A class of non-weight modules of 𝑈_𝑝(𝖘𝖑₂) and Clebsch–Gordan type formulas

References

[1] H. H. Andersen, J. C. Jantzen and W. Soergel, Representations of Quantum Groups at a 𝑝th Root of Unity and of Semisimple Groups in Characteristic 𝑝: Independence of 𝑝, Astérisque 220, Société Mathématique de France, Paris, 1994. Search in Google Scholar

[2] R. Bezrukavnikov, I. Mirković and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), no. 3, 945–991. 10.4007/annals.2008.167.945Search in Google Scholar

[3] K. A. Brown and I. Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (2001), no. 4, 733–779. 10.1007/s002090100274Search in Google Scholar

[4] C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris 1989), Progr. Math. 92, Birkhäuser, Boston (1990), 471–506. Search in Google Scholar

[5] V. G. Drinfel’d, Hopf algebras and the quantum Yang–Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258. 10.1142/9789812798336_0013Search in Google Scholar

[6] V. G. Drinfel’d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2, American Mathematical Society, Providence (1987), 798–820. 10.1142/9789812798336_0014Search in Google Scholar

[7] Y. He, Y.-a. Cai and R. Lü, A class of new simple modules for s⁢ln+1 and the Witt algebra, J. Algebra 541 (2020), 415–435. 10.1016/j.jalgebra.2019.09.011Search in Google Scholar

[8] J. C. Jantzen, Lectures on Quantum Groups, Grad. Stud. Math. 6, American Mathematical Society, Providence, 1996. 10.1090/gsm/006Search in Google Scholar

[9] M. Jimbo, A 𝑞-difference analogue of U⁢(g) and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. 10.1007/BF00704588Search in Google Scholar

[10] C. Kassel, Quantum Groups, Grad. Texts in Math. 155, Springer, New York, 1995. 10.1007/978-1-4612-0783-2Search in Google Scholar

[11] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), no. 2, 237–249. 10.1016/0001-8708(88)90056-4Search in Google Scholar

[12] G. Lusztig, Modular representations and quantum groups, Classical Groups and Related Topics (Beijing 1987), Contemp. Math. 82, American Mathematical Society, Providence (1989), 59–77. 10.1090/conm/082/982278Search in Google Scholar

[13] G. Lusztig, Introduction to Quantum Groups, Progr. Math. 110, Birkhäuser, Boston, 1993. Search in Google Scholar

[14] Y. Ma, Research on non-weight modules over quantum groups Uq⁢(s⁢l2) and Uq⁢(s⁢l3), Postdoctoral Fellowship Final Report, USTC, 2017. Search in Google Scholar

[15] J. Nilsson, Simple s⁢ln+1-module structures on U⁢(h), J. Algebra 424 (2015), 294–329. 10.1016/j.jalgebra.2014.09.036Search in Google Scholar

[16] J. Nilsson, U⁢(h)-free modules and coherent families, J. Pure Appl. Algebra 220 (2016), no. 4, 1475–1488. 10.1016/j.jpaa.2015.09.013Search in Google Scholar

[17] M. Ondrus, Whittaker modules for Uq⁢(s⁢l2), J. Algebra 289 (2005), no. 1, 192–213. 10.1016/j.jalgebra.2005.03.018Search in Google Scholar

[18] M. Ondrus, Tensor products and Whittaker vectors for quantum groups, Comm. Algebra 35 (2007), no. 8, 2506–2523. 10.1080/00927870701326387Search in Google Scholar

[19] D. Pedersen, Irreducible quantum group modules with finite dimensional weight spaces. I, preprint (2015), https://arxiv.org/abs/1504.07042. Search in Google Scholar

[20] D. Pedersen, Irreducible quantum group modules with finite dimensional weight spaces. II, preprint (2015), https://arxiv.org/abs/1506.08011. Search in Google Scholar

[21] F. J. Plaza Martín and C. Tejero Prieto, Construction of simple non-weight s⁢l⁢(2)-modules of arbitrary rank, J. Algebra 472 (2017), 172–194. 10.1016/j.jalgebra.2016.10.012Search in Google Scholar

[22] M. Rosso, Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), no. 4, 581–593. 10.1007/BF01218386Search in Google Scholar

[23] X. Tang and Y. Xu, On representations of quantum groups Uq⁢(fm⁢(K,H)), Bull. Aust. Math. Soc. 78 (2008), no. 2, 261–284. 10.1017/S0004972708000701Search in Google Scholar