Module algebra structures of nonstandard quantum group $ X_{q}(A_{1}) $ on the quantum plane

Dong Su; Fengxia Gao; Zhenzhen Gao; Dong Su; Fengxia Gao; Zhenzhen Gao

doi:10.3934/era.2025157

Electronic Research Archive

2025, Volume 33, Issue 6: 3543-3560. doi: 10.3934/era.2025157

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Research article

Module algebra structures of nonstandard quantum group $ X_{q}(A_{1}) $ on the quantum plane

1.
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
2.
School of Science, Henan Institute of Technology, Xinxiang 453003, China

Received: 03 March 2025 Revised: 20 May 2025 Accepted: 30 May 2025 Published: 09 June 2025

In this paper, for $ n\geq2 $ and $ n\neq3 $, the module algebra structures of $ X_{q}(A_{1}) $ on the quantum $ n $-space were discussed, where the quantum $ n $-space is denoted by $ A_{q}(n) $. In particular, a complete list of $ X_{q}(A_{1}) $-module algebra structures on the quantum plane $ A_{q}(2) $ was produced and the isomorphism classes of these structures were described.
- nonstandard quantum group,
- quantum $ n $-space,
- Hopf action,
- module algebra,
- weight
Citation: Dong Su, Fengxia Gao, Zhenzhen Gao. Module algebra structures of nonstandard quantum group $ X_{q}(A_{1}) $ on the quantum plane[J]. Electronic Research Archive, 2025, 33(6): 3543-3560. doi: 10.3934/era.2025157

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Abstract

In this paper, for $ n\geq2 $ and $ n\neq3 $, the module algebra structures of $ X_{q}(A_{1}) $ on the quantum $ n $-space were discussed, where the quantum $ n $-space is denoted by $ A_{q}(n) $. In particular, a complete list of $ X_{q}(A_{1}) $-module algebra structures on the quantum plane $ A_{q}(2) $ was produced and the isomorphism classes of these structures were described.

References

[1]	M. Ge, G. Liu, K. Xue, New solutions of Yang-Baxter equations: Birman-Wenzl algebra and quantum group structures, J. Phys. A: Math. Gen., 24 (1991), 2679. https://doi.org/10.1088/0305-4470/24/12/008 doi: 10.1088/0305-4470/24/12/008
[2]	C. N. Yang, M. L. Ge, Braid Group, Knot Theory and Statistical Mechanics, World Scientific Publishing Company, 1991. https://doi.org/10.1142/0796
[3]	A. Aghamohammadi, V. Karimipour, S. Rouhani, The multiparametric non-standard deformation of $A_{n-1}$, J. Phys. A: Math. Gen., 26 (1993), 75. https://doi.org/10.1088/0305-4470/26/3/002 doi: 10.1088/0305-4470/26/3/002
[4]	N. Jing, M. Ge, Y. Wu, A new quantum group associated with a 'nonstandard' braid group representation, Lett. Math. Phys., 21 (1991), 193–203. https://doi.org/10.1007/BF00420369 doi: 10.1007/BF00420369
[5]	C. Cheng, S. Yang, Weak Hopf algebras corresponding to non-standard quantum groups, Bull. Korean Math. Soc., 54 (2017), 463–484. https://doi.org/10.4134/BKMS.b160029 doi: 10.4134/BKMS.b160029
[6]	D. Su, S. Yang, Representations of the small nonstandard quantum groups $\overline{X}_{q}(A_1)$, Commun. Algebra, 47 (2019), 5039–5062. https://doi.org/10.1080/00927872.2019.1612412 doi: 10.1080/00927872.2019.1612412
[7]	M. E. Sweedler, Hopf Algebras, W. A. Benjamin, 1969.
[8]	M. Beattie, A direct sum decomposition for the Brauer group of $H$-module algebras, J. Algebra, 43 (1976), 686–693. https://doi.org/10.1016/0021-8693(76)90134-4 doi: 10.1016/0021-8693(76)90134-4
[9]	R. J. Blattner, S. Montgomery, A duality theorem for Hopf module algebras, J. Algebra, 95 (1985), 153–172. https://doi.org/10.1016/0021-8693(85)90099-7 doi: 10.1016/0021-8693(85)90099-7
[10]	S. Montgomery, Hopf Algebras and Their Actions on Rings, American Mathematical Society, 1993. https://doi.org/10.1090/cbms/082
[11]	B. Drabant, A. Van Daele, Y. Zhang, Actions of multiplier hopf algebras, Commun. Algebra, 27 (1999), 4117–4172. https://doi.org/10.1080/00927879908826688 doi: 10.1080/00927879908826688
[12]	C. Kassel, Quantum Groups, Springer, 1995. https://doi.org/10.1007/978-1-4612-0783-2
[13]	A. Klimyk, K. Schmüdgen, Quantum Groups and Their Representations, Springer, 1997. https://doi.org/10.1007/978-3-642-60896-4
[14]	L. Castellani, J. Wess, Quantum Groups and their Applications in Physics, IOS Press, 1996.
[15]	S. Duplij, S. Sinel'shchikov, Classification of $U_{q}(sl_{2})$-module algebra structures on the quantum plane, J. Math. Phys. Anal. Geom., 6 (2010), 1–25.
[16]	S. Duplij, S. Sinel'shchikov, On $U_{q}(sl_{2})$-actions on the quantum plane, Acta Polytech., 50 (2010), 25–29. http://doi.org/10.14311/1259 doi: 10.14311/1259
[17]	S. Duplij, Y. Hong, F. Li, $U_{q}(sl(m+1))$-module algebra structures on the coordinate algebra of a quantum vector space, J. Lie Theory, 25 (2015), 327–361.
[18]	K. Chan, C. Walton Y. H. Wang, J. J. Zhang, Hopf actions on filtered regular algebras, J. Algebra, 397 (2014), 68–90. https://doi.org/10.1016/j.jalgebra.2013.09.002 doi: 10.1016/j.jalgebra.2013.09.002
[19]	N. Hu, Quantum divided power algebra, $q$-derivatives, and some new quantum groups, J. Algebra, 232 (2000), 2000,507–540. https://doi.org/10.1006/jabr.2000.8385 doi: 10.1006/jabr.2000.8385
[20]	V. A. Artamonov, Actions of pointed Hopf algebras on quantum polynomials, Russ. Math. Surv., 55 (2000), 1137–1138. https://doi.org/10.1070/rm2000v055n06ABEH000337 doi: 10.1070/rm2000v055n06ABEH000337
[21]	K. R. Goodearl, E. S. Letzter, Quantum $n$-space as a quotient of classical $n$-space, Trans. Am. Math. Soc., 352 (2000), 5855–5876. https://doi.org/10.1090/S0002-9947-00-02639-8 doi: 10.1090/S0002-9947-00-02639-8
[22]	J. Alev, M. Chamarie, Derivations et automorphismes de quelques algebras quantiques, Commun. Algebra, 20 (1992), 1787–1802. https://doi.org/10.1080/00927879208824431 doi: 10.1080/00927879208824431
[23]	V. A. Artamonov, Quantum polynomial algebras, J. Math. Sci., 87 (1997), 3441–3462. https://doi.org/10.1007/BF02355445
[24]	V. A. Artamonov, R. Wisbauer, Homological properties of quantum polynomials, Algebras Representation Theory, 4 (2001), 219–247. https://doi.org/10.1023/A:1011458821831 doi: 10.1023/A:1011458821831
[25]	D. Su, Module algebra structures of nonstandard quantum group $X_{q}(A_{1})$ on ${\mathbb{C}}_{q}[x, y, z]$, preprint, arXiv: 2504.19415.

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