HYPERBOLIC KAC-MOODY ALGEBRAS Gr. Tsagas and X. Agra¯otou Abstract The classi¯cation of solvable Lie algebras is still a hopeless problem. In 1945, Malcev reduced the solvable case to the nilpotent one and in [7] to each nilpotent Lie algebra n of ¯nite dimension, a Kac-Moody algebra g(A) has been associated (n is of type A). The aim of this paper is to connect the classi¯cation of nilpotent Lie algebra of maximal rank and of Hyperbolic type H2 under the methode of Kac-Moody algebras. We will follow very closely Kac's book [4]. AMS Subject Classi¯cation: 17B30. Key words: nilpotent, Kac-Moody algebras, hyperbolic algebras, root spaces. 1 Kac-Moody algebras. De¯nition 1.1 Let A(aij) 1 · i; j · l be a square matrix with entries in Z with the following properties: (i) aii = 2, (ii) aij · 0 if i =6 j, (iii) if aij = 0, then aji = 0, i =6 j for i; j = 1; :::; l. Then the square matrix A is called generalized Cartan matrix (denoted by g.C.m.). De¯nition 1.2 A generalized Cartan matrix A is called a matrix of hyperbolic type if it is indecomposable, symmetrizable of inde¯nite type and if every proper connected subdiagram of S(A) is of ¯nite or a±ne type. De¯nition 1.3 Let L be a Lie algebra with 3l generators xi, hi, yi (i = 1; : : : ; l) and A be a g.C.m. associated to the Lie algebras L, satisfying the relations: Editor Gr.Tsagas Proceedings of the Workshop on Global Analysis, Di®erential Geometry and Lie Algebras, 1996, 104-107 °c 1998 Balkan Society of Geometers, Geometry Balkan Press Hyperbolic Kac-Moody Algebras 105 (i) [hi; hj] = 0, [xi; yj] = ±ijhi; (ii) [hi; xj] = aijxj,[hi; yj] = ¡aijyj; 1¡aij 1¡aij (iii) (ad xi) xj = 0, (ad yi) yj = 0, for i; j = l; :::; l. Then the Lie algebra L is called a Kac-Moody algebra. Any Kac-Moody algebra can be written as follows: L = L+ © h © L¡; n where L+ = © L®, L¡ = © L®, h = © Khi. ®2¢+ ®2¢¡ i=1 2 The positive part of an Hyperbolic Kac-Moody algebra. De¯nition 2.1 Let W be the Weyl group of g(A), that means that the set generated by all the reflections ri; which are de¯ned by ri®j=®j ¡ aij®i (aij elements of A). A root ® 2 ¢ is called real if there exists w 2 W such that w(®) is a simple root. We re re denote by ¢+ the set of all positive real roots, where ¢+ = W (®i). De¯nition 2.2 A root a which is not real is called an imaginary root. Then the set im ¡ v of all positive imaginary roots are ¢+ = W ( C ). By de¯nition re [ im ¢+ = ¢+ ¢+ : Let ti 2 Dern+ be de¯ned by tiej = ±ijej, where ej are the generators of the Hyperbolic Kac-Moody algebra of type A. Let T be the maximal torus on n+, then the root space decomposition corresponding to T is: ® n+ = © n ; ®2¢+ where ® n = fx 2 n+ ; tx = ®(t)x; (8)t 2 T g; ¤ ® ¢+ = f® 2 T ; n =6 (0)g; mult(®) = dim n® of ® 2 ¡Cv; ([4]). Theorem 2.1 [7] Let n be a nilpotent Lie algebra of maximal rank. Then the mapping ® Gl(n): I ! L+(n)= © n ®2I is a bijection from the set of all Gl -orbits of all the ideals I(A) onto Nil(max; n) (the set of all the nilpotent Lie algebras of maximal rank). 106 Gr. Tsagas and X. Agra¯otou 3 Nilpotent Lie algebras of maximal rank and of type of the Hyperbolic Kac- Moody Algebra of type H2. Let A be the g.C.m. of type Hz which is: ( ¶ 2 ¡3 A = ¡3 2 The positive part of the hyperbolic Kac-Moody algebra of type A = H2 is the Lie algebra n+ de¯ned by the generators fe1; e2g and the relations: [e1 [e1 [e1 [e1 [e1; e2]]]]] = 0 = [e1 [e1 [e1 [e1 [e2; e1]]]]] : The maximal torus is T = Ct1 © Ct2. The root system with multiplicity till the height · 6 (ht(k1®1 + k2®2) = k1 + k2). The reflections are: ( ¶ ( ¶ ¡1 0 ¡1 0 r = ; r = ; 1 0 1 2 0 1 re f g ¢+ = ®1; ®2; 3®1 + ®2; ®1 + 3®2 ; im f ¢+ = ®1 + ®2; 2®1 + ®2; ®1 + 2®2; 2®1 + 2®2; 3®1 + 2®2; 2®1 + 3®2; 4®1 + 2®2; 3®1 + 3®2; 2®1 + 4®2g : hThe Basic chain is: N = f2®1 + 2®2; 3®1 + 2®2; 2®1 + 3®2; 4®1 + 2®2; 3®1 + 3®2; 2®1 + 4®2g: The automorphism group of the Dynkin diagram is < (1; 2) >. Finally we found 9 algebras, 3 continuous families with 1 parameter and 7 contin- uous families with 2 parameters. References (1) [1] X.Agra¯otou, Nilpotent Lie algebras of maximal rank and of Kac-Moody type D4 , Communications in Algebra (to appear). [2] G.Favre, L.Sautharoubane, Nilpotent Lie algebras of classical simple type, Journal of Algebra (to appear). [3] G.Favre, Gr.Tsagas, Nilpotent Lie algebras of maximal rank and of type F4,(to appear). [4] V.Kac, In¯nite dimensional Lie algebras and introduction, Progress in mathemat- ics, BirkhÄauser. Hyperbolic Kac-Moody Algebras 107 (1) (1) (1) [5] V.Kanagavel, Nilpotent Lie algebras of type A2 , B2 , G2 , Groups, Algebras and Geometries (to appear). [6] L. Southaroubane, Kac-Moody Lie algebras and the classi¯cation of nilpotent Lie algebras of maximal rank, Canadian J.Math 34, 1982, 1215-1239. [7] L.Southaroubane, Kac-Moody Lie algebras and the universal element for the cat- egory of nilpotent Lie algebras, Math. Ann. 263, 1983, 365-370. [8] Gr.Tsagas, X.Agra¯otou, Nilpotent Lie algebras of maximal rank and of type E6, E7, (to appear). Authors' address: Gr. Tsagas and X. Agra¯otou Aristotle University of Thessaloniki School of Technology, Department of Mathematics and Physics Thessaloniki 540 06, Greece..