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Crystal Bases of Modified \(\imath \)quantum Groups of Certain Quasi-Split Types

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Abstract

In order to see the behavior of \(\imath \)canonical bases at \(q = \infty \), we introduce the notion of \(\imath \)crystals associated to an \(\imath \)quantum group of certain quasi-split type. The theory of \(\imath \)crystals clarifies why \(\imath \)canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of \(\imath \)crystals whose projective limit can be thought of as the \(\imath \)canonical basis of the modified \(\imath \)quantum group at \(q = \infty \).

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Acknowledgements

The author is grateful to the referees for valuable comments. This work was supported by JSPS KAKENHI Grant Numbers JP20K14286 and JP21J00013.

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  1. Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Osaka, 558-8585, Japan

    Hideya Watanabe

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  1. Hideya Watanabe
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Correspondence to Hideya Watanabe.

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Watanabe, H. Crystal Bases of Modified \(\imath \)quantum Groups of Certain Quasi-Split Types. Algebr Represent Theor 27, 1–76 (2024). https://doi.org/10.1007/s10468-023-10207-z

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