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Categorification of tensor powers of the vector representation of \(U_q({\mathfrak {gl}}(1|1))\)

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Abstract

We consider the monoidal subcategory of finite-dimensional representations of \(U_q(\mathfrak {gl}(1|1))\) generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both \(U_q(\mathfrak {gl}(1|1))\) and the intertwining operators using subquotient categories of the BGG category .

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Notes

  1. We use this notation because \(W_{\mathfrak {p}}\) and \(W_{\mathfrak {q}}\) will correspond later to two parabolic subalgebras \({\mathfrak {p}},{\mathfrak {q}}\subset \mathfrak {gl}_n\).

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Acknowledgments

The present work is part of the author’s Ph.D. thesis. The author would like to thank his advisor Catharina Stroppel for her help and support. The author would also like to thank the anonymous referee for many helpful comments.

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  1. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstrasse 1, 79104, Freiburg, Germany

    Antonio Sartori

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Correspondence to Antonio Sartori.

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This work has been supported by the Graduiertenkolleg 1150, funded by the Deutsche Forschungsgemeinschaft.

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Sartori, A. Categorification of tensor powers of the vector representation of \(U_q({\mathfrak {gl}}(1|1))\) . Sel. Math. New Ser. 22, 669–734 (2016). https://doi.org/10.1007/s00029-015-0202-1

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