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# 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 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.

### 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 (1997-2011); 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
• Worship on Geometry, representation theory and Clusters, Leicester 2013, co-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)

##### 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