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Based modules over the \(\imath \)quantum group of type AI

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Abstract

This paper studies classical weight modules over the \(\imath \)quantum group \(\textbf{U}^{\imath }\) of type AI. We introduce the notion of based \(\textbf{U}^{\imath }\)-modules by generalizing the notion of based modules over quantum groups (quantized enveloping algebras). We prove that each finite-dimensional irreducible classical weight \(\textbf{U}^{\imath }\)-module with integer highest weight is a based \(\textbf{U}^{\imath }\)-module. As a byproduct, a new combinatorial formula for the branching rule from \(\mathfrak {sl}_n\) to \(\mathfrak {so}_n\) is obtained.

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References

  1. Appel, A., Vlaar, B.: Universal \(K\)-matrices for quantum Kac-Moody algebras. Represent. Theory 26, 764–824 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balagović, M., Kolb, S.: The bar involution for quantum symmetric pairs. Represent. Theory 19, 186–210 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balagović, M., Kolb, S.: Universal \(K\)-matrix for quantum symmetric pairs. J. Reine Angew. Math. 747, 299–353 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, H., Wang, W.: A New Approach to Kazhdan–Lusztig Theory of Type B via Quantum Symmetric Pairs, Astérisque (402), vii+134 (2018)

  5. Bao, H., Wang, W.: Canonical bases arising from quantum symmetric pairs. Invent. Math. 213(3), 1099–1177 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, H., Wang, W.: Canonical bases arising from quantum symmetric pairs of Kac-Moody type. Compos. Math. 157(7), 1507–1537 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berman, C., Wang, W.: Formulae of \(\imath \)-divided powers in \(U_q(\mathfrak{sl} _2)\). J. Pure Appl. Algebra 222(9), 2667–2702 (2018)

    Article  MathSciNet  Google Scholar 

  8. De Commer, K., Matassa, M.: Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices, Adv. Math. 366, 107029, 100 pp (2020)

  9. Dixmier, J.: Enveloping Algebras, Revised reprint of the 1977 translation. Graduate Studies in Mathematics, 11. American Mathematical Society, Providence, RI, 1996. xx+379 pp

  10. Ehrig, M., Stroppel, C.: Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality. Adv. Math. 331, 58–142 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gavrilik, A.M., Klimyk, A.U.: \(q\)-deformed orthogonal and pseudo-orthogonal algebras and their representations. Lett. Math. Phys. 21(3), 215–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases, Graduate Studies in Mathematics, 42. American Mathematical Society, Providence, RI. xviii+307 (2002)

  13. Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, xii+169 (1972)

  14. Jang, I.-S., Kwon, J.-H.: Flagged Littlewood-Richardson tableaux and branching rule for classical groups. J. Combin. Theory Ser. A 181, 105419 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jantzen, J. C.: Lectures on Quantum Groups, Graduate Studies in Mathematics, 6. American Mathematical Society, Providence, RI, viii+266 (1996)

  16. Kashiwara, M.: Crystalizing the \(q\)-analogue of universal enveloping algebras. Comm. Math. Phys. 133(2), 249–260 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kashiwara, M.: On crystal bases of the \(q\)-analogue of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53(2), 165–184 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kolb, S.: Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Koornwinder, T.H.: Orthogonal polynomials in connection with quantum groups, Orthogonal polynomials (Columbus, OH, 1989), 257–292, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 294, Kluwer Acad. Publ., Dordrecht (1990)

  21. Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra 220(2), 729–767 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lusztig, G.: Introduction to Quantum Groups, Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser/Springer, New York xiv+346, (2010)

  24. Naito, S., Sagaki, D.: An approach to the branching rule from sl2n(C) to sp2n(C) via Littelmann’s path model. J. Algebra 286(1), 187–212 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Noumi, M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123(1), 16–77 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Regelskis, V., Vlaar, B.: Quasitriangular coideal subalgebras of \(U_q(\mathfrak{g} )\) in terms of generalized Satake diagrams. Bull. Lond. Math. Soc. 52(4), 693–715 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sartori, A., Tubbenhauer, D.: Webs and \(q\)-Howe dualities in types \(BCD\). Trans. Am. Math. Soc. 371(10), 7387–7431 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stokman, J.V.: Generalized Onsager algebras. Algebr. Represent. Theory 23(4), 1523–1541 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  29. Watanabe, H.: Crystal basis theory for a quantum symmetric pair \((\textbf{U},\textbf{U} ^\jmath )\). Int. Math. Res. Not. IMRN 22, 8292–8352 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Watanabe, H.: Global crystal bases for integrable modules over a quantum symmetric pair of type AIII. Represent. Theory 25, 27–66 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Watanabe, H.: Classical weight modules over \(\imath \)quantum groups. J. Algebra 578, 241–302 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thanks anonymous referees for helpful comments. This work was supported by JSPS KAKENHI Grant Number JP20K14286.

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  1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Hideya Watanabe

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Watanabe, H. Based modules over the \(\imath \)quantum group of type AI. Math. Z. 303, 43 (2023). https://doi.org/10.1007/s00209-022-03189-z

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