# 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 13 publications

Graded Frobenius cluster categories
Grabowski, J.E., Pressland, M. 30/09/2016 In: arxiv.org. p. 1-21. 21 p.
Journal article

Graded quantum cluster algebras of infinite rank as colimits
Grabowski, J.E., Gratz, S. 14/10/2015 In: arxiv.org. 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