Paul Martin: Reference Section : Projects...

The blob algebras (this page is a work in progress!)

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 (!):

• Libedinsky and Plaza, Blob algebra approach to modular representation theory, (arxiv) Proc. Lond. Math. Soc. Vol. 121 (2020) Issue3, 656-701.
• Espinoza, Plaza, Blob algebra and two-color Soergel calculus, JPAA 2019
• Bowman, Cox, Hazi, Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras, arXiv:2005.02825
• Lehrer, Lyu, Generalised Temperley-Lieb algebras of type G(r,1,n), JPAA 2023
• Plaza and Ryom-Hansen, Graded cellular bases ...
• Reeves, A tensor space representation of the symplectic blob algebra
• Reeves, Tilting modules for the symplectic blob algebra
• S Ryom-Hansen, Cell structures on the blob algebra
• A Gainutdinov, Jacobsen, Saleur, Vasseur, A physical approach to the classification of indecomposable Virasoro representations from the blob algebra, Nuclear Physics B 2013
• Bowman, Cox, Speyer, A family of graded decomposition numbers for diagrammatic Cherednik algebras, Okinawa preprint 2024
• Lobos, Plaza, Ryom-Hansen, The nil-blob algebra: An incarnation of type tilde-A_1 Soergel calculus and of the truncated blob algebra, J Alg 2021
• Ostrik, (not directly on blob algebra, but...) Module categories, weak Hopf algebras and modular invariants, arXiv 2001

• Saleur, The blob algebra and the periodic TL algebra, LMP
• Saleur, On an algebraic approach to higher dimensional statistical mechanics, CMP
  Some works following on from this:
  • Alcaraz, Ram, Rittenberg, Cyclic representations of the periodic TL algebra
  • Ibarra, Morris, Montoya Vega, Temperley-Lieb categories on non-orientable surfaces arXiv 2025
• Woodcock, On the structure of the blob algebra, J Alg 2000
• Woodcock, Generalised blob algebras and alcove geometry, LMS JCM
  Some works following on from this:
  • Maturana, Ryom-Hansen, Graded cellular basis and Jucys-Murphy elements for generalized blob algebras, JPAA 2020
  • Bowman, The many graded cellular bases of Hecke algebras, arXiv:1702.06579
• Cox, Graham and Martin The blob algebra in positive characteristic
• R Green, O King, A Parker et al, (Symplectic blob algebra...),
  Towers of recollement and bases for diagram algebras: Planar diagrams and a little beyond, J Alg 2007
  On the non-generic representation theory of the symplectic blob algebra, arXiv 2008
  A presentation for the symplectic blob algebra, JAA 2012
• J de Gier et al
• T tom Dieck
• A Doikou
• S Ryom-Hansen, Virtual algebraic Lie theory: Tilting modules and Ringel duals for blob algebras, Proc LMS 2004
• Dan Levy
• Haring-Oldenburg, The reduced BMW algebra of Coxeter type-B
• A Hazi et al
• ...

• 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
  • F Alcaraz, A Ram and V Rittenberg, Cyclic representations of the periodic TL algebra

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