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Representations and Nilpotent Orbits of Lie Algebraic Systems pp 19–72Cite as

Birkhäuser

On Cacti and Crystals

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Part of the book series: Progress in Mathematics ((PM,volume 330))

Abstract

In the present work we study actions of various groups generated by involutions on the category \(\mathscr O^{int}_q({\mathfrak {g}})\) of integrable highest weight \(U_q({\mathfrak {g}})\)-modules and their crystal bases for any symmetrizable Kac–Moody algebra \({\mathfrak {g}}\). The most notable of them are the cactus group and (yet conjectural) Weyl group action on any highest weight integrable module and its lower and upper crystal bases. Surprisingly, some generators of cactus groups are anti-involutions of the Gelfand–Kirillov model for \(\mathscr O^{int}_q({\mathfrak {g}})\) closely related to the remarkable quantum twists discovered by Kimura and Oya (Int Math Res Notices, 2019).

To Anthony Joseph, with admiration

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Notes

  1. 1.

    The difference between our notation and that of [7, 27] is in the linear automorphism of \(\mathcal A_q({\mathfrak {g}})\) defined on homogeneous elements x by \(x\mapsto q^{\frac 12(|x|,|x|)-(|x|,\rho )}x\).

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Acknowledgements

The first two authors are grateful to Anton Alekseev and Université de Genève, Switzerland, for their hospitality. An important part of this work was done during the second author’s stay at the Weizmann Institute of Science, Israel and during the conference in honor of Anthony Joseph’s 75th birthday, at the Weizmann Institute and at the University of Haifa. The authors would like to use this opportunity to thank Maria Gorelik and Anna Melnikov for organizing that wonderful event.

This work was partially supported by a BSF grant no. 2016363 (A. Berenstein), the Simons foundation collaboration grant no. 245735 (J. Greenstein) and by the Minerva foundation with funding from the Federal German Ministry for Education and Research (J.-R. Li).

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, USA

    Arkady Berenstein

  2. Department of Mathematics, University of California, Riverside, CA, USA

    Jacob Greenstein

  3. Department of Mathematics, The Weizmann Institute of Science, Rehovot, Israel

    Jian-Rong Li

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  1. Arkady Berenstein
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Correspondence to Arkady Berenstein .

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Editors and Affiliations

  1. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Maria Gorelik

  2. Department of Mathematics, University of Haifa, Haifa, Israel

    Vladimir Hinich

  3. Department of Mathematics, University of Haifa, Haifa, Israel

    Anna Melnikov

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Berenstein, A., Greenstein, J., Li, JR. (2019). On Cacti and Crystals. In: Gorelik, M., Hinich, V., Melnikov, A. (eds) Representations and Nilpotent Orbits of Lie Algebraic Systems. Progress in Mathematics, vol 330. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23531-4_2

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