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On Chevalley's Z-Form Indian J. Pure Appl. Math., 46(5): 695-700, October 2015 °c Indian National Science Academy DOI: 10.1007/s13226-015-0134-7 ON CHEVALLEY’S Z-FORM M. S. Raghunathan National Centre for Mathematics, Indian Institute of Technology, Mumbai 400 076, India e-mail: [email protected] (Received 28 October 2014; accepted 18 December 2014) We give a new proof of the existence of a Chevalley basis for a semisimple Lie algebra over C. Key words : Chevalley basis; semisimple Lie algebras. 1. INTRODUCTION Let g be a complex semisimple Lie algebra. In a path-breaking paper [2], Chevalley exhibitted a basis B of g (known as Chevalley basis since then) such that the structural constants of g with respect to the basis B are integers. The basis moreover consists of a basis of a Cartan subalgebra t (on which the roots take integral values) together with root vectors of g with respect to t. The structural constants were determined explicitly (up to signs) in terms of the structure of the root system of g. Tits [3] provided a more elegant approach to obtain Chevalley’s results which essentially exploited geometric properties of the root system. Casselman [1] used the methods of Tits to extend the Chevalley theorem to the Kac-Moody case. In this note we prove Chevalley’s theorem through an approach different from those of Chevalley and Tits. Let G be the simply connected algebraic group corresponding to g. Let T be a maximal torus (the Lie subalgebra t corresponding to it is a Cartan subalgebra of G). Let © be the root system of G with respect to T and ∆ a simple root system. For ' 2 ©, let G(') be the 3-dimensional subgroup (isomorphic to SL(2)) corresponding to '. For an element v 6= 0 in the root space g', there is a unique natural isomorphism f';v of SL(2) on G(') taking the diagonal group into T and the induced Lie algebra map taking e12 to v. Set f(';v)(e12 ¡ e21) = s';v. The key result we prove in this note is the following. 696 M. S. RAGHUNATHAN ® ˜ For each ® 2 ∆, let e® be a non-zero element in g and set s® = s®;e® . Let W be the sugroup of G generated by fs®j® 2 ∆g. It is well known that W˜ ½ N(T), the normalizer of T. Then W˜ \ T = T(2), the group of 2-torsion elements in T. Further if w; w0 2 W˜ and ®; ®0 2 ∆ with 0 0 w(®) = w(® ), Ad(w)(e®) = §Ad(w )(e®0 ). One deduces then the existence of a Chevalley basis from the above two assertions. The second assertion above is in fact equivalent to the existence of the Chevalley basis and the first can be deduced from the existence. The first assertion seems to be of some interest in itself. It says that while the sequence f1g ! T ! N(T) ! N(T)=T ! f1g does not split it comes close to to doing so. 2. REPRESENTATIONS OF SL(2; C) We recall in this section some basic facts about finite dimensional (holomorphic) representations of the SL(2; C) (or simply SL(2)), the group of (2 £ 2)-matrices over C of determinant 1 (under multiplication). D will denote the group of diagonal matrices in SL(2). For t 2 C¤, d(t) denotes ¡1 2 the diagonal matrix with d(t)11 = t and d(t)22 = t . We set C = V and denote by ½ the natural representation of SL(2) on V . We denote by sl(2; C) (or simply sl(2)), the Lie algebra fX²M(2; C)j trace(X) = 0g of SL(2). We let ½˙ denote the natural representation of sl(2) on V (induced by ½). For 1 · i; j · 2, let eij be the matrix whose (k l)-th entry is ±ik ¢ ±jl. Set s = e12 ¡ e21, e+ = e12, e¡ = e21 and h = e11 ¡ e22. Then s 2 SL(2) is the “Weyl” element; it normalizes 2 D and s = ¡(identity). Further, fe+; h; e¡g is a basis of sl(2); and we have [h; e+] = 2 ¢ e+, [h; e¡] = ¡2 ¢ e¡ and [e+; e¡] = h. For an integer m ¸ 0 we denote by ½m (resp. ½˙m) the representation of SL(2) (resp. sl(2)) induced by ½ on the symmetric m-th power Sm(V ) of V . If fu; vg is the standard basis of V = C2, p q m ¤ fu ¢ v j p + q = mg is a basis of S (V ). Let t²C and s²SL(2) be the matrix e+ ¡ e¡. One then has, as is easily seen by inspection: Lemma 2.1 — p q p¡q p q ½m(d(t))(u ¢ v ) = t ¢ u ¢ v p q p+1 q¡1 ½˙m(e¡)(u ¢ v ) = q ¢ u ¢ v p q p¡1 q+1 ½˙m(e+)(u ¢ v ) = p ¢ u ¢ v p q p q ½˙m(h)(u ¢ v ) = 2 ¢ (p ¡ q) ¢ u ¢ v ON CHEVALLEY’S Z-FORM 697 p q p q p p §1 §(p¡q) p q ½m(s)(u ¢ v ) = (¡1) ¢ u ¢ v = (¡1) ¢ (q!=p!) ¢ ½˙m(e¨) (u ¢ v ) according as p ¸ q or q ¸ p. For any integer m ¸ 0, ½m is an irreducible representation of SL(2) of dimension m + 1 and, up to equivalence, it is the unique irreducible representation of dimension m + 1. Moreover every representation ¿ of SL(2) is a direct sum of irreducible representations; thus ¿ is a direct sum of copies of the “½m”s. One deduces from the lemma above the following: Corollary 2.2 — Let ¿ be an irreducible representation of SL(2) on a vector space E of dimension m + 1. If z is an eigen vector for ¿(h) in E with eigen value k, (m+k)=2 k ¿(s)(z) = (¡1) ¢ ((m ¡ k)=2)! ¢ ¿(e¨) (z)=((m + k)=2)! In particular, if z²E is a D-fixed vector, ¿(s)(z) = §z. 3. SEMISIMPLE LIE GROUPS,MAXIMAL TORI,ROOTS 3.1 Maximal torus, roots and root-spaces Let G be a connected simply connected semisimple algebraic group over C and g its Lie algebra. Let T be a maximal torus in G and t ½ g the Lie subalgebra of g corresponding to t. A root of G (with respect to T) is a non-trivial eigen character ' : T ! C¤ for the adjoint action of T on the Lie algebra g. We denote by X(T) the group of characters on T and by © ½ X(T) the set of roots. For ' 2 ©, g' = fv 2 gjAd(t)(v) = '(t) ¢ v; 8t 2 Tg, the eigenspace corresponding to ', “the root- ` ' space of '” is of dimension 1 and g = t © '2© g . The Killing form < :; : > on g, restricted to t is ¤ ¤ ¡1 non-degenerate. Hence it defines an isomorphism A of t on its dual t . For ¸ 2 t , set A (¸) = H¸. ¤ ¤ We have then for H 2 t and ¸ 2 t , < H¸; H > = ¸(H). For ¸; ¹ 2 t , set < ¸; ¹ >=< H¸;H¹ >. 3.2 Simple roots; the Weyl group Let N(T) be the normalizer of T (in G) and W = N(T)=T, the Weyl group. The inner conjugation action of N(T) on T factors through to an action of W on T and hence on X(T); the set © is stable under this action. The set f< '; ' >1=2 j ' 2 ©g of “root-lengths” has at most two elements and the Weyl group acts transitively on the set of all roots of the same length. We fix a lexicographic (total) order in X(T) and denote by ©+ (resp. ©¡, resp. ∆) the system of positive (resp. negative, resp. simple) roots. Let Ω be the set of roots dominant for the simple system ∆. The cardinality of Ω is the same as that of the set of root-lengths. For ¯ 2 Ω, P¯ is the parabolic subgroup determined by ¯: it is the sugroup of G corresponding to the Lie sub-algebra generated by fg'j' 2 ©+g, t and fg'j ' 2 ©; < '; ¯ >= 0g. 698 M. S. RAGHUNATHAN 3.3 The 3-dimensional sub-algebras g® ® ¡® For ® in ©, let h® = 2 ¢ H®= < H®;H® >. Now for any 0 6= v+ 2 g ; there is a unique v¡ 2 g with [v+; v¡] = h®. Define i®;v : sl(2) ! g by setting i®;v(e§) = v§; i®;v(h) = h®: it is a Lie ® ¡® algebra isomorphism of sl(2) on the Lie subalgebra g © C ¢ h® © g = g(®); the Lie subgroup G(®) corresponding to g(®) is an algebraic subgroup and i®;v integrates to an isomorphism f®;v of GL(2) on G(®) - we have assumed that G is simply connected. We set s®;v = f®;v(s). As 2 2 s = ¡(Identity) in SL(2), s®;v is the unique non-trivial central element in G(®) and is of order 2. It is easy to see that s®;v belongs to N(T) and that it acts trivially on ft 2 Tj®(t) = 1g, the kernel of ®. 3.4 The group W˜ For each ® in ∆ we now fix elements e§® with [e®; e¡®] = h® and set i®;e® = i®, f®;e® = f® and ˜ ˜ ˜ s®;e® = s®. Let W be the subgroup of G generated by S = fs®j® 2 ∆g. Then W maps onto W under the projection map ¼ : N(T) ! W. Let w® = ¼(s®) and S = ¼(S˜). Then (W; S) is a Coxeter system. Let B be the kernel of ¼ : W˜ ! W; B = T \ W˜ . We have with this notation the first key result. Proposition 3.5 — B = T2, the subgroup of 2-torsion elements in T.
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