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Infinite root systems. (Systèmes de racines infinis.) (French) Zbl 0880.17019
The author’s stated goal in this work is to create sets of axioms for root systems that are sufficiently general to include those of Kac-Moody algebras and also those that appear in the generalization of these algebras by Borcherds or in their almost-\(K\)-split forms, and compatible with the axiomatization of infinite root systems due to R. V. Moody and A. Pianzola [Trans. Am. Math. Soc. 315, 661-696 (1989; Zbl 0676.17011)].
The root systems studied in this work are associated with a matrix from a class that is more general than that of the generalized Cartan matrices associated with Kac-Moody algebras. Its diagonal entries can be \(0,1,2\) or negative, and the axioms do not require the entries to be integers, but they must still be rational and satisfy certain other restrictions. Aside from this, the axioms given here for a generating system of roots over \(K\) are quite similar to those of Moody and Pianzola for a set of root data. (Here \(K\) is either a field of characteristic 0 or the ring of integers.)
From the axioms considerable information is derived concerning the structure of these root systems. The invariance of the conditions on these systems under passage to subsystems is verified, and the theorem of the conjugacy of bases is proved. The final chapter is devoted to a study of quotient root systems.
Reviewer: G.Brown (Boulder)

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17B65 Infinite-dimensional Lie (super)algebras
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References:
[1] J. Bausch , Étude et classification des automorphismes d’ordre fini et de première espèce des algèbres de Kac-Moody affines , Revue de l’Institut Elie Cartan 11, Département de Mathématiques de l’Université de Nancy I ( 1989 ). MR 1021053 | Zbl 0698.17016 · Zbl 0698.17016
[2] J. Bausch et G. Rousseau , Involutions de première espèce des algèbres affines , Revue de l’Institut Elie Cartan 11, Département de Mathématiques de l’Université de Nancy I ( 1989 ). MR 1021054 | Zbl 0705.17015 · Zbl 0705.17015
[3] N. Bourbaki , Algèbre commutative chapitre 1, modules plats , Hermann, Paris. Zbl 0108.04002 · Zbl 0108.04002
[4] N. Bourbaki , Groupes et algèbres de Lie , Hermann, Paris. Zbl 0244.22007 · Zbl 0244.22007
[5] R. Borcherds , Generalized Kac-Moody algebras , J. of algebra 115 ( 1989 ), 501-512. MR 89g:17004 | Zbl 0644.17010 · Zbl 0644.17010 · doi:10.1016/0021-8693(88)90275-X
[6] V. Back , N. Bardy , H. Ben-Messaoud et G. Rousseau , Formes presque-déployées d’algèbres de Kac-Moody : Classification et racines relatives , Journal of Algebra 171 ( 1995 ), 43-96. MR 96d:17022 | Zbl 0823.17034 · Zbl 0823.17034 · doi:10.1006/jabr.1995.1004
[7] K. Brown , Buildings , Springer Verlag, Berlin ( 1989 ). MR 90e:20001 | Zbl 0715.20017 · Zbl 0715.20017
[8] F. Bruhat et J. TITS , Groupes réductifs sur un corps local, n^\circ 1 et n^\circ 2 , Publ. Math. I.H.E.S., 41 ( 1972 ), 5-252 ; 60 ( 1984 ), 5-184. Numdam | MR 48 #6265 | Zbl 0254.14017 · Zbl 0254.14017 · doi:10.1007/BF02715544 · numdam:PMIHES_1972__41__5_0 · eudml:103918
[9] J.-Y. Hée , Systèmes de racines sur un anneau commutatif totalement ordonné , Geometriae Dedicata 37 ( 1991 ), 65-102. MR 92f:17006 | Zbl 0721.17019 · Zbl 0721.17019 · doi:10.1007/BF00150405
[10] J.-Y. Hée , Torsion de groupes munis d’une donnée radicielle , Thèse d’Etat, Orsay ( 1993 ).
[11] V. G. Kac , Infinite dimensional Lie algebras , troisième édition, Cambridge University Press ( 1990 ). MR 92k:17038 | Zbl 0716.17022 · Zbl 0716.17022
[12] R. V. Moody et A. Pianzola , On infinite root systems , Trans. of the A.M.S, 315 n^\circ 2 ( 1989 ) 661-696. MR 90a:17017 | Zbl 0676.17011 · Zbl 0676.17011 · doi:10.2307/2001300
[13] D. H. Peterson et V. G. KAC , Infinite flag varieties and conjugacy theorems , Proc. Natl. Acad. Sci. USA, 80 ( 1983 ) 1778-1782. MR 84g:17017 | Zbl 0512.17008 · Zbl 0512.17008 · doi:10.1073/pnas.80.6.1778
[14] J. Ramagge , A realisation of certain KM groups of types II and III , Journal of Algebra 171 ( 1995 ) 713-806. MR 96f:22020b | Zbl 0841.22015 · Zbl 0841.22015 · doi:10.1006/jabr.1995.1036
[15] G. Rousseau , Espaces affines symétriques et algèbres affines , Revue de l’Institut Elie Cartan 11, Département de Mathématiques de l’Université de Nancy I ( 1989 ). MR 1021055 · Zbl 0705.17017
[16] G. Rousseau , L’immeuble jumelé d’une forme presque déployée d’une algèbre de Kac-Moody , Bull. Soc. Math. Belg. 42 ( 1990 ) 673-694. MR 95m:20031 | Zbl 0737.17010 · Zbl 0737.17010
[17] H. Rubenthaler , Construction de certaines algèbres remarquables dans les algèbres de Lie semi-simples , Journal of Algebra 81 ( 1983 ) 268-278. MR 84g:17010 | Zbl 0527.17003 · Zbl 0527.17003 · doi:10.1016/0021-8693(83)90221-1
[18] P. Slodowy , Beyond Kac-Moody algebras and inside , Can. Math. Soc. Conf. Proc. 5 ( 1986 ) 361-371. MR 832211 | Zbl 0582.17011 · Zbl 0582.17011
[19] J. Tits , Classification of algebraic groups and discontinuous subgroups , Boulder 1965 , Proc. of Symposia in pure math. n^\circ 9 ( 1966 ), 33-62. Zbl 0238.20052 · Zbl 0238.20052
[20] Z. B. Vinberg , Discrete linear groups generated by reflections , Math. USSR-Izvestija 5 ( 1971 ), 1083-1119. Zbl 0256.20067 · Zbl 0256.20067 · doi:10.1070/IM1971v005n05ABEH001203
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