Projective class rings of the category of Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra

Yaguo Guo; Shilin Yang; Yaguo Guo; Shilin Yang

doi:10.3934/era.2023256

Electronic Research Archive

2023, Volume 31, Issue 8: 5006-5024. doi: 10.3934/era.2023256

Previous Article Next Article

Research article

Projective class rings of the category of Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra

Yaguo Guo ,
Shilin Yang ^,

Faculty of Science, Beijing University of Technology, Beijing 100124, China

Received: 18 March 2023 Revised: 30 June 2023 Accepted: 05 July 2023 Published: 13 July 2023

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra $ \mathcal{\bar{A}} $ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over $ \mathcal{\bar{A}} $ is described explicitly by generators and relations.
- Yetter-Drinfeld module,
- tensor product,
- the projective class ring
Citation: Yaguo Guo, Shilin Yang. Projective class rings of the category of Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra[J]. Electronic Research Archive, 2023, 31(8): 5006-5024. doi: 10.3934/era.2023256

Related Papers:

Abstract

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra $ \mathcal{\bar{A}} $ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over $ \mathcal{\bar{A}} $ is described explicitly by generators and relations.

References

[1]	N. Andruskiewitsch, H. J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order$p^3$, J. Algebra, 209 (1998), 658–691. http://doi.org/10.1006/jabr.1998.7643 doi: 10.1006/jabr.1998.7643
[2]	D. N. Yetter, Quantum groups and representations of monoidal categories, Math. Proc. Cambridge, 108 (1990), 261–290. http://doi.org/10.1017/S0305004100069139 doi: 10.1017/S0305004100069139
[3]	N. Andruskiewitsch, H. J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math., 154 (2000), 1–45. http://doi.org/10.1006/aima.1999.1880 doi: 10.1006/aima.1999.1880
[4]	N. Andruskiewitsch, H. J. Schneider, Pointed Hopf algebras, in New Directions in Hopf algebras, Cambridge: Cambridge University Press, (2002), 1–68.
[5]	N. Andruskiewitsch, H. J. Schneider, On the classification of finite dimensional pointed Hopf algebras, Ann. Math., 171 (2010), 375–417. http://doi.org/10.4007/annals.2010.171.375 doi: 10.4007/annals.2010.171.375
[6]	N. Andruskiewitsch, G. Carnovale, G. A. García, Finite dimensional pointed Hopf algebras over finite simple groups of Lie type Ⅰ. Non-semisimple classes in $PSL_n(q)$, J. Algebra, 442 (2015), 36–65. http://doi.org/10.1016/j.jalgebra.2014.06.019 doi: 10.1016/j.jalgebra.2014.06.019
[7]	G. A. García, J. M. J. Giraldi, On Hopf algebras over quantum subgroups, J. Pure Appl. Algebra, 223 (2019), 738–768. https://doi.org/10.1016/j.jpaa.2018.04.018 doi: 10.1016/j.jpaa.2018.04.018
[8]	N. Hu, R. Xiong, Some Hopf algebras of dimension 72 without the Chevalley property, preprint, arXiv: 1612.04987.
[9]	Y. Shi, Finite dimensional Nichols algebras over Kac-Paljutkin algebra $H_8$, Rev. Unión Mat. Argent., 60 (2019), 265–298. http://doi.org/10.33044/revuma.v60n1a17 doi: 10.33044/revuma.v60n1a17
[10]	R. Xiong, On Hopf algebras over the unique $12$-dimensional Hopf algebra without the dual Chevalley property, Commun. Algebra, 47 (2019) 1516–1540. http://doi.org/10.1080/00927872.2018.1508582 doi: 10.1080/00927872.2018.1508582
[11]	Y. Zheng, Y. Gao, N. Hu, Finite dimensional Hopf algebras over the Hopf algebra $H_{b:1}$ of Kashina, J. Algebra, 567 (2021), 613–659. http://doi.org/10.1016/j.jalgebra.2020.09.035 doi: 10.1016/j.jalgebra.2020.09.035
[12]	Y. Zheng, Y. Gao, N. Hu, Finite dimensional Hopf algebras over the Hopf algebra $H_{d:-1, 1}$ of Kashina, J. Pure Appl. Algebra, 225 (2021), 106527. https://doi.org/10.1016/j.jpaa.2020.106527 doi: 10.1016/j.jpaa.2020.106527
[13]	D. E. Radford, On oriented quantum algebras derived from representations of the quantum double of a finite dimensional Hopf algebra, J. Algebra, 270 (2003), 670–695. http://doi.org/10.1016/j.jalgebra.2003.07.006 doi: 10.1016/j.jalgebra.2003.07.006
[14]	H. Zhu, H. Chen, Yetter-Drinfeld modules over the Hopf-Ore extension of the group algebra of dihedral group, Acta Math. Sin., 28 (2012), 487–502. http://doi.org/10.1007/s10114-011-9777-4 doi: 10.1007/s10114-011-9777-4
[15]	R. Xiong, Some classification results on finite dimensional Hopf algebras, Ph.D thesis, East China Normal University, 2019.
[16]	Y. Zhang, The Ore extensions of Hopf algebras and their related topics, Ph.D thesis, Beijing University of Technology, 2020.
[17]	J. Chen, S. Yang, D. Wang, Y. Xu, On $4n$-dimension neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras, preprint, arXiv: 1809.00514.
[18]	S. Yang, Y. Zhang, Ore extensions for the Sweedler's Hopf algebra $H_4$, Mathematics, 8 (2020), 1293. http://doi.org/10.3390/math8081293 doi: 10.3390/math8081293
[19]	Y. Guo, S. Yang, The Grothendieck ring of Yetter-Drinfeld modules over a class of $2n^2$-dimension Kac-Paljutkin Hopf algebras, Comm. Algebra, 51 (2023). https://doi.org/10.1080/00927872.2023.2213340 doi: 10.1080/00927872.2023.2213340
[20]	Y. Guo, S. Yang, Projective class rings of a kind of category of Yetter-Drinfeld modules, AIMS Mathematics, 8 (2023), 10997–11014. http://doi.org/10.3934/math.2023557 doi: 10.3934/math.2023557
[21]	Y. Li, N. Hu, The Green rings of the $2$-rank Taft algebra and its two relatives twisted, J. Algebra, 410 (2014), 1–35. http://doi.org/10.1016/j.jalgebra.2014.04.006 doi: 10.1016/j.jalgebra.2014.04.006
[22]	H. X. Chen, A class of noncommutative and noncocommutative Hopf algebras: the quantum version, Comm. Algebra, 27 (1999), 5011–5032. http://doi.org/10.1080/00927879908826745 doi: 10.1080/00927879908826745
[23]	H. X. Chen, H. S. E. Mohammed, W. Lin, H. Sun, The projective class rings of a family of pointed Hopf algebras of rank two, Bull. Belg. Math. Soc. Simon Stevin, 23 (2016), 693–711. http://doi.org/10.36045/bbms/1483671621 doi: 10.36045/bbms/1483671621
[24]	N. Hu, Quantum group structure associated to the quantum affine space, Algebra Colloq., 11 (2004), 483–492.
[25]	G. Feng, N. Hu, Y. Li, Drinfeld doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial, Pac. J. Math., 312 (2021), 421–456. http://doi.org/10.2140/pjm.2021.312.421 doi: 10.2140/pjm.2021.312.421
[26]	S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, 1993. http://doi.org/10.1090/cbms/082
[27]	M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge: Cambridge University Press, 1995. http://doi.org/10.1017/CBO9780511623608
[28]	M. Lorenz, Representations of finite dimensional Hopf algebra, J. Algebra, 188 (1997), 476–505. https://doi.org/10.1006/jabr.1996.6827 doi: 10.1006/jabr.1996.6827

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)