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Paul Martin: Reference Section
The blob algebras (this page is a work in progress!)
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.
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:
Some other 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
- 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