Abstract
In this paper, we describe the irreducible representations and give a dimension formula for the framisation of the Temperley–Lieb algebra. We then prove that the framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. This allows us to construct a basis for it. We also study in a similar way the complex reflection Temperley–Lieb algebra.
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We are grateful to Dimoklis Goundaroulis, Jesús Juyumaya, Aristides Kontogeorgis and Sofia Lambropoulou for introducing us to this whole range of problems, and for many fruitful conversations. We would also like to thank Tamás Hausel for his interesting questions that led us to provide some extra results on the Yokonuma–Hecke algebra. Finally, we would like to thank Loïc Poulain d’Andecy and Nicolas Jacon for our useful conversations on the second part of this paper. This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS. The second author gratefully acknowledges financial support of EPSRC through the Grant Ep/I02610x/1.
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Chlouveraki, M., Pouchin, G. Representation theory and an isomorphism theorem for the framisation of the Temperley–Lieb algebra. Math. Z. 285, 1357–1380 (2017). https://doi.org/10.1007/s00209-016-1751-5
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DOI: https://doi.org/10.1007/s00209-016-1751-5