The compact exceptional Lie algebra $\mathfrak g^c_2$ as a twisted ring group
HTML articles powered by AMS MathViewer
- by Cristina Draper;
- Proc. Amer. Math. Soc. 152 (2024), 3679-3688
- DOI: https://doi.org/10.1090/proc/16821
- Published electronically: July 17, 2024
- HTML | PDF | Request permission
Abstract:
A new highly symmetrical model of the compact Lie algebra $\mathfrak {g}^c_2$ is provided as a twisted ring group for the group $\mathbb {Z}_2^3$ and the ring $\mathbb {R}\oplus \mathbb {R}$. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in $\mathfrak {su}(2)$ and of the Gell-Mann matrices in $\mathfrak {su}(3)$. As a bonus, the split Lie algebra $\mathfrak {g}_2$ is also seen as a twisted ring group.References
- Ilka Agricola, Old and new on the exceptional group $G_2$, Notices Amer. Math. Soc. 55 (2008), no. 8, 922–929. MR 2441524
- Helena Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), no. 1, 188–224. MR 1713433, DOI 10.1006/jabr.1998.7850
- Pilar Benito, Cristina Draper, and Alberto Elduque, Models of the octonions and $G_2$, Linear Algebra Appl. 371 (2003), 333–359. MR 1997380, DOI 10.1016/S0024-3795(03)00478-6
- Pilar Benito, Cristina Draper, and Alberto Elduque, Lie-Yamaguti algebras related to ${\mathfrak {g}}_2$, J. Pure Appl. Algebra 202 (2005), no. 1-3, 22–54. MR 2163399, DOI 10.1016/j.jpaa.2005.01.003
- Y. A. Bahturin and M. V. Tvalavadze, Group gradings on $G_2$, Comm. Algebra 37 (2009), no. 3, 885–893. MR 2503183, DOI 10.1080/00927870802278529
- A. J. Calderón Martín, C. Draper, and C. Martín González, Gradings on the real forms of the Albert algebra, of $\mathfrak {g}_2$, and of $\mathfrak {f}_4$, J. Math. Phys. 51 (2010), no. 5, 053516, 21. MR 2666994, DOI 10.1063/1.3326238
- Cristina Draper Fontanals, Notes on $G_2$: the Lie algebra and the Lie group, Differential Geom. Appl. 57 (2018), 23–74. MR 3758361, DOI 10.1016/j.difgeo.2017.10.011
- Cristina Draper and Cándido Martín, Gradings on ${\mathfrak {g}}_2$, Linear Algebra Appl. 418 (2006), no. 1, 85–111. MR 2257580, DOI 10.1016/j.laa.2006.01.017
- C. Draper, T. Meyer, and J. Sánchez-Ortega, Graded contractions of the $\mathbb Z_2^3$-grading on $\mathfrak g_2$, J. Algebra (2024), DOI 10.1016/j.jalgebra.2024.05.049.
- Alberto Elduque, Gradings on octonions, J. Algebra 207 (1998), no. 1, 342–354. MR 1643126, DOI 10.1006/jabr.1998.7474
- Alberto Elduque and Mikhail Kochetov, Gradings on simple Lie algebras, Mathematical Surveys and Monographs, vol. 189, American Mathematical Society, Providence, RI; Atlantic Association for Research in the Mathematical Sciences (AARMS), Halifax, NS, 2013. MR 3087174, DOI 10.1090/surv/189
- Alberto Elduque and Mikhail Kochetov, Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra, J. Algebra 366 (2012), 165–186. MR 2942648, DOI 10.1016/j.jalgebra.2012.05.008
- Alberto Elduque and Adrián Rodrigo-Escudero, Clifford algebras as twisted group algebras and the Arf invariant, Adv. Appl. Clifford Algebr. 28 (2018), no. 2, Paper No. 41, 15. MR 3790094, DOI 10.1007/s00006-018-0862-y
- D. S. Passman, What is a group ring?, Amer. Math. Monthly 83 (1976), no. 3, 173–185. MR 390033, DOI 10.2307/2977018
- D. S. Passman, The Algebraic Structure of Group Rings, Reprint of the 1977 original. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1985. xiv+734 pp. ISBN:0-89874-789-9
- Robert A. Wilson, On the compact real form of the Lie algebra $\mathfrak {g}_2$, Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 1, 87–91. MR 2575374, DOI 10.1017/S030500410999020X
Bibliographic Information
- Cristina Draper
- Affiliation: Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Spain
- MR Author ID: 651907
- ORCID: 0000-0002-2998-7473
- Email: cdf@uma.es
- Received by editor(s): July 22, 2023
- Received by editor(s) in revised form: February 4, 2024, and February 9, 2024
- Published electronically: July 17, 2024
- Additional Notes: This work was supported by the Spanish Ministerio de Ciencia e Innovación through projects PID2019-104236GB-I00/AEI/10.13039/501100011033 and PID2020-118452GB-I00, all of them with FEDER funds, and by the group FQM-336 by Junta de Andalucía
- Communicated by: Sarah Witherspoon
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3679-3688
- MSC (2020): Primary 17B25, 16S35; Secondary 20C05, 17B70
- DOI: https://doi.org/10.1090/proc/16821
- MathSciNet review: 4781965