BACK or Mathematics Department Home

Paul Martin: Reference Section : Projects...

Consider the Brauer algebra B_n as a subalgebra of the partition algebra. (See here for the partition algebra.) Consider in particular the diagram basis (certain pictures of partitions of a set of n+n elements drawn in a rectangular frame), and the algebra multiplication formulated in terms of a corresponding diagram juxtaposition. From this one sees that there is a subalgebra with basis the subset of non-crossing diagrams. This is the TL algebra - an algebra of considerable significance in many areas of mathematics and physics. The blob algebra is a generalisation of the TL algebra as we now indicate.

The non-crossing idea involves drawing diagrams on a rectangle. It is natural (particularly from a computational physics perspective) to consider a generalisation where one draws diagrams on a cylinder, and distinguishes closed loops (formed in diagram composition) that are non-contractible. Rather than address this algebra directly, it is convenient to study an algebra in which the cohomological data is encoded in a blob on the `seam' (a line drawn on the cylinder to cut it open, and hence return to the rectangle). This is the idea of the blob algebra. (Although it can also be characterised in a number of other interesting ways...)

One of the most interesting aspects of the study of the blob algebra concerns its representation theory. Its `reductive' representation theory over algebraically closed fields is quite completely understood (see [Cox et al] and references therein). But a number of other interesting questions about it and its generalisations remain open.

Some nice looking recent papers on the blob algebra, indicating a lot of interesting open problems (!):

Some original refs:
• Saleur, The blob algebra and the periodic TL algebra, LMP
• Saleur, On an algebraic approach to higher dimensional statistical mechanics, CMP
• Woodcock, On the structure of the blob algebra, J Alg
• Woodcock, Generalised blob algebras and alcove geometry, LMS JCM
• Cox, Graham and Martin The blob algebra in positive characteristic
• R Green et al, Symplectic blob algebra...
• J de Gier et al
• T tom Dieck
• A Doikou
• S Ryom-Hansen
• ...
Some other refs:
• These papers study a very mild generalisation of the blob:
• R Green, Generalised TL algebras and decorated tangles
• R Green, Decorated tangles and canonical bases