Representations of quantum superalgebra $ U_{r, s}(\mathrm{osp}(1, 2)) $ at the root of unity and its restrictions

Yujie Gao; Shilin Yang; Yujie Gao; Shilin Yang

doi:10.3934/math.2026268

AIMS Mathematics

2026, Volume 11, Issue 3: 6485-6498. doi: 10.3934/math.2026268

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Representations of quantum superalgebra $ U_{r, s}(\mathrm{osp}(1, 2)) $ at the root of unity and its restrictions

Yujie Gao ,
Shilin Yang ^,

School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing, 100124, China

Received: 26 November 2025 Revised: 22 February 2026 Accepted: 06 March 2026 Published: 13 March 2026
MSC : 16T20, 17B37, 20G42

In this paper, the simple modules for the quantum superalgebra $ U_{r, s}(\mathrm{osp}(1, 2)) $ with two parameters at the root of unity (i.e., $ q = (rs^{-1})^{1/2} $ is a root of unity) are completely determined up to isomorphism. Furthermore, the classification of finite-dimensional simple modules over the restricted quantum superalgebra $ \overline{U}_{r, s}(\mathrm{osp}(1, 2)) $ is given.
- two-parameter quantum group,
- quantum superalgebra,
- simple module
Citation: Yujie Gao, Shilin Yang. Representations of quantum superalgebra $ U_{r, s}(\mathrm{osp}(1, 2)) $ at the root of unity and its restrictions[J]. AIMS Mathematics, 2026, 11(3): 6485-6498. doi: 10.3934/math.2026268

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Abstract

In this paper, the simple modules for the quantum superalgebra $ U_{r, s}(\mathrm{osp}(1, 2)) $ with two parameters at the root of unity (i.e., $ q = (rs^{-1})^{1/2} $ is a root of unity) are completely determined up to isomorphism. Furthermore, the classification of finite-dimensional simple modules over the restricted quantum superalgebra $ \overline{U}_{r, s}(\mathrm{osp}(1, 2)) $ is given.

References

[1]	V. N. Tolstoy, Extremal projectors for quantized Kac-Moody superalgebras and some of their applications, In: Quantum Groups, Berlin: Springer, 1990. https://doi.org/10.1007/3-540-53503-9_45
[2]	R. B. Zhang, Finite-dimensional irreducible representations of the quantum supergroup $U_q(\mathrm{gl}(m/n))$, J. Math. Phys., 34 (1993), 1236–1254. https://doi.org/10.1063/1.530198 doi: 10.1063/1.530198
[3]	R. B. Zhang, Finite-dimensional representations of $U_q(C(n+1))$ at arbitrary $q$, J. Phys. A Math. Gen., 26 (1993), 7041–7059. https://doi.org/10.1088/0305-4470/26/23/042 doi: 10.1088/0305-4470/26/23/042
[4]	T. D. Palev, N. I. Stoilova, J. Van der Jeugt, Finite-dimensional representations of the quantum superalgebra $U_q[\mathrm{gl}(n/m)]$ and related $q$-identities, Commun. Math. Phys., 166 (1994), 367–378. https://doi.org/10.1007/BF02112320 doi: 10.1007/BF02112320
[5]	R. B. Zhang, The $\mathrm{gl}(M\|N)$ super Yangian and its finite-dimensional representations, Lett. Math. Phys., 37 (1996), 419–434. https://doi.org/10.1007/BF00312673 doi: 10.1007/BF00312673
[6]	R. B. Zhang, Symmetrizable quantum affine superalgebras and their representations, J. Math. Phys., 38 (1997), 535–543. https://doi.org/10.1063/1.531833 doi: 10.1063/1.531833
[7]	R. B. Zhang, Finite-dimensional representations of $U_q(\mathrm{osp}(1/2n))$ and its connection with quantum $\mathrm{so}(2n+1)$, Lett. Math. Phys., 25 (1992), 317–325. https://doi.org/10.1007/BF00398404 doi: 10.1007/BF00398404
[8]	D. Arnaudon, Centre and representations of small quantum superalgebras at root of unity, Czech. J. Phys., 46 (1996), 1121–1129. https://doi.org/10.1007/BF01690325 doi: 10.1007/BF01690325
[9]	D. Arnaudon, M. Bauer, Scasimir operator, scentre and representations of $\mathcal{U}_q(\mathrm{osp}(1\|2))$, Lett. Math. Phys., 40 (1997), 307–320. https://doi.org/10.1023/A:1007359625264 doi: 10.1023/A:1007359625264
[10]	X. Gomez, Projective indecomposable modules over the small quantum $\mathrm{osp}(1\|2)$, Czech. J. Phys., 50 (2000), 71–78. https://doi.org/10.1023/A:1022872914794 doi: 10.1023/A:1022872914794
[11]	B. Hou, Z. L. Zhang, B. L. Cai, The structure of quantum group $\mathcal{U}_q(\mathrm{osp}(1, 2, f))$, J. Math. Res. Appl., 31 (2011), 451–461. https://doi.org/10.3770/j.issn:1000-341X.2011.03.009 doi: 10.3770/j.issn:1000-341X.2011.03.009
[12]	C. Sun, W. Wang, S. Yang, The Harish-Chandra homomorphism and representations of quantum superalgebras $U_q(\mathrm{osp}(1, 2, \mathrm{c}))$, Filomat, 40 (2026), 1361–1376. https://orcid.org/0000-0002-7491-9699
[13]	G. Benkart, S. Witherspoon, Two-parameter quantum groups and Drinfel'd doubles, Algebr. Represent. Th., 7 (2004), 261–286. https://doi.org/10.1023/B:ALGE.0000031151.86090.2e doi: 10.1023/B:ALGE.0000031151.86090.2e
[14]	G. Benkart, S. Witherspoon, Representations of two-parameter quantum groups and Schur-Weyl duality, 2001. https://doi.org/10.48550/arXiv.math/0108038
[15]	N. Bergeron, Y. Gao, N. Hu, Drinfel'd doubles and Lusztig's symmetries of two-parameter quantum groups, J. Algebra, 301 (2006), 378–405. https://doi.org/10.1016/j.jalgebra.2005.08.030 doi: 10.1016/j.jalgebra.2005.08.030
[16]	X. Bai, N. Hu, Two-parameter quantum groups of exceptional type $E$-series and convex PBW-type basis, Algebra Colloq., 15 (2008), 619–636. https://doi.org/10.1142/S100538670800059X doi: 10.1142/S100538670800059X
[17]	N. Hu, Q. Shi, The two-parameter quantum group of exceptional type $G_2$ and Lusztig's symmetries, Pacific J. Math., 230 (2007), 327–345. https://doi.org/10.2140/pjm.2007.230.327 doi: 10.2140/pjm.2007.230.327
[18]	G. Benkart, S. Witherspoon, Representations of finite dimensional algebras and related topics in Lie theory and geometry, American Mathematical Society, 2004. https://doi.org/10.1090/fic/040
[19]	X. Bai, Two-parameter quantum groups of type $E$ and restricted two-parameter quantum groups of type $D_n$, East China Normal University, 2006. https://doi.org/10.7666/d.y896708
[20]	R. Chen, Restricted two-parameter quantum groups of type $C_n$, East China Normal University, 2008. https://doi.org/10.7666/d.y1373024
[21]	N. Hu, X. Wang, Convex PBW-type Lyndon basis and restricted two-parameter quantum groups of type $G_2$, Pacific J. Math., 241 (2009), 243–273. https://doi.org/10.2140/pjm.2009.241.243 doi: 10.2140/pjm.2009.241.243
[22]	N. Hu, X. Wang, Convex PBW-type Lyndon bases and restricted two-parameter quantum groups of type $B$, J. Geom. Phys., 60 (2010), 430–453. https://doi.org/10.1016/j.geomphys.2009.11.005 doi: 10.1016/j.geomphys.2009.11.005
[23]	X. Chen, N. Hu, X. Wang, Convex PBW-type Lyndon bases and restricted two-parameter quantum group of type $F_4$, Acta Math. Sin. English Ser., 39 (2023), 1053–1084. https://doi.org/10.1007/s10114-023-1536-9 doi: 10.1007/s10114-023-1536-9
[24]	N. Hu, X. Xu, Novel isoclasses of one-parameter exotic small quantum groups originating from a two-parameter framework, Bull. Sci. Math., 206 (2026), 103738. https://doi.org/10.1016/j.bulsci.2025.103738 doi: 10.1016/j.bulsci.2025.103738
[25]	G. Shi, Two-parameter quantum supergroups of the Lie superalgebra $\mathrm{osp}(1\|2n)$, J. East China Norm. Univ. Natur. Sci. Ed., 2011 (2011), 121–132.
[26]	F. Liu, N. Hu, N. Jing, Quantum supergroup $U_{r, s}(osp(1, 2))$, Scasimir operators and Dickson polynomials, J. Algebra Appl., 23 (2024), 2450003. https://doi.org/10.1142/S0219498824500038 doi: 10.1142/S0219498824500038

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