Bicrossed products of generalized Taft algebra and group algebras

Dingguo Wang; Xiangdong Cheng; Daowei Lu

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 801-816
  • ISSN: 0011-4642

Abstract

top
Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}(q)\bowtie k[G]$ between the generalized Taft algebra $H_{m,d}(q)$ and group algebra $k[G]$ is actually the smash product $H_{m,d}(q)\sharp k[G]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}(q)\bowtie k[ C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

How to cite

top

Wang, Dingguo, Cheng, Xiangdong, and Lu, Daowei. "Bicrossed products of generalized Taft algebra and group algebras." Czechoslovak Mathematical Journal 72.3 (2022): 801-816. <http://eudml.org/doc/298349>.

@article{Wang2022,
abstract = {Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_\{m,d\}(q)\bowtie k[G]$ between the generalized Taft algebra $H_\{m,d\}(q)$ and group algebra $k[G]$ is actually the smash product $H_\{m,d\}(q)\sharp k[G]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_\{m,d\}(q)\bowtie k[ C_\{n_1\}\times C_\{n_2\}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.},
author = {Wang, Dingguo, Cheng, Xiangdong, Lu, Daowei},
journal = {Czechoslovak Mathematical Journal},
language = {eng},
number = {3},
pages = {801-816},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bicrossed products of generalized Taft algebra and group algebras},
url = {http://eudml.org/doc/298349},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Wang, Dingguo
AU - Cheng, Xiangdong
AU - Lu, Daowei
TI - Bicrossed products of generalized Taft algebra and group algebras
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 801
EP - 816
AB - Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}(q)\bowtie k[G]$ between the generalized Taft algebra $H_{m,d}(q)$ and group algebra $k[G]$ is actually the smash product $H_{m,d}(q)\sharp k[G]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}(q)\bowtie k[ C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.
LA - eng
UR - http://eudml.org/doc/298349
ER -

References

top
  1. Agore, A. L., 10.1016/j.jpaa.2017.05.014, J. Pure Appl. Algebra 222 (2018), 914-930. (2018) Zbl1416.16033MR3720860DOI10.1016/j.jpaa.2017.05.014
  2. Agore, A. L., 10.3842/SIGMA.2018.027, SIGMA, Symmetry Integrability Geom. Methods Appl. 14 (2018), Article ID 027, 14 pages. (2018) Zbl1414.16027MR3778923DOI10.3842/SIGMA.2018.027
  3. Agore, A. L., Bontea, C. G., Militaru, G., 10.1007/s10468-012-9396-5, Algebr. Represent. Theory 17 (2014), 227-264. (2014) Zbl1351.16031MR3160722DOI10.1007/s10468-012-9396-5
  4. Agore, A. L., Chirvăsitu, A., Ion, B., Militaru, G., 10.1007/s10468-009-9145-6, Algebr. Represent. Theory 12 (2009), 481-488. (2009) Zbl1187.20023MR2501197DOI10.1007/s10468-009-9145-6
  5. Agore, A. L., Militaru, G., 10.1016/j.jalgebra.2013.06.012, J. Algebra 391 (2013), 193-208. (2013) Zbl1293.16026MR3081628DOI10.1016/j.jalgebra.2013.06.012
  6. Agore, A. L., Năstăsescu, L., 10.1007/s00013-019-01328-3, Arch. Math. 113 (2019), 21-36. (2019) Zbl1447.16025MR3960780DOI10.1007/s00013-019-01328-3
  7. Aguiar, M., Andruskiewitsch, N., 10.1090/conm/376, Algebraic Structures and Their Representations Contemporary Mathematics 376 (2005), 127-173. (2005) Zbl1100.16032MR2147019DOI10.1090/conm/376
  8. Bontea, C. G., 10.1007/s10587-014-0109-6, Czech. Math. J. 64 (2014), 419-431. (2014) Zbl1322.16022MR3277744DOI10.1007/s10587-014-0109-6
  9. Brzeziński, T., 10.1081/AGB-100001537, Commun. Algebra 29 (2001), 737-748. (2001) Zbl1003.16024MR1841995DOI10.1081/AGB-100001537
  10. Caenepeel, S., Ion, B., Militaru, G., Zhu, S., 10.1023/A:1009917210863, Algebr. Represent. Theory 3 (2000), 19-42. (2000) Zbl0957.16027MR1755802DOI10.1023/A:1009917210863
  11. Chen, X.-W., Huang, H.-L., Ye, Y., Zhang, P., 10.1016/j.jalgebra.2003.12.019, J. Algebra 275 (2004), 212-232. (2004) Zbl1071.16030MR2047446DOI10.1016/j.jalgebra.2003.12.019
  12. Cibils, C., 10.1007/BF02096879, Commun. Math. Phys. 157 (1993), 459-477. (1993) Zbl0806.16039MR1243707DOI10.1007/BF02096879
  13. Huang, H., Chen, H., Zhang, P., Generalized Taft algebras, Algebra Colloq. 11 (2004), 313-320. (2004) Zbl1079.16026MR2081190
  14. Keilberg, M., 10.1007/s10468-015-9540-0, Algebr. Represent. Theory 18 (2015), 1267-1297. (2015) Zbl1354.16042MR3422470DOI10.1007/s10468-015-9540-0
  15. Keilberg, M., 10.1080/00927872.2018.1461883, Commun. Algebra 46 (2018), 5146-5178. (2018) Zbl1414.16028MR3923748DOI10.1080/00927872.2018.1461883
  16. Lu, D., Ning, Y., Wang, D., 10.21136/CMJ.2020.0079-19, Czech. Math. J. 70 (2020), 959-977. (2020) Zbl07285973MR4181790DOI10.21136/CMJ.2020.0079-19
  17. Maillet, E., 10.24033/bsmf.617, Bull. Soc. Math. Fr. 28 (1900), 7-16 French \99999JFM99999 31.0144.02. (1900) MR1504357DOI10.24033/bsmf.617
  18. Majid, S., 10.1016/0021-8693(90)90099-A, J. Algebra 130 (1990), 17-64. (1990) Zbl0694.16008MR1045735DOI10.1016/0021-8693(90)90099-A
  19. Majid, S., 10.1017/CBO9780511613104, Cambridge University Press, Cambridge (1995). (1995) Zbl0857.17009MR1381692DOI10.1017/CBO9780511613104
  20. Michor, P. W., Knit product of graded Lie algebras and groups, Rend. Circ. Mat. Palermo (2) Suppl. 22 (1990), 171-175. (1990) Zbl0954.17508MR1061798
  21. Radford, D. E., 10.1090/S0002-9939-1975-0396652-0, Proc. Am. Math. Soc. 53 (1975), 9-15. (1975) Zbl0324.16009MR0396652DOI10.1090/S0002-9939-1975-0396652-0
  22. Taft, E. J., 10.1073/pnas.68.11.2631, Proc. Natl. Acad. Sci. USA 68 (1971), 2631-2633. (1971) Zbl0222.16012MR0286868DOI10.1073/pnas.68.11.2631
  23. Takeuchi, M., 10.1080/00927878108822621, Commun. Algebra 9 (1981), 841-882. (1981) Zbl0456.16011MR0611561DOI10.1080/00927878108822621
  24. Zappa, G., Sulla costruzione dei gruppi prodotto di dati sottogruppi permutabili tra loro, Atti 2. Congr. Un. Mat. Ital., Bologna 1942 (1942), 119-125 Italian. (1942) Zbl0026.29104MR0019090

NotesEmbed ?

top

You must be logged in to post comments.