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Representations of involutory subalgebras of affine Kac-Moody algebras. (English) Zbl 1514.17029

Summary: We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Cartan-Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of \(\mathbb{N}\)-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity and work out a detailed example in the case of the affine extension of \(\mathfrak{e}_8\).

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

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[1] Abramenko, P.; Mühlherr, B., Présentations de certaines \(BN\)-paires jumelées comme sommes amalgamées, C. R. Acad. Sci. Paris Sér. I Math., 325, 701-706 (1997) · Zbl 0934.20024 · doi:10.1016/S0764-4442(97)80044-4
[2] Berman, S., On generators and relations for certain involutory subalgebras of Kac-Moody Lie algebras, Commun. Algebra, 17, 3165-3185 (1989) · Zbl 0693.17012 · doi:10.1080/00927878908823899
[3] Carbone, L.; Feingold, AJ; Freyn Walter, W., A lightcone embedding of the twin building of a hyperbolic Kac-Moody group, SIGMA, 16, 045 (2020) · Zbl 1486.20068
[4] Damour, T.; Hillmann, C., Fermionic Kac-Moody billiards and supergravity, JHEP, 08, 100 (2009) · doi:10.1088/1126-6708/2009/08/100
[5] Damour, T.; Kleinschmidt, A.; Nicolai, H., Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B, 634, 319-324 (2006) · Zbl 1247.83225 · doi:10.1016/j.physletb.2006.01.015
[6] Damour, T.; Kleinschmidt, A.; Nicolai, H., \(K(E_{10})\), supergravity and fermions, JHEP, 08, 046 (2006) · doi:10.1088/1126-6708/2006/08/046
[7] de Buyl, S.; Henneaux, M.; Paulot, L., Hidden symmetries and Dirac fermions, Class. Quantum Gravity, 22, 3595-3622 (2005) · Zbl 1153.83371 · doi:10.1088/0264-9381/22/17/018
[8] de Buyl, S.; Henneaux, M.; Paulot, L., Extended \(E_8\) invariance of 11-dimensional supergravity, JHEP, 02, 056 (2006) · doi:10.1088/1126-6708/2006/02/056
[9] Duff, MJ; Liu, JT, Hidden space-time symmetries and generalized holonomy in M theory, Nucl. Phys. B, 674, 217-230 (2003) · Zbl 1097.81713 · doi:10.1016/j.nuclphysb.2003.09.019
[10] Duff, MJ; Stelle, KS, Multimembrane solutions of \(D{=}11\) supergravity, Phys. Lett. B, 253, 113-118 (1991) · doi:10.1016/0370-2693(91)91371-2
[11] Ghatei, D.; Horn, M.; Köhl, R.; Weiß, S., Spin covers of maximal compact subgroups of Kac-Moody groups and spin-extended Weyl groups, J. Group Theory, 20, 401-504 (2017) · Zbl 1390.20048 · doi:10.1515/jgth-2016-0034
[12] Goddard, P.; Olive, DI, Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys. A, 1, 303 (1986) · Zbl 0631.17012 · doi:10.1142/S0217751X86000149
[13] Hainke, G., Köhl, R., Levy, P.: “Generalized spin representations,” with an appendix by M. Horn and R. Köhl, Münster. J. Math. 8, 181-210 (2015) doi:10.17879/65219674985. arXiv:1403.4463 · Zbl 1402.17032
[14] Harring, P., Köhl, R.: “Fundamental groups of split real Kac-Moody groups and generalized real flag manifolds,” with appendices by T. Hartnick and R. Köhl and by J. Grüning and R. Köhl. Accepted for publication in Transf. Groups. arXiv:1905.13444
[15] Hilgert, J.; Neeb, K-H, Structure and Geometry of Lie Groups (2012), New York: Springer, New York · Zbl 1229.22008 · doi:10.1007/978-0-387-84794-8
[16] Hull, C.: Holonomy and symmetry in M theory. arXiv:hep-th/0305039
[17] Julia, B.; Nicolai, H., Conformal internal symmetry of 2d sigma models coupled to gravity and a Dilaton, Nucl. Phys. B, 482, 431 (1996) · Zbl 0925.81312 · doi:10.1016/S0550-3213(96)00551-2
[18] Kac, VG, Infinite Dimensional Lie Algebras (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0716.17022 · doi:10.1007/978-1-4757-1382-4
[19] Kac, V.G., Peterson, D.H.: Defining relations of certain infinite-dimensional groups. In: The Mathematical Heritage of Élie Cartan (Lyon, 1984). Astérisque, Numéro Hors Série, 165-208 (1985) · Zbl 0625.22014
[20] Kleinschmidt, A.; Nicolai, H., Gradient representations and affine structures in AE(n), Class. Quantum Gravity, 22, 4457-4488 (2005) · Zbl 1086.83032 · doi:10.1088/0264-9381/22/21/004
[21] Kleinschmidt, A.; Nicolai, H., IIA and IIB spinors from \(K(E_{10})\), Phys. Lett. B, 637, 107-112 (2006) · Zbl 1247.81176 · doi:10.1016/j.physletb.2006.04.007
[22] Kleinschmidt, A.; Nicolai, H., On higher spin realizations of \(K(E_{10})\), JHEP, 08, 041 (2013) · Zbl 1342.81191 · doi:10.1007/JHEP08(2013)041
[23] Kleinschmidt, A.; Nicolai, H., Standard model fermions and \(K(E_{10})\), Phys. Lett. B, 747, 251-254 (2015) · Zbl 1369.81118 · doi:10.1016/j.physletb.2015.06.005
[24] Kleinschmidt, A., Nicolai, H.: Higher spin representations of \(K(E_{10})\). In: Brink, L., Henneaux, M., Vasiliev, M. (eds.) Higher Spin Gauge Theories, pp. 25-38. World Scientific (2017).doi:10.1142/9789813144101_0003. arXiv:1602.04116 · Zbl 1358.81115
[25] Kleinschmidt, A.; Nicolai, H.; Palmkvist, J., \(K(E_9)\) from \(K(E_{10})\), JHEP, 06, 051 (2007) · doi:10.1088/1126-6708/2007/06/051
[26] Kleinschmidt, A., Nicolai, H., Viganò, A.: On spinorial representations of involutory subalgebras of Kac-Moody algebras. In: Gritsenko, V., Spiridonov, V. (eds.) Partition Functions and Automorphic Forms. Moscow Lectures, vol 5. Springer, Cham (2020). doi:10.1007/978-3-030-42400-8_4. arXiv:1811.11659 · Zbl 1454.81197
[27] Lautenbacher, R., Köhl, R.: Extending generalized spin representations. J. Lie Theory 28, 915-940 (2018). https://www.heldermann.de/JLT/JLT28/JLT284/jlt28045.htm. arXiv:1705.00118 · Zbl 1440.17018
[28] Marquis, T.: An introduction to Kac-Moody groups over fields10.4171/187. In: EMS Textbooks in Mathematics. European Mathematical Society (2018) · Zbl 1405.20003
[29] Meissner, KA; Nicolai, H., Standard model fermions and N=8 supergravity, Phys. Rev. D, 91, 065029 (2015) · doi:10.1103/PhysRevD.91.065029
[30] Meissner, KA; Nicolai, H., Standard model fermions and infinite-dimensional R-symmetries, Phys. Rev. Lett., 121, 091601 (2018) · doi:10.1103/PhysRevLett.121.091601
[31] Nicolai, H., Two-dimensional gravities and supergravities as integrable system, Lect. Notes Phys., 396, 231-273 (1991) · doi:10.1007/3-540-54978-1_12
[32] Nicolai, H.; Samtleben, H., On \(K(E_9)\), Q. J. Pure Appl. Math., 1, 180-204 (2005) · Zbl 1098.22010 · doi:10.4310/PAMQ.2005.v1.n1.a8
[33] Pressley, A.; Segal, G., Loop Groups (1986), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0618.22011
[34] Rotman, JJ, An Introduction to Homological Algebra (2009), Berlin: Springer, Berlin · Zbl 1157.18001 · doi:10.1007/b98977
[35] Slodowy, P., Singularitäten: Kac-Moody-Liealgebren, Assoziierte Gruppen und Verallgemeinerungen (1984), Bonn: Habilitationsschrift Universität, Bonn
[36] Tits, J., Buildings of Spherical Type and Finite BN-Pairs (1974), Berlin-New York: Springer, Berlin-New York · Zbl 0295.20047
[37] van Leeuwen, M.A.A., Cohen, A.M., Lisser, B.: LiE, A Package for Lie Group Computations. Available from http://wwwmathlabo.univ-poitiers.fr/ maavl/LiE/
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