Abstract
We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne’s universal monoidal category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows us to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.
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Acknowledgement
We thank Volodymyr Mazorchuk, Catharina Stroppel and Oded Yacobi for very useful discussions. We would also like to thank Joanna Meinel for useful comments and the referee for spotting several typos. This research was supported by the Australian Research Council grants DP140103239 and DP150103431.
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Presented by: Steffen Koenig
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Coulembier, K., Ehrig, M. The Periplectic Brauer Algebra III: The Deligne Category. Algebr Represent Theor 24, 993–1027 (2021). https://doi.org/10.1007/s10468-020-09976-8
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DOI: https://doi.org/10.1007/s10468-020-09976-8
Keywords
- Deligne category
- Thick tensor ideals
- Periplectic Lie superalgebra
- Categorification
- Diagram algebras
- Temperley-Lieb algebra
- Fock space