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Nilpotent Lie Algebras of Maximal Rank and of Type F4 As an Associated Generalized Cartan Matrix

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Nilpotent Lie Algebras of Maximal Rank and of Type F4 As an Associated Generalized Cartan Matrix NILPOTENT LIE ALGEBRAS OF MAXIMAL RANK AND OF TYPE F4 AS AN ASSOCIATED GENERALIZED CARTAN MATRIX Gr. Tsagas Abstract The aim of the present paper is to determined all Nilpotent Lie algebras of the maximal rank and rank F4. The number of such algebras is 43. Key words: Nilpotent Lie algebras Kac-Moody Algebra, Generalized Cartan Matrix 1 INTRODUCTION Let A = (Aij); i; j = 1; : : : ; n be a Generalized Cartan Matrix denoted briefly by G.C.M. From this matrix and a given root system we can construct Nilpotent Lie algebras of maximal rank having A = (Aij); i; j = 1; : : : ; n as a G.C.M. In order to obtain these we consider the positive part L+(A) of the Kac-Moody Lie algebras L(A) taken by the G.C.M., A = (Aij) and the given root system ¢. The aim of the present paper is to obtain all the Nilpotent Lie algebras of maximal rank whose G.C.M. is the Cartan matrix of the exceptional Lie algebras F4. Each of them is called of type F4. The cases An, Bn, Cn, Dn and G2 have been studied in ([9]) and ([21]). The cases for E6, E7 and E8 are studied in ([24]), ([25]) and ([26]). The whole paper contains seven paragraphs each of them is analyzed as follows. The second paragraph gives the general theory of Kac- Moody Lie algebras. The basic elements and properties of Nilpotent Lie algebras are given in the fourth paragraph. The relation between Kac- Moody Lie algebras and Nilpotent Lie algebras is given in the fourth paragraph. The ¯fth paragraph contains estimates and constructions of Nilpotent Lie algebras of maximal rank and of type F4. The sixth paragraph includes the determination of the ideals. The structure constants and some other properties of the fourtythree Nilpotent Lie algebra of maximal rank and of type F4 are included in the last paragraph. Editor Gr.Tsagas Proceedings of The Conference of Geometry and Its Applications in Technology and The Workshop on Global Analysis, Di®erential Geometry and Lie Algebras, 1999, 176-205 °c 2001 Balkan Society of Geometers, Geometry Balkan Press Nilpotent Lie algebras of maximal rank 177 2 Kac- Moody Lie algebra Let A = (Aij), i, j = 1; : : : ; n, be a square matrix of order n with entries in Z satisfaying: (i) Aii = 2; i = 1; : : : ; n; (ii) Aij · 0, if i =6 j, i, j = 1; : : : ; n; (iii) if Aij = 0, i =6 j, then Aij = 0: A = (Aij) is called Generalized Cartan Matrix denoted briefly by G. C. M. All through this pare the G. C. M. will be of order n . Two G. C. M. A and D are called equivalent if there exists σ 2 Gn, where Gn is the group of permutatios of f1; : : : ; ng, such that: Bij = Aσiσj; 8i; j = 1; :::; n: We consider the Lie algebra L(A), associated to the G. C. M. A(Aij), generated by the set fe1; : : : ; en; h1; : : : ; hn; f1; : : : ; fng satisfying: ½ 1, i = j [h ; h ] = 0; [e ; f ] = ± h ; i; j = 1; : : : ; n; ± = i j i j ij i ij 0, i =6 j [hi; ej] = Aijej; [hi; fj] = ¡Aijfj; i; j = 1; : : : ; n ¡Aij +1 ¡Aij +1 (adei) ej = 0; (adfi) fj = 0; i; j = 1; : : : ; n; i =6 j: Pn n n Let fa1; : : : ; ang be the canonical base of Z . If a 2 N ¡f0g, then a = diai, where i=1 di 2 N and at last one of them is di®erent zero. We denote by La (resp. L¡a) the subroctor space of L(A) generated by the elements ([ei1 ; : : : ; eir ]) (resp. fi) appears di times and the meaning between the two brackets is the following: ([X1;:::;Xk]) = (X1[X2;:::; ]) P n We assume that if a = diai 2 Z and all the dis are not the same sing, then La = (0). We denote: L0 = H = Lh1 © Kh2 © ::: © Khn The root system of L(A), denoted bbbby ¢, is de¯ned by: n ¢ = fa 2 Z a =6 0, La =6 (0)g The Lie algebra L(A), by means of ¢ t f0g, is grade, that means: L(A) = © La[La;L¯] ½ La+¯; 8a; ¯ 2 ¢ t f0g : a24tf0g The positive root system is de¯ned as follows: n ¢+ = fa 2 N a =6 0, La =6 (0)g 178 Gr. Tsagas The negative roots, denoted by ¢, are de¯ned by: n ¢¡ = ¡¢+ = f¡aa 2 N ; a =6 0;La = (0)g It is obvious that: ¢ = ¢¡ t f0g t ¢+: From the above we conclude that the Lie algebra L(A) can be written: L(A) = L¡(A) © H © L+(A) where: L¡(A) = © La;L+(A) = © La a2¢+ a2¢¡ The Lie algebra L(A), associated to G. C. M., generated by: fe1; : : : ; en; h1; : : : ; hn; f1; : : : ; fng and de¯ned above, is called Kac- Moody Lie algebra. P P Remark 1 If a = diai, then we denote by jaj = di which is called height of a. We denote by: k f 2 j j g f 2 4 j j · g ¢+ = a ¢+ a = k ; ¢½ = a + a p n f g and therefore we have ¢+ = a1; : : : ; an . 3 Nilpotent Lie algebras. Let g be a Nilpotent Lie algebra of dimension m over the algebraically closed ¯eld K of characteristic zero. We denote by Derg and Autg the derivation Lie algebra and automorphism group of g respectively. A torus T on g is a commutative subalgebra of Derg consisting of semi- simple endomorphisms. A torus T is called maximal, if it is not contained strictly in any other torus. A torus de¯nes a representation in g, that means: T xg ! g; (t; x) ! tx From the fact that T is a commutative family of semi- simple endomorphisms and the properties of K, we conclude that the elements of T can be diagonalized simulta- neously. Therefore g is decomposed into a direct sum of root spaces, that means: g = © g¯ ¯2T B where T ¤ is the dual of the vector space T and: g¯ = fx 2 gtx = ¯(t)x; 8t 2 T g: Nilpotent Lie algebras of maximal rank 179 The root system of g associated to T , denoted by R(T ), is de¯ned by: R(T ) = f¯ 2 T ¤g¯ =6 (0)g: From now on we assume that g is a Nilpotent Lie algebra. We suppose that T is a f g ¤ maximal torus on g and dim T = k. Let ¯1; :::; ¯k be a base of T whose dual base of T is ft1; : : : ; tng, that means: ¯i(tj) = ±ij: The vectors fx1; : : : ; xkg of g with the property: ti(xj) = ±ijxj is called T minimal system of generators or briefly T ¡ msg. Hence we have: ¯ g i = Kxi f g and therefore x1; : : : ; xk are root vectors for T . ¯i, i = 1; : : : ; k, is called root of xi, i = 1; : : : ; k, respectively. We have the following propositions ([21]). Proposition 1 If g is a Nilpotent Lie algebra, then the following two statements are equivalent: (1) fx1; : : : ; xkg is a minimal system of generators; 2 2 2 2 (2) fx1 +c g; : : : ; xk +c gg is a base for the vector space g=c g, where c g = [g; g]: The type of g is de¯ned the dimension of g=c2g. Proposition 2 Let g be a Nilpotent Lie algebra of type (1). Let T be a maximal f g ¡ torus on g, x1; : : : ; xk T mg system, ¯i the root of xi. The dimension of T is f g equal to the rank of ¯1; : : : ; ¯k : Proposition 3 Let gbe a Nilpotent Lie algebra of type s. The dimension of the maximal torus T on g is called rank of g. If k is the rank, then we have k · s: 4 Connection between Nilpotent Lie algebras and Kac- Moody Lie algebras Let g be a Nilpotent Lie algebra of type n which is the dimension of the Lie algebra gc2g. If the rank of g is n, then g is called maximal rank. Let g be a Nilpotent Lie algebra. Let T be a maximal torus on g. Let (x1; : : : ; xn) be a T ¡ mg, for those elements we have: ¡Aij +1 (adxi) ; xj = 0; then Aij 2 Z¡ t f0g : If we put Aij = 2, then using Aij i = 1j from the above we have the matrix: A = (Aij) 180 Gr. Tsagas with the properties: (1)Aij = 2, i = 1; : : : ; n (2)Aij · 0, i, j = 1; : : : ; n, i =6 j; (3) If Aij = 0, then Aij = 0, when i =6 j; i; j = 1; : : : ; n: This is the G. C. M. associated to g. Let A = (Aij) be a G. C. M. we assume that the of positive roots 4+ are given, whose number is ¯nite, that means: ( ) Xn 4 v v v 2 f g n + = a = di ai di N , a1; : : : ; an base of Z i=1 the number of roots av, v = 1; :::; m, is ¯nite. We denote by L+(A) the positive part of the Kac- Moody Lie algebra L(A) as- sociated to A and 4+. Therefore L+(A) is a Lie algebra generated by fe1; : : : ; eng satisfying only the relations: ¡Aij +1 (1) (adei) ej = 0, i =6 j, i = 1; :::; n (2) L+(A) is grated by: L+(A) = © La,[La;L¯] ½ La+¯, 8a; ¯ 2 ¢4+ a24+ We refer the following propositions ([21]): Proposition 4 Let L+(A) be the positive part of the Kac- Moody Lie algebra L(A) associated to G. C. M. A. Then we have: m C L+(A) = © La; j®j¸m m where C L+(A) is the nth.
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