\(\mathfrak{sl}_2\) triples whose nilpositive elements are in a space which is spanned by the real root vectors in rank 2 symmetric hyperbolic Kac-Moody Lie algebras. (English) Zbl 07926397
Summary: In analogy with the theory of nilpotent orbits in finite-dimensional semisimple Lie algebras, it is known that the principal \(\mathfrak{sl}_2\) subalgebras can be constructed in hyperbolic Kac-Moody Lie algebras. We obtained a series of \(\mathfrak{sl}_2\) subalgebras in rank 2 symmetric hyperbolic Kac-Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss \(\mathfrak{sl}_2\) modules obtained by the action of the \(\mathfrak{sl}_2\) subalgebras on the original Lie algebras.
MSC:
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
References:
| [1] | D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold, New York, 1993. Zbl 0972.17008 MR 1251060 · Zbl 0972.17008 |
| [2] | E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, in American Math-ematical Society Translations II Series 6, American Mathematical Society, 1957, 111-245. · Zbl 0077.03404 · doi:10.1090/trans2/006/02 |
| [3] | M. R. Gaberdiel, D. I. Olive, and P. C. West, A class of Lorentzian Kac-Moody algebras, Nuclear Phys. B 645 (2002), 403-437. Zbl 0999.17033 MR 1938316 · Zbl 0999.17033 · doi:10.1016/S0550-3213(02)00690-9 |
| [4] | R. Howe and E.-C. Tan, Nonabelian harmonic analysis, Universitext, Springer, New York, 1992. Zbl 0768.43001 MR 1151617 · Zbl 0768.43001 · doi:10.1007/978-1-4613-9200-2 |
| [5] | V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. Zbl 0716.17022 MR 1104219 · Zbl 0716.17022 · doi:10.1017/CBO9780511626234 |
| [6] | S.-J. Kang and D. J. Melville, Rank 2 symmetric hyperbolic Kac-Moody algebras, Nagoya Math. J. 140 (1995), 41-75. Zbl 0846.17025 MR 1369479 · Zbl 0846.17025 · doi:10.1017/S0027763000005419 |
| [7] | B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032. Zbl 0099.25603 MR 0114875 · Zbl 0099.25603 · doi:10.2307/2372999 |
| [8] | H. Nicolai and D. I. Olive, The principal SO(1, 2) subalgebra of a hyperbolic Kac-Moody algebra, Lett. Math. Phys. 58 (2001), 141-152. Zbl 1012.17017 MR 1876250 · Zbl 1012.17017 · doi:10.1023/A:1013389001951 |
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