Personal tools

You are here: Home / Algebra

# Algebra

This group has a strong research record in many interconnecting areas of algebra with particular emphasis on homological algebra, representation theory and their interaction with non-commutative geometry.

Homological algebra is a relatively new subject which is just over sixty years of age; however it has by now become classical. It provides a common language for experts in topology, algebraic geometry, category theory and various branches of algebra. It is also fundamental for theoretical physics, particularly those aspects related with string theory and deformation quantization.

Representation theory is a much older subject which contains some of the most beautiful and symmetric constructions in the whole of mathematics. The representation theory of various algebraic objects can be viewed as a natural extension of their structure theory, and has innumerable applications, including within quantum mechanics, crystallography and physical chemistry. Representations of group algebras, Lie algebras, finite-dimensional associative algebras and quivers are all studied within this group, together with their connections with algebraic geometry, differential geometry and theoretical physics.

One of the exciting new areas in mathematics, which was created some 30 years ago, is non-commutative geometry. The idea here is to study non-commutative rings as if they were rings of functions on some imaginary 'non-commutative spaces'. This simple idea has turned out to be very fruitful and led to numerous advances in algebra and neighbouring fields. One of the especially useful spin-offs is the theory of quantum groups.

Non-commutative geometry also figures prominently in the study of operad algebras (such as A-infinity, L-infinity, or C-infinity-algebras). It turns out that a surprising amount of classical geometric techniques and concepts can be carried over in the framework of non-commutative geometry.  Non-commutative geometric objects arise in many different settings and projects by members of this group look at particular classes of algebras (cluster algebras, diagram algebras, Schur algebras, self-injective algebras) and their derived categories.

### Some Recent Publications

1. EL Green, S Schroll, S., 2017. Special multiserial algebras are quotients of symmetric special multiserial algebras. Journal of Algebra, 473, pp.397-405.
2. EL Green, S Schroll, 2016. Multiserial and special multiserial algebras and their representations. Advances in Mathematics, 302, pp.1111-1136.
3. Djament, A., Pirashvili, T. and Vespa, C., 2016. Cohomologie des foncteurs polynomiaux sur les groupes libres. Documenta Mathematica, 21, pp.205-222.
4. R Kurdiani, T Pirashvili, Functor homology and homology of commutative monoids. InSemigroup Forum 2016 Feb 1 (Vol. 92, No. 1, pp. 102-120). Springer US.
5. T Ashton, A Mudrov , Representations of quantum conjugacy classes of orthosymplectic groups. Journal of Mathematical Sciences. 2016 Mar 1;213(5):637-50.
6.  I Gálvez-Carrillo, L Lombardi, A Tonks. An A Operad in Spineless Cacti. Mediterranean journal of mathematics. 2015 Nov 1;12(4):1215-26.
7.  T Pirashvili, Projective and injective symmetric categorical groups and duality. Proceedings of the American Mathematical Society. 2015;143(3):1315-23.
8.  T Ashton, A Mudrov, R-matrix and Mickelsson algebras for orthosymplectic quantum groups. Journal of Mathematical Physics. 2015 Aug;56(8):081701.
9.  S Schroll, Trivial extensions of gentle algebras and Brauer graph algebras, Journal of Algebra, 444, (2015), pp.183-200.
10.  N Snashall, R Taillefer, Classification of symmetric special biserial algebras with at most one non-uniserial indecomposable projective. Proc. Edinburgh Math. Soc. 58 (2015), 739–767.
11.  R. Marsh, S Schroll, The geometry of Brauer graph algebras and cluster mutations, Journal of Algebra, 419, (2014), pp. 141-166.
12.  EL Green, S Schroll, N Snashall, Group actions and coverings of Brauer graph algebras, Glasgow Mathematical Journal 56 (02), 439-464 (2014).
13. AA Baranov, AA Osinovskaya, ID Suprunenko, Modular representations of the special linear groups with small weight multiplicities,  Journal of Algebra 397, 225-251, (2014).
14.  RJ Marsh, S Schroll, A circular order on edge-coloured trees and RNA m-diagrams, Advances in Applied Mathematics 54, 11-26 (2014).
15.  AA Baranov, J Rowley, Inner ideals of simple locally finite Lie algebras, Journal of Algebra 379, 11-30 (2013).
16. A Mudrov, On dynamical adjoint functor, Applied Categorical Structures, 1-20 (2013).
17. A Mudrov, Non-Levi closed conjugacy classes of SOq (N), Journal of Mathematical Physics 54 (8), 081701 (2013).
18. T Pirashvili, On Strongly Perfect Lie Algebras, Communications in Algebra 41 (5), 1619-1625 (2013).
19. M Hartl, T Pirashvili, C Vespa, Polynomial Functors from Algebras Over a Set-Operad and Nonlinear Mackey Functors, Int Math Res Notice,
20. AI Mudrov, On Dynamical Galois Extensions, Communications in Algebra 40 (8), 2867-2892 (2012).
21. A Parker, N Snashall, A family of Koszul self-injective algebras with finite Hochschild cohomology,  Journal of Pure and Applied Algebra 216 (5), 1245-1252 (2012).
22. S Schroll, N Snashall, Hochschild cohomology and support varieties for tame Hecke algebras, The Quarterly Journal of Mathematics 62 (4), 1017-1029 (2011).
23. N Snashall, R Taillefer, The Hochschild cohomology ring of a class of special biserial algebras, Journal of Algebra and Its Applications 9 (01), 73-122 (2010).
24. K Erdmann, S Schroll, On the Hochschild cohomology of tame Hecke algebras, Archiv der Mathematik 94 (2), 117-127 (2010).
25. T Pirashvili, MJ Redondo, Universal coefficient theorem in triangulated categories, Algebras and Representation Theory 11 (2), 107-114 (2008)
26. HJ Baues, M Jibladze, T Pirashvili, Quadratic algebra of square groups, Advances in Mathematics 217 (3), 1236-1300 (2008).

R Kurdiani, T Pirashvili, Functor homology and homology of commutative monoids. In Semigroup Forum 2016 Feb 1 (Vol. 92, No. 1, pp. 102-120). Springer US.

### Recent activities

• Workshop on Algebraic Structures in Geometry and Physics, July 2008; organised by Lazarev
• LMS Regional Meeting and Workshop on Derived Categories in Algebra, Topology and Geometry 2009; organised by Pirashvili and Snashall
• 24th British Topology Meeting, Leicester 2009; co-organised by Lazarev
• BLOC and REPNET Meetings (1997-2014); organised by Schroll and Snashall
• BMC Workshop on Algebraic Topology, Leicester 2011; organised by Lazarev
• BMC Workshop on Representations of Algebras, Leicester 2011; organised by Snashall
• Workshop on Geometry, Representation Theory and Clusters, Leicester 2013, co-organised by Schroll
• Workshop on Cluster Algebras and Finite Dimensional Algebras, Leicester 2015, organised by Schroll.
• Leicester is the grant holder for the LMS scheme 3 grant BLOC, 4 meetings per year (Snashall).

#### Current PhD students

Joanne Leader (supervisor: Snashall), Zeki Mirza (supervisor: Pirashvili), Tom Ashton (supervisor: Mudrov), Drew Duffield (supervisor: Schroll)

##### Recent PhDs:
• "Cohomology of Lambda Rings" by Michael Robinson (2010) (supervisor: Pirashvili)
• "Inner Ideals of Simple Locally Finite Lie Algebras" by Jamie Rowley (2012) (supervisor: Baranov)
• "Hochschild cohomology and periodicity of tame weakly symmetric algebras" by Ahlam Fallatah (2012) (supervisor: Snashall)

#### Grants

The Algebra group has received funding from:

Contact details

Department of Mathematics
University of Leicester
Leicester LE1 7RH
United Kingdom

Tel.: +44 (0)116 252 3917
Fax: +44 (0)116 252 3915

Campus Based Courses