Representation rings of extensions of Hopf algebra of Kac-Paljutkin type

Dong Su; Shilin Yang; Dong Su; Shilin Yang

doi:10.3934/era.2024240

Electronic Research Archive

2024, Volume 32, Issue 9: 5201-5230. doi: 10.3934/era.2024240

Previous Article Next Article

Research article

Representation rings of extensions of Hopf algebra of Kac-Paljutkin type

Dong Su ¹,
Shilin Yang ^{2
,
,}

1.
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
2.
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China

Received: 23 April 2024 Revised: 19 July 2024 Accepted: 29 August 2024 Published: 05 September 2024

In this paper, we focus on studying two classes of finite dimensional $ \Delta $-associative algebras, which are extensions of a family of $ 2n^2 $-dimensional Kac-Paljutkin type semi-simple Hopf algebras $ H_{2n^2} $. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.
- $ \Delta $-associative algebra,
- tensor product,
- representation ring
Citation: Dong Su, Shilin Yang. Representation rings of extensions of Hopf algebra of Kac-Paljutkin type[J]. Electronic Research Archive, 2024, 32(9): 5201-5230. doi: 10.3934/era.2024240

Related Papers:

Abstract

In this paper, we focus on studying two classes of finite dimensional $ \Delta $-associative algebras, which are extensions of a family of $ 2n^2 $-dimensional Kac-Paljutkin type semi-simple Hopf algebras $ H_{2n^2} $. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.

References

[1]	H. Chen, F. V. Oystaeyen, Y. Zhang, The Green rings of Taft algebras, Proc. Amer. Math. Soc., 142 (2014), 765–775. https://doi.org/10.1090/S0002-9939-2013-11823-X doi: 10.1090/S0002-9939-2013-11823-X
[2]	Z. Wang, L. Li, Y. Zhang, Green rings of pointed rank one Hopf algebras of nilpotent type, Algebr. Represent. Theory, 17 (2014), 1901–1924. https://doi.org/10.1007/s10468-014-9484-9 doi: 10.1007/s10468-014-9484-9
[3]	Z. Wang, L. Li, Y. Zhang, Green rings of pointed rank one Hopf algebras of non-nilpotent type, J. Algebra, 449 (2016), 108–137. https://doi.org/10.1016/j.jalgebra.2015.11.002 doi: 10.1016/j.jalgebra.2015.11.002
[4]	R. Suter, Modules over $\mathfrak{U}_{q}(sl_{2})$, Commun. Math. Phys., 163 (1994), 359–393. https://doi.org/10.1007/bf02102012 doi: 10.1007/bf02102012
[5]	H. Kondo, Y. Saito, Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to $sl_{2}$, J. Algebra, 330 (2011), 103–129. https://doi.org/10.1016/j.jalgebra.2011.01.010 doi: 10.1016/j.jalgebra.2011.01.010
[6]	H. Chen, Finite-dimensional representations of a quantum double, J. Algebra, 251 (2002), 751–789. https://doi.org/10.1006/jabr.2002.9144 doi: 10.1006/jabr.2002.9144
[7]	H. Chen, H. S. E. Mohammed, H. Sun, Indecomposable decomposition of tensor products of modules over Drinfeld doubles of Taft algebras, J. Pure Appl. Algebra, 221 (2017), 2752–2790. https://doi.org/10.1016/j.jpaa.2017.01.007 doi: 10.1016/j.jpaa.2017.01.007
[8]	D. Su, S. Yang, Representation rings of small quantum groups $\overline{U}_{q}(sl_{2})$, J. Math. Phys., 58 (2017), 091704. https://doi.org/10.1080/00927872.2019.1612412 doi: 10.1080/00927872.2019.1612412
[9]	H. Chen, The Green ring of Drinfeld double $D(H_{4})$, Proc. Am. Math. Soc., 142 (2014), 765–775. https://doi.org/10.1007/s10468-013-9456-5 doi: 10.1007/s10468-013-9456-5
[10]	H. Sun, H. S. E. Mohammed, W. Lin, H. Chen, Green rings of Drinfeld doubles of Taft algebras, Commun. Algebra, 48 (2020), 3933–3947. https://doi.org/10.1080/00927872.2020.1752225 doi: 10.1080/00927872.2020.1752225
[11]	R. Yang, S. Yang, The Grothendieck rings of Wu-Liu-Ding algebras and their Casimir numbers (Ⅰ), J. Algebra Appl., 21 (2022), 2250178. https://doi.org/10.1142/S021949882250178X doi: 10.1142/S021949882250178X
[12]	R. Yang, S. Yang, The Grothendieck rings of Wu-Liu-Ding algebras and their Casimir numbers (Ⅱ), Commun. Algebra, 49 (2021), 2041–2073. https://doi.org/10.1080/00927872.2020.1862139 doi: 10.1080/00927872.2020.1862139
[13]	J. Wu, Note on the coradical filtration of $D(m, d, \varepsilon)$, Commun. Algebra, 44 (2016), 4844–4850. https://doi.org/10.1080/00927872.2015.1113295 doi: 10.1080/00927872.2015.1113295
[14]	J. Wu, G. Liu, N. Ding, Classification of affine prime regular Hopf algebras of GK-dimension one, Adv. Math., 296 (2016), 1–54. https://doi.org/10.1016/j.aim.2016.03.037 doi: 10.1016/j.aim.2016.03.037
[15]	J. Chen, S. Yang, D. Wang, Grothendieck rings of a class of Hopf algebras of Kac-Paljutkin type, Front. Math. China, 16 (2021), 29–47. https://doi.org/10.1007/s11464-021-0893-x doi: 10.1007/s11464-021-0893-x
[16]	Y. Guo, S. Yang, The Grothendieck ring of Yetter-Drinfeld modules over a class of $2n^{2}$-dimensional Kac-Paljutkin Hopf algebras, Commun. Algebra, 51 (2023), 4517–4566. https://doi.org/10.1080/00927872.2023.2213340 doi: 10.1080/00927872.2023.2213340
[17]	H. Sun, H. Chen, Y. Zhang, Representations of the small quasi-quantum group, preprint, arXiv: 2305.06083. https://doi.org/10.48550/arXiv.2305.06083
[18]	C. Cibils, The projective class ring of basic and split Hopf algebras, K-Theory, 17 (1999), 385–393. https://doi.org/10.1023/A:1007703709638 doi: 10.1023/A:1007703709638
[19]	M. Herschend, On the representation ring of a quiver, Algebra Represent. Theory, 12 (2009), 513–541. https://doi.org/10.1007/s10468-008-9118-1 doi: 10.1007/s10468-008-9118-1
[20]	Y. Li, N. Hu, The Green rings of the $2$-rank Taft algebras and its two relatives twisted, J. Algebra, 410 (2014), 1–35. https://doi.org/10.1016/j.jalgebra.2014.04.006 doi: 10.1016/j.jalgebra.2014.04.006
[21]	L. Li, Y. Zhang, The Green rings of the Generalized Taft algebras, Contemp. Math, 585 (2013), 275–288. https://doi.org/10.1090/conm/585/11618 doi: 10.1090/conm/585/11618
[22]	D. Su, S. Yang, Green rings of weak Hopf algebras based on generalized Taft algebra, Period. Math. Hung., 76 (2018), 229–242. https://doi.org/10.1007/s10998-017-0221-0 doi: 10.1007/s10998-017-0221-0
[23]	Z. Wang, L. You, H. Chen, Representations of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one, Algebra Represent. Theory, 18 (2015), 801–830. https://doi.org/10.1007/s10468-015-9517-z doi: 10.1007/s10468-015-9517-z
[24]	Q. Xie, J. Sun, Non-weight modules over $N = 1$ Lie superalgebras of Block type, Forum Math., 35 (2023), 1279–1300. https://doi.org/10.1515/forum-2022-0267 doi: 10.1515/forum-2022-0267
[25]	R. Yang, S. Yang, Representations of a non-pointed Hopf algebra, AIMS Math., 6 (2021), 10523–10539. https://doi.org/10.3934/math.2021611 doi: 10.3934/math.2021611
[26]	D. Su, S. Yang, Green rings of $\Delta$-associative algebras, J. Beijing Univ. Technol., 43 (2017), 1275–1282. https://doi.org/10.11936/bjutxb2016070014 doi: 10.11936/bjutxb2016070014
[27]	D. Pansera, A class of semisimple Hopf algebras acting on quantum polynomial algebras, in Rings, Modules and Codes, 727 (2019), 303–316. https://doi.org/10.1090/conm/727/14643
[28]	F. Li, Weak Hopf algebras and some new solutions of the quantum Yang-Baxter equation, J. Algebra, 208 (1998), 72–100. https://doi.org/10.1006/jabr.1998.7491 doi: 10.1006/jabr.1998.7491
[29]	M. E. Sweedler, Hopf Algebras, W. A. Benjamin, New York, 1969.
[30]	M. Auslander, I. Reiten, S. O. Smal$\phi$, Representation Theory of Artin Algebras, University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511623608
[31]	I. Assem, D. Simson, A. Skowroński, Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, Cambridge University Press, Cambridge, 2006. https://doi.org/10.1017/CBO97805116192 12
[32]	T. Li, Q. Wang, Structure preserving quaternion full orthogonalization method with applications, Numer. Linear Algebra Appl., 30 (2023), e2495. https://doi.org/10.1002/nla.2495 doi: 10.1002/nla.2495
[33]	T. Li, Q. Wang, Structure preserving quaternion biconjugate gradient method, SIAM J. Matrix Anal. Appl., 45 (2024), 306–326. https://doi.org/10.1137/23M1547299 doi: 10.1137/23M1547299
[34]	X. Zhang, W. Ding, T. Li. Tensor form of GPBiCG algorithm for solving the generalized Sylvester quaternion tensor equations, J. Franklin Inst., 360 (2023), 5929–5946. https://doi.org/10.1016/j.jfranklin.2023.04.009 doi: 10.1016/j.jfranklin.2023.04.009

Reader Comments

Your name:*

Email:*
© 2024 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)