Invited talk

London Algebra Colloquium
Invited talk

Glasgow Mathematical Journal (Journal)
Editorial activity

Algebra-Geometry seminar, Newcastle University
Invited talk

Visit of Dr Stephane Launois, University of Kent

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

Visit of Sira Gratz, Leibniz Universität Hannover, Germany