Lie algebras with prescribed $\mathfrak {sl}_3$ decomposition
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- by Georgia Benkart and Alberto Elduque PDF
- Proc. Amer. Math. Soc. 140 (2012), 2627-2638 Request permission
Abstract:
In this work, we consider Lie algebras $\mathcal {L}$ containing a subalgebra isomorphic to $\mathfrak {sl}_3$ and such that $\mathcal {L}$ decomposes as a module for that $\mathfrak {sl}_3$ subalgebra into copies of the adjoint module, the natural three-dimensional module and its dual, and the trivial one-dimensional module. We determine the multiplication in $\mathcal {L}$ and establish connections with structurable algebras by exploiting symmetry relative to the symmetric group $\mathsf {S}_4$.References
- B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), no. 2, 133–156. MR 507909, DOI 10.1007/BF01351677
- B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, J. Algebra 161 (1993), no. 1, 1–19. MR 1245840, DOI 10.1006/jabr.1993.1202
- Yuri Bahturin and Georgia Benkart, Some constructions in the theory of locally finite simple Lie algebras, J. Lie Theory 14 (2004), no. 1, 243–270. MR 2040179
- Georgia Benkart and Oleg Smirnov, Lie algebras graded by the root system $\rm BC_1$, J. Lie Theory 13 (2003), no. 1, 91–132. MR 1958577
- S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), no. 2, 323–347. MR 1161095, DOI 10.1007/BF02100608
- Alberto Elduque and Susumu Okubo, Lie algebras with $S_4$-action and structurable algebras, J. Algebra 307 (2007), no. 2, 864–890. MR 2275376, DOI 10.1016/j.jalgebra.2006.08.033
- Alberto Elduque and Susumu Okubo, $S_4$-symmetry on the Tits construction of exceptional Lie algebras and superalgebras, Publ. Mat. 52 (2008), no. 2, 315–346. MR 2436728, DOI 10.5565/PUBLMAT_{5}2208_{0}4
- John R. Faulkner, Structurable superalgebras of classical type, Comm. Algebra 38 (2010), no. 9, 3268–3310. MR 2724219, DOI 10.1080/00927870903200885
- N. Jacobson, Exceptional Lie algebras, Lecture Notes in Pure and Applied Mathematics, vol. 1, Marcel Dekker, Inc., New York, 1971. MR 0284482
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
Additional Information
- Georgia Benkart
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 34650
- Email: benkart@math.wisc.edu
- Alberto Elduque
- Affiliation: Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain
- MR Author ID: 208418
- Email: elduque@unizar.es
- Received by editor(s): January 3, 2001
- Received by editor(s) in revised form: March 7, 2011
- Published electronically: December 9, 2011
- Additional Notes: Part of this work was done during a visit of the first author to the University of Zaragoza, supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2007-67884-C04-02).
The second author was supported by the Spanish Ministerios de Educación y Ciencia and Ciencia e Innovación and FEDER (MTM 2007-67884-C04-02 and MTM2010-18370-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de Álgebra) - Communicated by: Gail R. Letzter
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2627-2638
- MSC (2010): Primary 17B60; Secondary 17A30
- DOI: https://doi.org/10.1090/S0002-9939-2011-11120-1
- MathSciNet review: 2910750
Dedicated: In memory of Hyo Chul Myung