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Invariants of Real Forms of Affine Kac-Moody Lie Algebras
Journal of Algebra 223, 208–236 (2000) doi:10.1006/jabr.1999.8030, available online at http://www.idealibrary.com on
Invariants of Real Forms of Affine Kac-Moody Lie Algebras Punita Batra School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005, India E-mail: [email protected] Communicated by Georgia Benkart Received July 2, 1998
The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finitedimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification. © 2000 Academic Press
1. INTRODUCTION Elie Cartan classified finite-dimensional real semisimple Lie algebras in a long 1913 paper. From then until the late 1970s, the proof of the classification was gradually simplified so that it took only about 25 pages. In a 1996 paper [6], Knapp modified some diagrams that had arisen in connection with the analysis of discrete series, calling them Vogan diagrams, and showed how Vogan diagrams could be used to prove the classification in under 3 pages. The theory of Vogan diagrams for real semisimple Lie algebras includes three theorems worth noting here. Details may be found in [5]. The first is that if two real semisimple Lie algebras, in the presence of some choices, have matching Vogan diagrams, then the Lie algebras are isomorphic. The second is that any diagram that looks like a Vogan diagram comes from 208 0021-8693/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.
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some real semisimple Lie algebra and a system of choices. The third is that any Vogan diagram can be transformed, by changing the ordering, into a diagram in which each simple component has at most one noncompact imaginary root and that root occurs at most twice in the largest root of that simple component. As a result, the classification comes down to deciding which of some really easy Vogan diagrams correspond to isomorphic Lie algebras, that can be done readily by hand. This paper deals with indecomposable nontwisted affine Kac–Moody Lie algebras and their real forms. The real forms of this kind of Lie algebra have been classified as a result of work by Guy Rousseau and H. Ben Messaoud (see Section 3.6). What is sought is a theory of Vogan diagrams for them. The real forms for a given complex Lie algebra of this type break into two kinds, called “almost split” and “almost compact.” The almost split ones correspond in one-one fashion with all but one of the almost compact ones, the exceptional almost compact form being the so-called compact form. In this paper we introduce Vogan diagrams for the almost compact real forms of an indecomposable nontwisted affine Kac–Moody Lie algebra. The underlying Dynkin diagram is merely the completed diagram of an ordinary indecomposable Dynkin diagram. Here some two-element orbits will be marked, and some of the one-element orbits will be shaded. These markings and shadings are possible as a result of an extensive theory about the real forms that is already in place. The main theorem of this paper (Theorem 5.2) is that two real forms that have matching Vogan diagrams are isomorphic. It is not true that the ordering can be changed in such a way that at most one root is shaded. This fact makes it important to introduce the notion of equivalence of Vogan diagrams, in which two such diagrams are equivalent if they correspond to different orderings for the same real form. It is a fairly easy matter to work out representatives of the equivalence classes of Vogan diagrams, and these are listed for some nontwisted affine Kac–Moody Lie algebras in the Appendix. A count of the number of these diagrams yields a match with the number of almost compact real forms in the known classification, as it should. As is true in the finite-dimensional case, the theory of Vogan diagrams does more than give a faster proof of the classification. It allows one to identify, for a given Lie algebra, which standard Lie algebra is isomorphic to it, without tracking down isomorphisms whose existence is known but for which there are no formulas. This paper is organized as follows. Section 2 recites known facts about (complex) indecomposable nontwisted affine Kac–Moody Lie algebras. Section 3 records aspects of the classification of the real forms and then introduces Vogan diagrams, making the necessary justifications. Section 4 dis-
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cusses some low-rank examples, showing explicitly what almost compact real forms are and what their Vogan diagrams look like. Section 5 states and proves the main theorem. The Appendix gives representatives of equivalence classes of Vogan diagrams when the given complex simple Lie algebra is of type An , Bn , Cn , and D4 , proving that there are no others.
2. PRELIMINARIES 2.1. Affine Kac–Moody Lie Algebras Let I = 1; n + 1, n ∈ , be an interval in . Let A = aij i; j∈I be a matrix with integer coefficients. A is called a generalized Cartan matrix if it satisfies the following conditions: • aii = 2 for i = 1; : : : ; n + 1. • aij are nonpositive integers for i 6= j. • aij = 0 implies aji = 0. Let Ç = ÇA be the free complex Lie algebra generated by a basis of the standard Cartan subalgebra È and the elements ei ; fi for i ∈ I with the following defining relations [3]:
adei
ei ; fj = δij α∨i ;
i; j ∈ I
h; h0 = 0;
h; h0 ∈ È
h; ei = αi ; hei ;
i ∈ I; h ∈ È
h; fi = −αi ; hfi ;
i ∈ I; h ∈ È
1−aij
ej = 0;
i 6= j
adfi 1−aij fj = 0;
i 6= j:
The algebra Ç is called a Kac–Moody Lie algebra. The algebra Ç is an affine Kac–Moody Lie algebra if i.e., after the indices are reordered • A is an indecomposable matrix, A cannot be written in the form 0A1 A0 : 2
= 0.
n+1 n+1 • There exists a vector ai i=1 , with ai all positive such that Aai i=1
Then A is called a Cartan matrix of affine type. The affine algebra associated 1 with a generalized Cartan matrix of type Xn is called a nontwisted affine Kac–Moody Lie algebra.
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2.2. Dynkin Diagram Associated with a Generalized Cartan Matrix We associate with a generalized Cartan matrix A a graph SA called the Dynkin diagram of A as follows. If aij aji ≤ 4 and aij ≥ aji , the vertices i and j are connected by aij lines, and these lines are equipped with an arrow pointing toward i if aij > 1. If aij aji > 4, the vertices i and j are connected by a bold-faced line equipped with an ordered pair of integers aij ; aji . It is clear that A is indecomposable if and only if SA is a connected graph. A is determined by the Dynkin diagram SA and an enumeration of its vertices. 2.3. A Realization of Nontwisted Affine Kac–Moody Lie Algebras Let L = t; t −1 be the algebra P of Laurent polynomials in t. The residue of a Laurent polynomial P = k∈ ck t k (where all but a finite number of ck are 0) is defined as Res P = c−1 . ˚ be a finite-dimensional simple Lie algebra over of type Xn . Let Ç ˚ is an infinite-dimensional complex Lie algebra with Then L˚ Ç = L ⊗ Ç the bracket P ⊗ x; Q ⊗ y = PQ ⊗ x; y;
˚: P; Q ∈ Ly x; y ∈ Ç
˚ and exFix a nondegenerate, invariant, symmetric bilinear form ·; · in Ç Ç by tend this form to an L-valued form ·; ·t on L˚ P ⊗ x; Q ⊗ yt = PQx; y;
˚: P; Q ∈ Ly x; y ∈ Ç
Ç by The derivation t j dtd of L extends to L˚ tj
d dP P ⊗ x = t j ⊗ x; dt dt
˚: P ∈ Ly x ∈ Ç
; bt ; a; b ∈ L˚ Ç defines a 2-cocycle on L˚ Ç Therefore ψa; b = Res da dt (Co 1), (Co 2) [3, p. 97]. ˜ Ç the central extension of the Lie algebra L˚ We denote by L˚ Ç associ˜ Ç = L˚ ated with the cocycle ψ. Explicitly L˚ Ç ⊕ c with the bracket a + λc; b + µc = a; b + ψa; bc;
a; b ∈ L˚ Ç; λ; µ ∈ :
ˆ Ç the Lie algebra, which is obtained by adjoining Finally, we denote by L˚ ˆ Ç = ˜ Ç a derivation which acts on L˚ to L˚ Ç as t dtd and kills c. Explicitly L˚ L˚ Ç ⊕ c ⊕ d with the bracket defined by t k ⊗ x + λc + µd; t j ⊗ y + λ1 c + µ1 d = t j+k ⊗ x; y + µjt j ⊗ y − µ1 kt k ⊗ x + kδj; −k x; yc;
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ˆ Ç ˚y λ; µ; λ1 ; µ1 ∈ y k; j ∈ [3, Sect. 7.2.2]. [Ka, 7.2.2]. L˚ where x; y ∈ Ç is a nontwisted affine Kac–Moody Lie algebra associated with the affine 1 ˆ Ç, matrix A of type Xn [3, Theorem 7.4]. Here c = c is the center of L˚ ˆ Ç. and L˚ Ç ⊕ c is the derived algebra of L˚ Notation.
ˆ Ç = Ç and t k ⊗ x will be written as t k x. From now on L˚
2.4. Invariant Bilinear Form of Ç The normalized invariant form of Ç [3, Sect. 6.2] can be described as ˚ and extend ·; · to follows. Take the normalized invariant form ·; · on Ç all of Ç by P ⊗ x; Q ⊗ y = Res t −1 PQx; y;
˚y P; Q ∈ L x; y ∈ Ç
c ⊕ d; L˚ Ç = 0; c; c = d; d = 0; c; d = 1. It is a nondegenerate, symmetric, invariant, bilinear form of Ç [3, Sect. 7.5]. 2.5. Roots of Ç ˚ ⊂È ˚∗ be the root system of the finite-dimensional Lie algebra Ç ˚, Let 1 let α1 ; α2 ; : : : ; αn be the root basis, let H1 ; H2 ; : : : ; Hn the coroot basis, and let Ei , Fi i = 1; 2; : : : ; n the Chevalley generators. Let θ be the L ˚ . Let Ç ˚ = α∈1∪0 ˚α be the root highest root of the finite-root system 1 Ç ˚ ˚. Let ω ˚. We choose space decomposition of Ç ˚ be the Cartan involution of Ç 2 ˚θ such that F0 ; ωF ˚ 0 = − θ;θ and set E0 = −ωF ˚ 0 . Then we F0 ∈ Ç have, due to [3, Theorem 2.2e], E0 ; F0 = −θ∨ : ˚ + c + d is an ˆ Ç. Here È = È Now we come to the Lie algebra Ç = L˚ ˚∗ to a n + 2-dimensional commutative subalgebra in Ç. We extend λ ∈ È ∗ ˚ linear functional on È by setting λ; c = λ; d = 0, so that È is identified with a subspace in È∗ . We denote by δ the linear functional on È defined = 0, δ; d = 1. Set by δÈ+c ˚ en+1 = tE0 ;
fn+1 = t −1 F0
ei = Ei ; fi = Fi i = 1; 2; : : : ; n: Remark 1. In [3], Kac uses e0 for en+1 and f0 for fn+1 . We see from [3, Sect. 7.4.1] that en+1 ; fn+1 =
2 c − θ∨ : θ; θ
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Now we describe the root system and the root space decomposition of Ç with respect to È: ˚ ∪ jδ; where j ∈ − 0: 1 = jδ + γ; where j ∈ ; γ ∈ 1 M L˚ Çα ; Ç=È⊕ α∈1
˚ ˚γ , L˚ Çjδ = t j È. where L˚ Çjδ+γ = t j Ç We set π = α1 ; α2 ; : : : ; αn ; αn+1 = δ − θ ; 2 ∨ ∨ ∨ ∨ ∨ ∨ c−θ : π = α1 ; α2 ; : : : ; αn ; αn+1 = θ; θ Then by [3, Proposition 6.4a] A = α∨i ; αj n+1 i; j=1 :
In other words, È; π; π ∨ is a realization of the affine matrix A. 2.6. Cartan Involution of Ç The map ei 7→ −fi ; fi 7→ −ei ; i ∈ Iy h 7→ −h; h ∈ È can be uniquely extended to an involution ω of Ç. ω is called the Cartan involution of Ç. 2.7. Weyl Group For each i ∈ I, we define a fundamental reflection ri of the space È∗ by ri α = α − α; α∨i αi
∀α ∈ È∗ :
The subgroup W = GLÈ∗ generated by ri , i ∈ I is called the Weyl group of Ç. 3. CLASSIFICATION OF REAL FORMS OF NONTWISTED AFFINE KAC–MOODY LIE ALGEBRAS 3.1. Cartan and Borel Subalgebras of Ç We define a group G acting on the Lie algebra Ç via the adjoint representation Adx G 7→ AutÇ. It is generated by the subgroups Uα for α ∈ ±π and AdUα = expadÇα . A maximal adÇ -diagonalizable subalgebra of Ç is called a Cartan subalgebra. Every Cartan subalgebra of Ç is AdG-conjugate to the standard Cartan subalgebra È [4]. A Borel subalgebra of Ç is a maximal completely solvable subalgebra. It is always conjugated by AdG to b+ or b− , where L L + − b = È ⊕ α>0 Çα and b = È ⊕ α<0 Çα . But b+ and b− are not conjugated under AdG. So there are exactly two conjugacy classes of Borel subalgebra: the positive and the negative ones [4].
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3.2. Real Forms of Ç A real form of Ç is a Lie algebra Ç over such that there exists an isomorphism from Ç to Ç ⊗ [1]. If we replace with in the definition of Ç, we obtain a real form Ç which is called split. A real form of Ç corresponds to a semi-linear involution of Ç (this means to an automorphism τ ¯ for λ ∈ (semi-linear)). of Ç such that τ2 = Id and τλx = λτx A linear or semilinear automorphism of Ç is said to be of the first kind if it transforms a Borel subalgebra into a Borel subalgebra of the same sign. A linear or semilinear automorphism of Ç is said to be of the second kind if it transforms a Borel subalgebra into a Borel subalgebra of the opposite sign. Let Ç be a real form of Ç, and fix an isomorphism from Ç to Ç ⊗ . Then the Galois group 0 = Gal/ acts on Ç and the corresponding group G. We identify Ç with the fixed point set Ç0 . Definition 3.1. If 0 consists of first kind automorphisms we say that Ç is almost split. Otherwise if the nontrivial element of 0 is a second kind automorphism, we say that Ç is almost compact. (In other words, a real form Ç of Ç is said to be almost split if for each γ in 0 the action of γ on Ç is of first kind, otherwise Ç is said to be almost compact [10].) 3.3. Automorphisms of Ç Let È be the standard Cartan subalgebra of Ç. Paper [4] defines a group e which acts on G and Ç. In fact H b = AdH e is isomorphic to ∗ I and H e corresponds to hi i∈I , it acts on Çα by multiplication if the elementQh of H P by the scalar i hi ni if α = i ni αi . e ∝ G of interior automorphisms of Ç is the The group IntÇ = AdH e and G. Its derived group is the image of the semi-direct product of H adjoint group AdG (denoted by IntÇ0 ). As G is transitive on the Cartan subalgebra, the group IntÇ does not depend on the choice of È. We consider the group AutA of permutations ρ of I such that aρiρj = aij for i; j ∈ I. We regard AutA as a subgroup of AutÇ0 by requiring ρei = eρi , ρfi = fρi . The Cartan involution ω commutes with AutA. We define the group of outer automorphisms of Ç to be ExtÇ = Id; ω × AutA: 3.3.1. Transvections Following [1], we denote by Tr the set of transvections of Ç which means linear maps of Ç in Ç of the form φ = exp ψ, where ψ is a linear map of Ç into c, zero on Ç0 . So we have φx = x + ψx.
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Then according to [1, p. 187], we have the following decomposition of the group of automorphisms of Çx AutÇ = ExtÇ ∝ IntÇ × Tr:
(3.1)
If σ is an automorphism of Ç of the first kind, then ω ◦ σ is a second kind automorphism and conversely [1, p. 17]. From (3.1) it follows that any automorphism of Ç is either an automorphism of the first kind or an automorphism of the second kind. 3.4. Semi-linear Automorphisms Definition 3.2. Following [1, p. 190], we define by Aut Ç the group of automorphisms of Ç which are -linear or semi-linear. The group AutÇ is normal in Aut Ç and is of index 2. A semi-linear automorphism of order 2 of Ç is called a semi-involution of Ç. For any semi-involution σ 0 we have the decomposition Aut Ç = 1; σ 0 ∝ AutÇ. We denote by σn0 the conjugation of Ç with respect to the standard split real form. We call σn0 the standard normal semi-involution of Ç. This commutes with the standard Cartan involution ω. Let ω0s = σn0 ω = ωσn0 . Then ω0s is called the standard Cartan semiinvolution of Ç. Its algebra of fixed points is the standard compact real form of Ç. A Cartan semi-involution of Ç is a semi-involution ω0 conjugate to ω0s by an element of Aut Ç. Then ω0 is a semi-involution of the second kind and the associated real form is called the compact real form Ë1 of Ç. Let σ 0 be a semi-involution and ω0 be a Cartan semi-involution. Then using [1, Proposition 4.5] and after having supposed that σ 0 and ω0 stabilize the same Cartan subalgebra È, one may suppose by conjugating by G that ω0 commutes with σ 0 . Definition 3.3. Let σ 0 be a semi-involution of Ç of the second kind 0 and let Ç = Çσ be the corresponding almost compact real form. A Cartan semi-involution ω0 which commutes with σ 0 is called a Cartan semiinvolution for σ 0 or Ç . The involution σ = σ 0 ω0 (resp. its restriction ω0 to Ç ) is called a Cartan involution of σ 0 (resp. of Ç ). The algebra of fixed points Ë0 = Çσ is called a maximal compact subalgebra of Ç . We have the Cartan decomposition Ç = Ë0 ⊕ Ð0 and Ë1 = Ë0 ⊕ iÐ0 , where Ð0 is the eigenspace of ω0 for the eigen value −1 [1, pp. 199–200]. 3.5. Maximally Compact Cartan Subalgebra of an Almost Compact Form For X; Y in Ç we define X; Y σ = −X; σY
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where ·; · is the nondegenerate, symmetric, invariant, bilinear form defined as in Section 2.4. It is easy to check that it is positive definite and symmetric form. Lemma 3.4. If Ç is an almost compact real form of Ç and σ is a Cartan involution of Ç , then adX∗ = −adσX
for all X ∈ Ç ;
where adjoint :∗ is defined relative to the inner product ·; ·σ . Proof.
We have adσXY; Zσ = −σX; Y ; σZ = Y; σX; σZ = Y; σX; σZ = Y; σX; Z = −Y; adXZσ = −adX∗ Y; Zσ :
Proposition 3.5. Let Ô0 be a maximal abelian subspace of Ë0 . Then È0 = ZÇ Ô0 is a σ-stable Cartan subalgebra of the almost compact real form Ç of the form È0 = Ô0 ⊕ Á0 , with Á0 ⊆ Ð0 . Proof. È0 = ZÇ Ô0 = x ∈ Ç x; Ô0 = 0: If x ∈ È0 and y ∈ Ô0 , then σx; y = σx; σσy = σx; σy = σx; y = 0: So È0 is σ-stable and hence is a vector space direct sum È0 = Ô0 ⊕ Á0 , where Á0 = È0 ∩ Ð0 . We have È0 ; È0 = Á0 ; Á0 and Á0 ; Á0 ⊆ Ô0 , since Á0 ⊆ Ð0 and È0 ∩ Ë0 = Ô0 . Thus È0 ; È0 is abelian. Since È0 is σ-stable and finite-dimensional, using the inner product ·; ·σ on È0 , it is easy to check that any ideal in È0 has a complementary ideal. So È0 is reductive. By [5, Corollary 1.53], È0 ; È0 is semisimple. Thus the semisimple and abelian Lie algebra È0 ; È0 must be zero. Consequently È0 is abelian.
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It is clear that È0 is maximal abelian in Ç . Now if x ∈ Ô0 , then by Lemma 3.4 we have adx∗ = −adx. So the members of adÇ Ô0 are skew adjoint. If x ∈ Á0 , then by Lemma 3.4 we have adx∗ = adx. So the members of adÇ Á0 are self-adjoint. Since Ô0 commutes with Á0 , adÈ0 is certainly diagonable on Ç ; hence È0 is a Cartan subalgebra of Ç . We say that a σ-stable Cartan subalgebra of an almost compact real form Ç of the form È0 = Ô0 ⊕ Á0 ; with Ô0 ⊆ Ë0 and Á0 ⊆ Ð0 , is maximally compact if dim Ô0 is as large as possible. Proposition 3.6. A maximally compact Cartan subalgebra È0 = Ô0 ⊕ Á0 of an almost compact real form Ç has the property that all of the roots are real on Á0 and imaginary on Ô0 . Proof. The values of the roots on a member H of È0 are the eigenvalues of adH. For H ∈ Á0 , these are real since adH is self-adjoint by Lemma 3.4. For H ∈ Ô0 , these are purely imaginary since adH is skew adjoint by Lemma 3.4. 3.6. Classification of Real Forms Following [2, Theorem 4.4], we obtain under AutÇ a one-to-one correspondence between the conjugacy classes of (linear) involutions of Ç of the second kind and the conjugacy classes of almost split real forms of Ç. Theorem 3.7 [9]. We consider 1. The semi-involutions σ 0 , of Ç of the second kind. 2. The involutions θ, of Ç of the first kind. 3. The relation σ 0 ≈ θ if and only if (a) ω0 = θσ 0 = σ 0 θ is a Cartan semi-involution. (b) θ and σ 0 stabilize the same Cartan subalgebra È. (c) È is contained in a minimal σ 0 -stable positive parabolic subalgebra. Then this relation induces a bijection between the conjugacy classes under AutÇ of semi-involutions of the second kind and conjugacy classes of involutions of the first kind. Thus we obtain under AutÇ a one-to-one correspondence between the conjugacy classes of (linear) involutions of Ç of the first kind (including identity) and the conjugacy classes of almost compact real forms of Ç. The compact real form is unique; it corresponds to identity. A classification of involutions of an affine Kac–Moody Lie algebra is given by Levstein [8]. A classification of automorphisms of finite order of affine Kac–Moody Lie 1 algebra of type An is given by Kobayashi [7].
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3.7. Vogan Diagrams Let Ç be an almost compact real form of Ç corresponding to the semiinvolution σ 0 of Ç of the second kind. Let σ be the Cartan-involution of Ç , and let Ç = Ë0 ⊕ Ð0 be the corresponding Cartan decomposition. We write Ç = Ë ⊕ Ð for the complexification of the Cartan decomposition. Let Ô0 be a maximal abelian subspace of Ë0 . Then È0 = ZÇ Ô0 is a σstable Cartan subalgebra of Ç of the form È0 = Ô0 ⊕ Á0 , with Á0 ⊆ Ð0 by Proposition 3.5. This È0 is a maximally compact Cartan subalgebra of Ç because Ô0 is as large as possible. By Proposition 3.6, we note that the roots of Ç; È are imaginary on Ô0 and real on Á0 . A root is real if it takes real values on È0 (i.e., it vanishes on Ô0 ), imaginary if it takes purely imaginary values on È0 (i.e., it vanishes on Á0 ), and complex otherwise. For any root α, σα is the root σαH = ασ −1 H. If α is imaginary, then σα = α. Thus Çα is σ-stable, and we have Çα = Çα ∩ Ë ⊕ Çα ∩ Ð. Since Çα is 1-dimensional, Çα ⊆ Ë or Çα ⊆ Ð. We call an imaginary root α compact if Çα ⊆ Ë, noncompact if Çα ⊆ Ð. Proposition 3.8. Let È0 be a σ-stable Cartan subalgebra of Ç . Then there are no real roots if and only if È0 is maximally compact. Proof. The Cayley transform construction dα [5, p. 334] tells us that if È0 has a real root α, then we can construct a new Cartan subalgebra whose Ô0 is strictly larger. Then È0 could not have been maximally compact. For the converse, we write È0 = Ô0 ⊕ Á0 , and let 1 = 1Ç; È be the set of roots. From the expansion M Çα : (3.2) Ç=È⊕ α∈1
Suppose there are no real roots. Then we obtain from (3.2) M M Çα = È ⊕ Çα = È ZÇ Ô0 = È ⊕ α∈1 αÔ0 =0
α∈1 αreal
and Ë0 ∩ ZÇ Ô0 = Ë0 ∩ È = Ô0 . Therefore Ô0 is maximal abelian in Ë0 , and È0 is maximally compact. Let Ç be an almost compact real form of Ç. Let σ be the Cartan involution and let Ç = Ë0 ⊕ Ð0 be the corresponding Cartan decomposition. Let È0 = Ô0 ⊕ Á0 be a maximally compact σ-stable Cartan subalgebra of Ç , with complexification È = Ô ⊕ Á. We let 1 = 1Ç; È be the set of roots. By Proposition 3.8, there are no real roots, i.e., no roots that vanish on Ô. We choose a positive system 1+ for 1 that takes iÔ0 before Á0 [5, p. 339]. Since σ is +1 on Ô0 and −1 on Á0 and since there are no real roots, σ1+ = 1+ . Therefore σ permutes the simple roots. It must fix the simple roots that are imaginary and permute in 2-cycles the simple roots that are complex.
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By the Vogan diagram of the triple Ç ; È0 ; 1+ , we mean the Dynkin diagram of 1+ with the 2-element orbits under σ labeled by an arrow and with the 1-element orbits painted or not, depending on whether the corresponding imaginary simple root is noncompact or compact. 3.7.1. Equivalence of Vogan Diagrams We define the equivalence of Vogan diagrams to be the equivalence relation generated by the following two operations: 1.
Application of an automorphism of the Dynkin diagram.
2. Change in the positive system by reflection in a simple, noncompact root, i.e., by a vertex which is colored in the Vogan diagram. As a consequence of reflection by a simple, noncompact root α, the rule for single and triple lines is that we leave α colored and its immediate neighbor is changed to the opposite color. The rule for double lines is that if α is the smaller root, then there is no change in the color of its immediate neighbor, but we leave α colored. If α is the bigger root, then we leave α colored and its immediate neighbor is changed to the opposite color. If two Vogan diagrams are not equivalent to each other, we call them non-equivalent. 4. EXAMPLES 1
4.1. Real Forms of A1 = ÓÌ2; t; t −1 ⊕ c ⊕ d Here, the generalized Cartan matrix is Dynkin diagram is
2 −2
−2 2
;
and the corresponding
. The generators are e1 = f1 =
0
1
0
0
0
0
1
0
! = e;
e2 = t
! = f;
f2 = t
−1
0
0
1
0
! = tf;
0
1
0
0
! = t −1 e:
1
And the Cartan subalgebra of A1 is h ⊕ c ⊕ d;
where h =
1
0
0
−1
! :
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The Cartan semi-involution ω0s of A1 is ω0s x t n e 7→ −t −n f t n f 7→ −t −n e t n h 7→ −t −n h it n e 7→ it −n f it n f 7→ it −n e it n h 7→ it −n h c 7→ −c d 7→ −d ic 7→ ic id 7→ id: 4.2. Almost Compact Real Forms of ÓÌ2; t; t −1 ⊕ c ⊕ d Using the involutions given by Kobayashi [7], we get the following four almost compact real forms of ÓÌ2; t; t −1 ⊕ c ⊕ d. 1. Involution of the first kind φ x e 7→ −e; f 7→ −f; tf 7→ −tf; t −1 e 7→ −t −1 e; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e + f; ie − f ; ih; tf + t −1 e; itf − t −1 e; th − t −1 h; ith + t −1 h ⊕ ic ⊕ id: 2. The involution of the first kind φ x e 7→ e; f 7→ f; tf 7→ −tf; t −1 e 7→ −t −1 e; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e − f; ie + f ; ih; tf + t −1 e; itf − t −1 e; th + t −1 h; ith − t −1 h ⊕ ic ⊕ id: 3. The involution of the first kind φ x e 7→ tf; f 7→ t −1 e; tf 7→ e; t e 7→ f; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e − t −1 e; ie + t −1 e; f − tf; if + tf ; h; th + t −1 h; ith − t −1 h −1
⊕ ic ⊕ id:
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4. The compact real form, corresponding to Cartan semi-involution ω0s , is given by n t e − t −n f; it n e + t −n f ; t n f − t −n e; it n f + t −n e; t n h − t −n h; it n h + t −n hy n ∈ ⊕ ic ⊕ id: 4.3. Vogan Diagrams Corresponding to Almost Compact Forms (1), (2) and (4) of ÓÌ2; t; t −1 ⊕ c ⊕ d Here, φ is the complexified Cartan involution for the real form. So its fixed point set will give us Ë0 . The maximally compact Cartan subalgebra È0 = ih + ic + id = Ô0 . We define simple roots as follows: α1 h = 2; α1 c = 0; α1 d = 0: The corresponding root vector is e. α2 h = −2; α2 c = 0; α2 d = 1: The corresponding root vector is tf . So the Vogan diagram corresponding to (1) is the following, because e and tf are in Ð: . The Vogan diagram corresponding to (2) is the following, because e is in Ë and tf is in Ð: . The Vogan diagram corresponding to (4) is the following, because e and tf are in Ë: .
4.4. Vogan Diagrams Corresponding to Almost Compact Forms (3) of ÓÌ2; t; t −1 ⊕ c ⊕ d The maximally compact Cartan subalgebra is È0 = h + ic + id = Ô0 ⊕ Á0 , where Ô0 = ic + id and Á0 = h. We define simple roots as follows: α1 h = 2; α2 h = −2;
α1 c = 0; α2 c = 0;
α1 d = 1: α2 d = 1:
So the Vogan diagram is the following, because α1 and α2 are complex roots: .
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4.5. Real Forms of A2 = ÓÌ3; t; t −1 ⊕ c ⊕ d Here, the generalized Cartan matrix and corresponding Dynkin diagram are .
The generators are 0 1 0 e1 = 0 0 0 ; 0
0
0 f1 = 1 0
0 0
0 0;
0
0
0
0 0 0 e2 = 0 0 1 ; 0 0 0 0 0 0 f2 = 0 0 0 ; 0 1 0
0 0 0 e3 = t 0 0 0 ; 1 0 0 0 0 1 f3 = t −1 0 0 0 : 0 0 0
1
The Cartan subalgebra of A2 is h1 + h2 ⊕ c ⊕ d; where
1 0 0 h1 = 0 −1 0 ; 0 0 0
0 0 0 0; h2 = 0 1 0 0 −1 1
−1 0 0 h3 = 0 0 0 : 0 0 1
The Cartan semi-involution ω0s of A2 is the following: ω0s x t n e1 7→ −t −n f1 t n e2 7→ −t −n f2 t n e3 7→ −t −n f3 t n f1 7→ −t −n e1 t n f2 7→ −t −n e2 t n f3 7→ −t −n e3 t n h1 7→ −t −n h1 t n h2 7→ −t −n h2 t n h3 7→ −t −n h3
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it n e1 7→ it −n f1 it n e2 7→ it −n f2 it n e3 7→ it −n f3 it n f1 7→ it −n e1 it n f2 7→ it −n e2 it n f3 7→ it −n e3 it n h1 7→ it −n h1 it n h2 7→ it −n h2 it n h3 7→ it −n h3 c 7→ −c d 7→ −d ic 7→ ic id 7→ id: 4.6. Almost Compact Real Forms of ÓÌ3; t; t −1 ⊕ c ⊕ d Using the involutions given by Kobayashi [7], we get the following five almost compact real forms of ÓÌ3; t; t −1 ⊕ c ⊕ d. 1. Involution of the first kind φ x e1 7→ e1 , f1 7→ f1 , e2 7→ −e2 , f2 7→ −f2 , e3 7→ −e3 , f3 7→ −f3 , c 7→ c, d 7→ d. The corresponding semiinvolution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n − f1 t −n , ie1 t n + f1 t −n , f1 t n − e1 t −n , if1 t n + e1 t −n , e2 t n + f2 t −n , ie2 t n − f2 t −n , f2 t n + e2 t −n , if2 t n − e2 t −n , e3 t n + f3 t −n , ie3 t n − f3 t −n y n ∈ ⊕ ic ⊕ id. 2. Involution of the first kind φ x e1 7→ e1 , f1 7→ f1 , e2 7→ e2 , f2 7→ f2 , e3 7→ −e3 , f3 7→ −f3 , c 7→ c, d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n + −1n+1 f1 t −n , ie1 t n − −1n+1 f1 t −n , e2 t n + −1n+1 f2 t −n , ie2 t n − −1n+1 f2 t −n , e3 t n − −1n+1 f3 t −n , ie3 t n + −1n+1 f3 t −n ; n ∈ ⊕ic ⊕ id. 3. The compact real form, corresponding to the Cartan semiinvolution ω0s ; is given by e1 t n − f1 t −n , ie1 t n + f1 t −n , h1 t n − h1 t −n , ih1 t n + h1 t −n , e2 t n − f2 t −n , ie2 t n + f2 t −n , h2 t n − h2 t −n , ih2 t n + h2 t −n , e3 t n − f3 t −n , ie3 t n + f3 t −n , h3 t n − h3 t −n , ih3 t n + h3 t −n ; n ∈ ⊕ ic ⊕ id.
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4. Involution of the first kind φ x e1 7→ −f1 , f1 7→ −e1 , e2 7→ −f2 , f2 7→ −e2 , e3 7→ −t 2 f3 , f3 7→ −t −2 e3 , c 7→ c, d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n + e1 t −n , ie1 t n − e1 t −n , f1 t n + f1 t −n , if1 t n − f1 t −n , e2 t n + e2 t −n , ie2 t n − e2 t −n , f2 t n + f2 t −n , if2 t n − f2 t −n , e3 t n + e3 t −n+2 , ie3 t n − e3 t −n+2 , f3 t n + f3 t −n+2 ; if3 t n − f3 t −n+2 ;n ∈ ⊕ ic ⊕ id. 5. Involution of the first kind φ x e1 7→ −f1 , f1 7→ −e1 , e2 7→ −f2 , f2 7→ −e2 , e3 7→ t 2 f3 , f3 7→ t −2 e3 , c 7→ c, 4d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n − −1n+1 e1 t −n , ie1 t n + −1n+1 e1 t −n , f1 t n − −1n+1 f1 t −n , if1 t n + −1n+1 f1 t −n , e2 t n − −1n+1 e2 t −n , ie2 t n + −1n+1 e2 t −n , f2 t n − −1n+1 f2 t −n , if2 t n + −1n+1 f2 t −n , e3 t n + −1n+1 e3 t −n+2 , ie3 t n − −1n+1 e3 t −n+2 , f3 t n + −1n+1 f3 t −n+2 , if3 t n − −1n+1 f3 t −n+2 ; n ∈ ⊕ ic ⊕ id. 4.7. Vogan Diagrams Corresponding to Almost Compact Real Forms (1), (2) and (3) of ÓÌ3; t; t −1 ⊕ c ⊕ d Here, φ is the complexified Cartan involution for the real form. So its fixed point set will give us Ë0 . The maximally compact Cartan subalgebra È0 = ih1 + ih2 + ic + id = Ô0 . We define simple roots as follows: α1 h1 = 2; α1 h2 = −1; α1 c = 0; α1 d = 0; with corresponding root vector e1 . α2 h1 = −1; α2 h2 = 2; α2 c = 0; α2 d = 0; with corresponding root vector e2 . α3 h1 = −1; α3 h2 = −1; α3 c = 0; α3 d = 1; with corresponding root vector e3 . So the Vogan diagram corresponding to (1) is the following, because e1 is in Ë and e2 , e3 are in Ð:
.
The Vogan diagram corresponding to (2) is the following, because e1 , e2 are in Ë and e3 is in Ð:
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.
The Vogan diagram corresponding to (3) is the following, because e1 , e2 , e3 are in Ë:
.
4.8. Vogan Diagrams Corresponding to Almost Compact Real Forms (4) and (5) of ÓÌ3; t; t −1 ⊕ c ⊕ d The maximally compact Cartan subalgebra is
1
0
È0 = 0 0
0
0
0 + −i
1 0
−2
0
i
0
0
0 + ic + id:
0
0
We see from φ’s that
0
Ô0 = −i 0
i
0
0
0 + ic + id;
0
0
1
Á0 = 0 0
and
0 1 0
We define simple roots as follows:
1
α1 0 0
0 1 0
0
0 =3; −2 α1 c = 0;
0
i
0
α1 −i
0
0 =1;
0
0
0
α1 d = 0:
0
0 :
−2
226 The corresponding 1 α2 0 0
punita batra root vector is itf3 + e2 : 0 0 0 1 0 α2 =−3; −i 0 −2 0 α2 c = 0;
The corresponding 1 α3 0 0
0
0
0 = 1;
0
0
α2 d = 0:
root vector is it −1 e3 +f2 : 0 0 0 −i 1 0 = 0; α 3 0 0 −2 α3 c = 0;
i
i 0 0
0 0 =−2; 0
α3 d = 1:
The corresponding root vector is e1 t + f1 t − ih1 t. So the Vogan diagram for (4) is the following, because e1 t + f1 t − ih1 t is in Ð:
.
The Vogan diagram for (5) is the following, because e1 t + f1 t − ih1 t is in Ë:
.
5. MAIN RESULTS Using [1, Theorem 4.6], we prove the following corollary. Corollary 5.1. If two Cartan semi-involutions θ and θ0 are conjugate by an element τ of IntÇ, then corresponding compact real forms Õ0 and Õ00 are also conjugate by τ.
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Proof. Each compact real form has an associated conjugation θ that determines it. From the construction of Õ0 = Ë0 ⊕ iÐ0 and Õ00 = Ë00 ⊕ iÐ00 , we see that this conjugation is a Cartan involution of Ç . So from the assumption of the corollary θ and θ0 are conjugate by an element of IntÇ . Since IntÇ = IntÇ, the corollary follows. f
Notation. Aut Ç is the group of linear and semi-linear automorphisms of Ç. Let Ç be an affine Kac–Moody Lie algebra. Let φ be a semi-linear automorphism of Ç. Let ·; · be the invariant, bilinear form defined as in Section 2.4. Then φX; φY = X; Y for X; Y ∈ Ç if and only if φ is in f Aut Ç [1, Proposition 3.4, Part R2]. Remark 2. It is easy to check that φX; φY = X; Y for X; Y ∈ Ç, for all Cartan semi-involutions φ of Ç. Theorem 5.2. Suppose two almost compact real forms of a nontwisted affine Kac–Moody Lie algebra Ç have equivalent Vogan diagrams. Then these two real forms are isomorphic. Remark 3. We may assume that the Vogan diagrams actually match. Proof. Suppose Ç1 and Ç2 are two almost compact real forms of Ç. As they are both real forms of Ç, they have the same complexification Ç. Let Õ0 = Ë0 ⊕ iÐ0 and Õ00 = Ë00 ⊕ iÐ00 be the compact real forms of Ç corresponding to Ç1 and Ç2 , respectively. Let θ and θ0 be the respective corresponding Cartan semi-involutions. By [1, Theorem 4.6], θ and θ0 are conjugate by IntÇ. So the corresponding compact real forms Õ0 and Õ00 are also conjugate by Corollary 5.1. So there exists x ∈ IntÇ such that xÕ00 = Õ0 . The real form xÇ2 of Ç is isomorphic to Ç2 and has Cartan decomposition xË00 ⊕ xÐ00 . Since xË00 ⊕ ixÐ00 = xÕ00 = Õ0 , there is no loss of generality in assuming that Õ00 = Õ0 . Then θÕ0 = Õ0
and
θ0 Õ0 = Õ0 :
(5.1)
Case 1. Let Á0 and Á00 be 0. Let È1 = Ô0 and È2 = Ô00 be, respectively, compact Cartan subalgebras of Ç1 and Ç2 . As Õ00 = Õ0 , Ô0 and Ô00 ⊆ Õ0 . So Ô0 and Ô00 are conjugate by an interior automorphism by [1, Proposition 4.9c]; i.e., there exists k ∈ IntÕ0 such that kÔ00 = Ô0 . Again replacing Ç2 by kÇ2 and arguing as above we may assume that Ô00 = Ô0 . Then the complexification of È1 and È2 is the same, denoted by È. Now that the complexifications Ç and È have been aligned, the root systems are the same. Let 1+ and 1+0 be the positive root systems given in the respective triples. Then there exists an automorphism of the first kind k0 such that k0 1+0 = 1+ . (This is the point where we are making use of the
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hypothesis that both real forms are almost compact.) Replacing Ç2 by k0 Ç2 and arguing as above we may assume that 1+0 = 1+ from the outset. Using the conjugacy of compact real forms of Ç, we construct X X X t n eα − fα t −n + it n eα + fα t −n + it n hα + hα t −n Õ0 = n; α
+
X n; α
n; α
n
t hα − hα t
−n
n; α
+ ic + id:
We define Yn; α = t n eα and Y−n; −α = t −n fα . Since the automorphisms of 1+0 defined by θ and θ0 are the same, it follows that θH = θ0 H
∀H ∈ È:
(5.2)
Since Á0 and Á00 are 0, all of the roots are imaginary. So if α is unpainted, then θYn; α = Yn; α = θ0 Yn; α ;
where n is fixed:
(5.3)
And if α is painted, then θYn; α = −Yn; α = θ0 Yn; α ; Since Ç=È⊕
M α>0
Çα ⊕
where n is fixed: M α<0
(5.4)
Çα ;
it follows that È and Yn; α and Yn; −α for α simple imaginary will generate Ç. So θ = θ0 and θid = θ0 id = id, and Ë = Ë 0 and Ð = Ð0 . So Ç1 = Ç2 = Õ0 ∩ Ë ⊕ iÕ0 ∩ Ð: Case 2. Let Á0 and Á00 be nonzero. The imaginary simple roots are handled as in Case 1. Suppose α ∈ 1 is a complex simple root. We write θYn; α = an; α Yθn;α . We have θ Õ0 ∩ spanYn; α ; Y−n; −α ⊆ Õ0 ∩ span Yθn; α ; Yθ−n; −α : From the compact real form Õ0 , θ Yn; α − Y−n; −α + iYn; α + Y−n; −α ⊆ Yθn; α − Yθ−n; −α + i Yθn; α + Yθ−n; −α : If z = x + yi where x; y are real, then xYn; α − Y−n; −α + yiYn; α + ¯ −n; −α : Thus the expression θzYn; α − zY ¯ −n; −α = Y−n; −α = zYn; α − zY
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229
¯ −n; −α Yθ−n; −α must be of the form wYθn;α − wY ¯ θ−n;−α : zan; α Yθn; α − za So a¯ n; α = a−n; −α : Using Remark 2 we get an; α a−n; −α = an; α Yθn;α ; a−n; −α Yθ−n;−α = θYn; α ; θY−n; −α = Yn; α ; Y−n; −α = 1: So we get an; α = 1:
(5.5)
Again Yn; α = θ2 Yn; α = θan; α Yθn;α = an; α aθn;α Yn; α . Therefore an; α aθn;α = 1:
(5.6) 1/2
For each pair of complex simple roots α and θα, choose square roots an; α 1/2 and aθn;α so that 1/2
1/2 an; α aθn;α = 1:
This is possible by (5.6). Similarly θ0 Yn; α = bn; α Yθn;α with bn; α = 1; 1/2
(5.7)
1/2
and define bn; α and bθn;α for α and θα simple so that 1/2
1/2 bn; α bθn;α = 1:
(5.8)
So by (5.5) and (5.7), we can define H and H 0 in Õ0 ∩ È by the conditions that αH = αH 0 = 0 for α imaginary simple, while for α and θα complex simple, we define 1/2 exp 21 αH = an; α;
exp
1 2
exp exp
1 2
1/2
θαH = aθn;α ; 1 2
1/2 αH 0 = bn; α; 1/2
θαH 0 = bθn;α :
Let us verify the identity θ0 ◦ Ad exp 21 H − H 0 = Ad exp 21 H − H 0 ◦ θ:
(5.9)
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If α is imaginary simple, then (5.9) above holds on Yn; α : If α is complex simple, then 0 θ0 ◦ Ad exp 21 H − H 0 Yn; α = θ0 e1/2αH−H Yn; α 1/2 −1/2 = an; α bn; α bn; α Yθn; α −1/2 1/2 = an; α bn; α θYn; α 1/2
−1/2
= aθn;α bθn;α θYn; α
= Ad exp 21 H − H 0 ◦ θYn; α : Applying (5.9) to Ë and then to Ð, we see that Ad exp 21 H − H 0 Ë ⊆ Ë 0 ;
(5.10)
Ad exp 21 H − H 0 Ð ⊆ Ð0 :
(5.11)
Then equality must hold in (5.10) and (5.11). Since the element Ad exp 21 H − H 0 carries Õ0 to itself, it must carry Ë0 = Õ0 ∩ Ë to Ë00 = Õ0 ∩ Ë 0 ; and Ð0 = Õ0 ∩ Ð to Ð00 = Õ0 ∩ Ð0 . Hence it must carry Ç1 = Ë0 ⊕ Ð0 to Ç2 = Ë00 ⊕ Ð00 : 6. APPENDIX 1
6.1. Non-equivalent Vogan Diagrams of An for n ≥ 2 1
The largest root in An is e1 − en+1 . So αn+1 in An acts like −e1 − en+1 . 1 So the completed Dynkin diagram for An is
.
When there is a compact Cartan subalgebra, there are the following Vogan diagrams:
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231
.
Here 1; 2; : : : ; n − 1; n, and 00 give one non-equivalent Vogan diagram. Again 10 and n0 give one non-equivalent Vogan diagram, 20 and n − 10 give one non-equivalent Vogan diagram, and so on. So if n is even, then the number of non-equivalent Vogan diagrams is 1 + 1 + n2 . And if n is odd, then the number of non-equivalent Vogan diagrams is + 1. 1 + 1 + n−1 2 When there are complex simple roots, then if n = 2rr ≥ 1 is even, we get the following two non-equivalent Vogan diagrams:
.
So if n = 2r, we get 2 + r + 2 = r + 4 non-equivalent Vogan diagrams. If n = 2r − 1 r ≥ 2 is odd, then we get the following five non-equivalent Vogan diagrams:
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.
So if n = 2r − 1, we get 2 + r − 1 + 1 + 5 = r + 7 non-equivalent Vogan 1 diagrams. Using [1], the number of almost compact real forms in A2r−1 is 1 also r + 7. And the number of almost compact real forms in A2r is also r + 4. 1
6.2. Non-equivalent Vogan Diagrams of Bn
1
The largest root in Bn is e1 + e2 . So αn+1 in Bn acts like −e1 + e2 . So 1 the completed Dynkin diagram for Bn is
.
There is one non-trivial automorphism, the flip of left two roots:
.
When there is a non-trivial automorphism, then none or one of the α2 ; α3 ; : : : ; αn ’s is painted. So we get n non-equivalent Vogan diagrams.
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When there is a compact Cartan subalgebra, then there are the following Vogan diagrams:
.
We write 0 or 1 of the n roots α1 ; α2 ; : : : ; αn is painted. We get diagrams 0, 1; : : : ; n: This gives n + 1 diagrams. For the diagram l0 , take l and paint αn+1 . We add 10 in the list of non-equivalent Vogan diagrams. Thus our list now contains n + n + 1 + 1 = 2n + 2 diagrams. By inductive argument we can show that 00 ; 20 ; : : : ; n0 ; are redundant. Using [1], 1 the number of almost compact real forms in Bn is also 2n + 2. 1
6.3. Non-equivalent Vogan Diagrams of Cn
1
The largest root in Cn is 2e1 . So the completed Dynkin diagram for Cn is . There is one non-trivial automorphism, the reflection about the middle root. This gives the following two diagrams if n is even:
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.
And the following single diagram if n is odd: . When there is a compact Cartan subalgebra, then there are the following Vogan diagrams:
. Here 00 and n are the same. All of 10 through n − 10 reduces to 00 . Also 1 and n − 1 are the same, 2 and n − 2 are the same, and so on. We get 0; 1; : : : ; n2 ; n and n0 if n is even, and ; n and n0 if n is odd. 0; 1; : : : ; n−1 2 From here n = 2r − 1 gives r + 2, and n = 2r gives r + 3 non-equivalent Vogan diagrams. So for n = 2r − 1, we get a total of r + 3 non-equivalent Vogan diagrams. For n = 2r, we get total r + 5 non-equivalent Vogan diagrams. Using [1], 1 the number of almost compact real forms in C2r−1 is also r + 3. And the 1 number of almost compact real forms in C2r is also r + 5. 1
6.4. Non-equivalent Vogan Diagrams of D4
1
The largest root in D4 is e1 + e2 . So α5 in D4 acts like −e1 + e2 . So 1 the completed Dynkin diagram for D4 is
.
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When there are complex simple roots, we have the following five nonequivalent Vogan diagrams:
.
When there is a compact Cartan subalgebra, we have the following Vogan diagrams:
.
Here 00 and 10 are the same. And 00 and 2 are the same. So we have four non-equivalent Vogan diagrams in this case. So we have a total 1 of nine non-equivalent Vogan diagrams in D4 . Using [1], the number of 1 almost compact real forms in D4 is also nine.
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punita batra ACKNOWLEDGMENTS
This work was done at SUNY at Stony Brook, NY under the guidance of Prof. A. W. Knapp. I would like to thank Prof. A. W. Knapp for his help and useful suggestions in preparing this paper. I would also like to thank Prof. G. Rousseau for a private communication about Theorem 3.7.
REFERENCES 1. J. Bausch and G. Rousseau, “Algebres de Kac–Moody affines (Automorphismes et Formes Reelles),” Institut Elie Cartan, Nancy, 1989. 2. V. Back, N. Bardy, H. Ben Messaoud, and G. Rousseau, Formes presque-d´eploy´ees des alg`ebres de Kac–Moody: Classification et racines relatives, J. Algebra 171 (1995), 43–96. 3. V. G. Kac, “Infinite Dimensional Lie Algebras,” 3rd Ed., Cambridge Univ. Press, London/New York, 1990. 4. V. G. Kac and D. H. Peterson, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad Sci. U.S.A. 80 (1983), 1778–1782. 5. A. W. Knapp, “Lie Groups Beyond an Introduction,” Progress in Mathematics, Vol. 140, Birkh¨auser, Boston, 1996. 6. A. W. Knapp, A quick proof of the classification of simple real Lie algebras, Proc. Amer. Math. Soc. 124, No. 10 (1996), 3257–3259. 1 7. Z. Kobayashi, Automorphisms of finite order of the affine Lie algebra Al , Tsukuba J. Math. 10, No. 2 (1986), 269–283. 8. F. Levstein, A classification of involutive automorphisms of an affine Kac–Moody algebra, J. Algebra 114 (1988), 489–518. 9. G. Rousseau and H. Ben Messaoud, Private communication. 10. G. Rousseau, Almost split K-forms of Kac–Moody algebras, in “Infinite dimensional Lie Algebras and Groups,” Marseille, 1988 (V. G. Kac, ed.), pp. 70–85, Advanced Series in Mathematical Physics, Vol. 7, World Scientific, Singapore, 1989.
Invariants of Real Forms of Affine Kac-Moody Lie Algebras Punita Batra School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400 005, India E-mail: [email protected] Communicated by Georgia Benkart Received July 2, 1998
The theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finitedimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification. © 2000 Academic Press
1. INTRODUCTION Elie Cartan classified finite-dimensional real semisimple Lie algebras in a long 1913 paper. From then until the late 1970s, the proof of the classification was gradually simplified so that it took only about 25 pages. In a 1996 paper [6], Knapp modified some diagrams that had arisen in connection with the analysis of discrete series, calling them Vogan diagrams, and showed how Vogan diagrams could be used to prove the classification in under 3 pages. The theory of Vogan diagrams for real semisimple Lie algebras includes three theorems worth noting here. Details may be found in [5]. The first is that if two real semisimple Lie algebras, in the presence of some choices, have matching Vogan diagrams, then the Lie algebras are isomorphic. The second is that any diagram that looks like a Vogan diagram comes from 208 0021-8693/00 $35.00 Copyright © 2000 by Academic Press All rights of reproduction in any form reserved.
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some real semisimple Lie algebra and a system of choices. The third is that any Vogan diagram can be transformed, by changing the ordering, into a diagram in which each simple component has at most one noncompact imaginary root and that root occurs at most twice in the largest root of that simple component. As a result, the classification comes down to deciding which of some really easy Vogan diagrams correspond to isomorphic Lie algebras, that can be done readily by hand. This paper deals with indecomposable nontwisted affine Kac–Moody Lie algebras and their real forms. The real forms of this kind of Lie algebra have been classified as a result of work by Guy Rousseau and H. Ben Messaoud (see Section 3.6). What is sought is a theory of Vogan diagrams for them. The real forms for a given complex Lie algebra of this type break into two kinds, called “almost split” and “almost compact.” The almost split ones correspond in one-one fashion with all but one of the almost compact ones, the exceptional almost compact form being the so-called compact form. In this paper we introduce Vogan diagrams for the almost compact real forms of an indecomposable nontwisted affine Kac–Moody Lie algebra. The underlying Dynkin diagram is merely the completed diagram of an ordinary indecomposable Dynkin diagram. Here some two-element orbits will be marked, and some of the one-element orbits will be shaded. These markings and shadings are possible as a result of an extensive theory about the real forms that is already in place. The main theorem of this paper (Theorem 5.2) is that two real forms that have matching Vogan diagrams are isomorphic. It is not true that the ordering can be changed in such a way that at most one root is shaded. This fact makes it important to introduce the notion of equivalence of Vogan diagrams, in which two such diagrams are equivalent if they correspond to different orderings for the same real form. It is a fairly easy matter to work out representatives of the equivalence classes of Vogan diagrams, and these are listed for some nontwisted affine Kac–Moody Lie algebras in the Appendix. A count of the number of these diagrams yields a match with the number of almost compact real forms in the known classification, as it should. As is true in the finite-dimensional case, the theory of Vogan diagrams does more than give a faster proof of the classification. It allows one to identify, for a given Lie algebra, which standard Lie algebra is isomorphic to it, without tracking down isomorphisms whose existence is known but for which there are no formulas. This paper is organized as follows. Section 2 recites known facts about (complex) indecomposable nontwisted affine Kac–Moody Lie algebras. Section 3 records aspects of the classification of the real forms and then introduces Vogan diagrams, making the necessary justifications. Section 4 dis-
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cusses some low-rank examples, showing explicitly what almost compact real forms are and what their Vogan diagrams look like. Section 5 states and proves the main theorem. The Appendix gives representatives of equivalence classes of Vogan diagrams when the given complex simple Lie algebra is of type An , Bn , Cn , and D4 , proving that there are no others.
2. PRELIMINARIES 2.1. Affine Kac–Moody Lie Algebras Let I = 1; n + 1, n ∈ , be an interval in . Let A = aij i; j∈I be a matrix with integer coefficients. A is called a generalized Cartan matrix if it satisfies the following conditions: • aii = 2 for i = 1; : : : ; n + 1. • aij are nonpositive integers for i 6= j. • aij = 0 implies aji = 0. Let Ç = ÇA be the free complex Lie algebra generated by a basis of the standard Cartan subalgebra È and the elements ei ; fi for i ∈ I with the following defining relations [3]:
adei
ei ; fj = δij α∨i ;
i; j ∈ I
h; h0 = 0;
h; h0 ∈ È
h; ei = αi ; hei ;
i ∈ I; h ∈ È
h; fi = −αi ; hfi ;
i ∈ I; h ∈ È
1−aij
ej = 0;
i 6= j
adfi 1−aij fj = 0;
i 6= j:
The algebra Ç is called a Kac–Moody Lie algebra. The algebra Ç is an affine Kac–Moody Lie algebra if i.e., after the indices are reordered • A is an indecomposable matrix, A cannot be written in the form 0A1 A0 : 2
= 0.
n+1 n+1 • There exists a vector ai i=1 , with ai all positive such that Aai i=1
Then A is called a Cartan matrix of affine type. The affine algebra associated 1 with a generalized Cartan matrix of type Xn is called a nontwisted affine Kac–Moody Lie algebra.
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2.2. Dynkin Diagram Associated with a Generalized Cartan Matrix We associate with a generalized Cartan matrix A a graph SA called the Dynkin diagram of A as follows. If aij aji ≤ 4 and aij ≥ aji , the vertices i and j are connected by aij lines, and these lines are equipped with an arrow pointing toward i if aij > 1. If aij aji > 4, the vertices i and j are connected by a bold-faced line equipped with an ordered pair of integers aij ; aji . It is clear that A is indecomposable if and only if SA is a connected graph. A is determined by the Dynkin diagram SA and an enumeration of its vertices. 2.3. A Realization of Nontwisted Affine Kac–Moody Lie Algebras Let L = t; t −1 be the algebra P of Laurent polynomials in t. The residue of a Laurent polynomial P = k∈ ck t k (where all but a finite number of ck are 0) is defined as Res P = c−1 . ˚ be a finite-dimensional simple Lie algebra over of type Xn . Let Ç ˚ is an infinite-dimensional complex Lie algebra with Then L˚ Ç = L ⊗ Ç the bracket P ⊗ x; Q ⊗ y = PQ ⊗ x; y;
˚: P; Q ∈ Ly x; y ∈ Ç
˚ and exFix a nondegenerate, invariant, symmetric bilinear form ·; · in Ç Ç by tend this form to an L-valued form ·; ·t on L˚ P ⊗ x; Q ⊗ yt = PQx; y;
˚: P; Q ∈ Ly x; y ∈ Ç
Ç by The derivation t j dtd of L extends to L˚ tj
d dP P ⊗ x = t j ⊗ x; dt dt
˚: P ∈ Ly x ∈ Ç
; bt ; a; b ∈ L˚ Ç defines a 2-cocycle on L˚ Ç Therefore ψa; b = Res da dt (Co 1), (Co 2) [3, p. 97]. ˜ Ç the central extension of the Lie algebra L˚ We denote by L˚ Ç associ˜ Ç = L˚ ated with the cocycle ψ. Explicitly L˚ Ç ⊕ c with the bracket a + λc; b + µc = a; b + ψa; bc;
a; b ∈ L˚ Ç; λ; µ ∈ :
ˆ Ç the Lie algebra, which is obtained by adjoining Finally, we denote by L˚ ˆ Ç = ˜ Ç a derivation which acts on L˚ to L˚ Ç as t dtd and kills c. Explicitly L˚ L˚ Ç ⊕ c ⊕ d with the bracket defined by t k ⊗ x + λc + µd; t j ⊗ y + λ1 c + µ1 d = t j+k ⊗ x; y + µjt j ⊗ y − µ1 kt k ⊗ x + kδj; −k x; yc;
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ˆ Ç ˚y λ; µ; λ1 ; µ1 ∈ y k; j ∈ [3, Sect. 7.2.2]. [Ka, 7.2.2]. L˚ where x; y ∈ Ç is a nontwisted affine Kac–Moody Lie algebra associated with the affine 1 ˆ Ç, matrix A of type Xn [3, Theorem 7.4]. Here c = c is the center of L˚ ˆ Ç. and L˚ Ç ⊕ c is the derived algebra of L˚ Notation.
ˆ Ç = Ç and t k ⊗ x will be written as t k x. From now on L˚
2.4. Invariant Bilinear Form of Ç The normalized invariant form of Ç [3, Sect. 6.2] can be described as ˚ and extend ·; · to follows. Take the normalized invariant form ·; · on Ç all of Ç by P ⊗ x; Q ⊗ y = Res t −1 PQx; y;
˚y P; Q ∈ L x; y ∈ Ç
c ⊕ d; L˚ Ç = 0; c; c = d; d = 0; c; d = 1. It is a nondegenerate, symmetric, invariant, bilinear form of Ç [3, Sect. 7.5]. 2.5. Roots of Ç ˚ ⊂È ˚∗ be the root system of the finite-dimensional Lie algebra Ç ˚, Let 1 let α1 ; α2 ; : : : ; αn be the root basis, let H1 ; H2 ; : : : ; Hn the coroot basis, and let Ei , Fi i = 1; 2; : : : ; n the Chevalley generators. Let θ be the L ˚ . Let Ç ˚ = α∈1∪0 ˚α be the root highest root of the finite-root system 1 Ç ˚ ˚. Let ω ˚. We choose space decomposition of Ç ˚ be the Cartan involution of Ç 2 ˚θ such that F0 ; ωF ˚ 0 = − θ;θ and set E0 = −ωF ˚ 0 . Then we F0 ∈ Ç have, due to [3, Theorem 2.2e], E0 ; F0 = −θ∨ : ˚ + c + d is an ˆ Ç. Here È = È Now we come to the Lie algebra Ç = L˚ ˚∗ to a n + 2-dimensional commutative subalgebra in Ç. We extend λ ∈ È ∗ ˚ linear functional on È by setting λ; c = λ; d = 0, so that È is identified with a subspace in È∗ . We denote by δ the linear functional on È defined = 0, δ; d = 1. Set by δÈ+c ˚ en+1 = tE0 ;
fn+1 = t −1 F0
ei = Ei ; fi = Fi i = 1; 2; : : : ; n: Remark 1. In [3], Kac uses e0 for en+1 and f0 for fn+1 . We see from [3, Sect. 7.4.1] that en+1 ; fn+1 =
2 c − θ∨ : θ; θ
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Now we describe the root system and the root space decomposition of Ç with respect to È: ˚ ∪ jδ; where j ∈ − 0: 1 = jδ + γ; where j ∈ ; γ ∈ 1 M L˚ Çα ; Ç=È⊕ α∈1
˚ ˚γ , L˚ Çjδ = t j È. where L˚ Çjδ+γ = t j Ç We set π = α1 ; α2 ; : : : ; αn ; αn+1 = δ − θ ; 2 ∨ ∨ ∨ ∨ ∨ ∨ c−θ : π = α1 ; α2 ; : : : ; αn ; αn+1 = θ; θ Then by [3, Proposition 6.4a] A = α∨i ; αj n+1 i; j=1 :
In other words, È; π; π ∨ is a realization of the affine matrix A. 2.6. Cartan Involution of Ç The map ei 7→ −fi ; fi 7→ −ei ; i ∈ Iy h 7→ −h; h ∈ È can be uniquely extended to an involution ω of Ç. ω is called the Cartan involution of Ç. 2.7. Weyl Group For each i ∈ I, we define a fundamental reflection ri of the space È∗ by ri α = α − α; α∨i αi
∀α ∈ È∗ :
The subgroup W = GLÈ∗ generated by ri , i ∈ I is called the Weyl group of Ç. 3. CLASSIFICATION OF REAL FORMS OF NONTWISTED AFFINE KAC–MOODY LIE ALGEBRAS 3.1. Cartan and Borel Subalgebras of Ç We define a group G acting on the Lie algebra Ç via the adjoint representation Adx G 7→ AutÇ. It is generated by the subgroups Uα for α ∈ ±π and AdUα = expadÇα . A maximal adÇ -diagonalizable subalgebra of Ç is called a Cartan subalgebra. Every Cartan subalgebra of Ç is AdG-conjugate to the standard Cartan subalgebra È [4]. A Borel subalgebra of Ç is a maximal completely solvable subalgebra. It is always conjugated by AdG to b+ or b− , where L L + − b = È ⊕ α>0 Çα and b = È ⊕ α<0 Çα . But b+ and b− are not conjugated under AdG. So there are exactly two conjugacy classes of Borel subalgebra: the positive and the negative ones [4].
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3.2. Real Forms of Ç A real form of Ç is a Lie algebra Ç over such that there exists an isomorphism from Ç to Ç ⊗ [1]. If we replace with in the definition of Ç, we obtain a real form Ç which is called split. A real form of Ç corresponds to a semi-linear involution of Ç (this means to an automorphism τ ¯ for λ ∈ (semi-linear)). of Ç such that τ2 = Id and τλx = λτx A linear or semilinear automorphism of Ç is said to be of the first kind if it transforms a Borel subalgebra into a Borel subalgebra of the same sign. A linear or semilinear automorphism of Ç is said to be of the second kind if it transforms a Borel subalgebra into a Borel subalgebra of the opposite sign. Let Ç be a real form of Ç, and fix an isomorphism from Ç to Ç ⊗ . Then the Galois group 0 = Gal/ acts on Ç and the corresponding group G. We identify Ç with the fixed point set Ç0 . Definition 3.1. If 0 consists of first kind automorphisms we say that Ç is almost split. Otherwise if the nontrivial element of 0 is a second kind automorphism, we say that Ç is almost compact. (In other words, a real form Ç of Ç is said to be almost split if for each γ in 0 the action of γ on Ç is of first kind, otherwise Ç is said to be almost compact [10].) 3.3. Automorphisms of Ç Let È be the standard Cartan subalgebra of Ç. Paper [4] defines a group e which acts on G and Ç. In fact H b = AdH e is isomorphic to ∗ I and H e corresponds to hi i∈I , it acts on Çα by multiplication if the elementQh of H P by the scalar i hi ni if α = i ni αi . e ∝ G of interior automorphisms of Ç is the The group IntÇ = AdH e and G. Its derived group is the image of the semi-direct product of H adjoint group AdG (denoted by IntÇ0 ). As G is transitive on the Cartan subalgebra, the group IntÇ does not depend on the choice of È. We consider the group AutA of permutations ρ of I such that aρiρj = aij for i; j ∈ I. We regard AutA as a subgroup of AutÇ0 by requiring ρei = eρi , ρfi = fρi . The Cartan involution ω commutes with AutA. We define the group of outer automorphisms of Ç to be ExtÇ = Id; ω × AutA: 3.3.1. Transvections Following [1], we denote by Tr the set of transvections of Ç which means linear maps of Ç in Ç of the form φ = exp ψ, where ψ is a linear map of Ç into c, zero on Ç0 . So we have φx = x + ψx.
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Then according to [1, p. 187], we have the following decomposition of the group of automorphisms of Çx AutÇ = ExtÇ ∝ IntÇ × Tr:
(3.1)
If σ is an automorphism of Ç of the first kind, then ω ◦ σ is a second kind automorphism and conversely [1, p. 17]. From (3.1) it follows that any automorphism of Ç is either an automorphism of the first kind or an automorphism of the second kind. 3.4. Semi-linear Automorphisms Definition 3.2. Following [1, p. 190], we define by Aut Ç the group of automorphisms of Ç which are -linear or semi-linear. The group AutÇ is normal in Aut Ç and is of index 2. A semi-linear automorphism of order 2 of Ç is called a semi-involution of Ç. For any semi-involution σ 0 we have the decomposition Aut Ç = 1; σ 0 ∝ AutÇ. We denote by σn0 the conjugation of Ç with respect to the standard split real form. We call σn0 the standard normal semi-involution of Ç. This commutes with the standard Cartan involution ω. Let ω0s = σn0 ω = ωσn0 . Then ω0s is called the standard Cartan semiinvolution of Ç. Its algebra of fixed points is the standard compact real form of Ç. A Cartan semi-involution of Ç is a semi-involution ω0 conjugate to ω0s by an element of Aut Ç. Then ω0 is a semi-involution of the second kind and the associated real form is called the compact real form Ë1 of Ç. Let σ 0 be a semi-involution and ω0 be a Cartan semi-involution. Then using [1, Proposition 4.5] and after having supposed that σ 0 and ω0 stabilize the same Cartan subalgebra È, one may suppose by conjugating by G that ω0 commutes with σ 0 . Definition 3.3. Let σ 0 be a semi-involution of Ç of the second kind 0 and let Ç = Çσ be the corresponding almost compact real form. A Cartan semi-involution ω0 which commutes with σ 0 is called a Cartan semiinvolution for σ 0 or Ç . The involution σ = σ 0 ω0 (resp. its restriction ω0 to Ç ) is called a Cartan involution of σ 0 (resp. of Ç ). The algebra of fixed points Ë0 = Çσ is called a maximal compact subalgebra of Ç . We have the Cartan decomposition Ç = Ë0 ⊕ Ð0 and Ë1 = Ë0 ⊕ iÐ0 , where Ð0 is the eigenspace of ω0 for the eigen value −1 [1, pp. 199–200]. 3.5. Maximally Compact Cartan Subalgebra of an Almost Compact Form For X; Y in Ç we define X; Y σ = −X; σY
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where ·; · is the nondegenerate, symmetric, invariant, bilinear form defined as in Section 2.4. It is easy to check that it is positive definite and symmetric form. Lemma 3.4. If Ç is an almost compact real form of Ç and σ is a Cartan involution of Ç , then adX∗ = −adσX
for all X ∈ Ç ;
where adjoint :∗ is defined relative to the inner product ·; ·σ . Proof.
We have adσXY; Zσ = −σX; Y ; σZ = Y; σX; σZ = Y; σX; σZ = Y; σX; Z = −Y; adXZσ = −adX∗ Y; Zσ :
Proposition 3.5. Let Ô0 be a maximal abelian subspace of Ë0 . Then È0 = ZÇ Ô0 is a σ-stable Cartan subalgebra of the almost compact real form Ç of the form È0 = Ô0 ⊕ Á0 , with Á0 ⊆ Ð0 . Proof. È0 = ZÇ Ô0 = x ∈ Ç x; Ô0 = 0: If x ∈ È0 and y ∈ Ô0 , then σx; y = σx; σσy = σx; σy = σx; y = 0: So È0 is σ-stable and hence is a vector space direct sum È0 = Ô0 ⊕ Á0 , where Á0 = È0 ∩ Ð0 . We have È0 ; È0 = Á0 ; Á0 and Á0 ; Á0 ⊆ Ô0 , since Á0 ⊆ Ð0 and È0 ∩ Ë0 = Ô0 . Thus È0 ; È0 is abelian. Since È0 is σ-stable and finite-dimensional, using the inner product ·; ·σ on È0 , it is easy to check that any ideal in È0 has a complementary ideal. So È0 is reductive. By [5, Corollary 1.53], È0 ; È0 is semisimple. Thus the semisimple and abelian Lie algebra È0 ; È0 must be zero. Consequently È0 is abelian.
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It is clear that È0 is maximal abelian in Ç . Now if x ∈ Ô0 , then by Lemma 3.4 we have adx∗ = −adx. So the members of adÇ Ô0 are skew adjoint. If x ∈ Á0 , then by Lemma 3.4 we have adx∗ = adx. So the members of adÇ Á0 are self-adjoint. Since Ô0 commutes with Á0 , adÈ0 is certainly diagonable on Ç ; hence È0 is a Cartan subalgebra of Ç . We say that a σ-stable Cartan subalgebra of an almost compact real form Ç of the form È0 = Ô0 ⊕ Á0 ; with Ô0 ⊆ Ë0 and Á0 ⊆ Ð0 , is maximally compact if dim Ô0 is as large as possible. Proposition 3.6. A maximally compact Cartan subalgebra È0 = Ô0 ⊕ Á0 of an almost compact real form Ç has the property that all of the roots are real on Á0 and imaginary on Ô0 . Proof. The values of the roots on a member H of È0 are the eigenvalues of adH. For H ∈ Á0 , these are real since adH is self-adjoint by Lemma 3.4. For H ∈ Ô0 , these are purely imaginary since adH is skew adjoint by Lemma 3.4. 3.6. Classification of Real Forms Following [2, Theorem 4.4], we obtain under AutÇ a one-to-one correspondence between the conjugacy classes of (linear) involutions of Ç of the second kind and the conjugacy classes of almost split real forms of Ç. Theorem 3.7 [9]. We consider 1. The semi-involutions σ 0 , of Ç of the second kind. 2. The involutions θ, of Ç of the first kind. 3. The relation σ 0 ≈ θ if and only if (a) ω0 = θσ 0 = σ 0 θ is a Cartan semi-involution. (b) θ and σ 0 stabilize the same Cartan subalgebra È. (c) È is contained in a minimal σ 0 -stable positive parabolic subalgebra. Then this relation induces a bijection between the conjugacy classes under AutÇ of semi-involutions of the second kind and conjugacy classes of involutions of the first kind. Thus we obtain under AutÇ a one-to-one correspondence between the conjugacy classes of (linear) involutions of Ç of the first kind (including identity) and the conjugacy classes of almost compact real forms of Ç. The compact real form is unique; it corresponds to identity. A classification of involutions of an affine Kac–Moody Lie algebra is given by Levstein [8]. A classification of automorphisms of finite order of affine Kac–Moody Lie 1 algebra of type An is given by Kobayashi [7].
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3.7. Vogan Diagrams Let Ç be an almost compact real form of Ç corresponding to the semiinvolution σ 0 of Ç of the second kind. Let σ be the Cartan-involution of Ç , and let Ç = Ë0 ⊕ Ð0 be the corresponding Cartan decomposition. We write Ç = Ë ⊕ Ð for the complexification of the Cartan decomposition. Let Ô0 be a maximal abelian subspace of Ë0 . Then È0 = ZÇ Ô0 is a σstable Cartan subalgebra of Ç of the form È0 = Ô0 ⊕ Á0 , with Á0 ⊆ Ð0 by Proposition 3.5. This È0 is a maximally compact Cartan subalgebra of Ç because Ô0 is as large as possible. By Proposition 3.6, we note that the roots of Ç; È are imaginary on Ô0 and real on Á0 . A root is real if it takes real values on È0 (i.e., it vanishes on Ô0 ), imaginary if it takes purely imaginary values on È0 (i.e., it vanishes on Á0 ), and complex otherwise. For any root α, σα is the root σαH = ασ −1 H. If α is imaginary, then σα = α. Thus Çα is σ-stable, and we have Çα = Çα ∩ Ë ⊕ Çα ∩ Ð. Since Çα is 1-dimensional, Çα ⊆ Ë or Çα ⊆ Ð. We call an imaginary root α compact if Çα ⊆ Ë, noncompact if Çα ⊆ Ð. Proposition 3.8. Let È0 be a σ-stable Cartan subalgebra of Ç . Then there are no real roots if and only if È0 is maximally compact. Proof. The Cayley transform construction dα [5, p. 334] tells us that if È0 has a real root α, then we can construct a new Cartan subalgebra whose Ô0 is strictly larger. Then È0 could not have been maximally compact. For the converse, we write È0 = Ô0 ⊕ Á0 , and let 1 = 1Ç; È be the set of roots. From the expansion M Çα : (3.2) Ç=È⊕ α∈1
Suppose there are no real roots. Then we obtain from (3.2) M M Çα = È ⊕ Çα = È ZÇ Ô0 = È ⊕ α∈1 αÔ0 =0
α∈1 αreal
and Ë0 ∩ ZÇ Ô0 = Ë0 ∩ È = Ô0 . Therefore Ô0 is maximal abelian in Ë0 , and È0 is maximally compact. Let Ç be an almost compact real form of Ç. Let σ be the Cartan involution and let Ç = Ë0 ⊕ Ð0 be the corresponding Cartan decomposition. Let È0 = Ô0 ⊕ Á0 be a maximally compact σ-stable Cartan subalgebra of Ç , with complexification È = Ô ⊕ Á. We let 1 = 1Ç; È be the set of roots. By Proposition 3.8, there are no real roots, i.e., no roots that vanish on Ô. We choose a positive system 1+ for 1 that takes iÔ0 before Á0 [5, p. 339]. Since σ is +1 on Ô0 and −1 on Á0 and since there are no real roots, σ1+ = 1+ . Therefore σ permutes the simple roots. It must fix the simple roots that are imaginary and permute in 2-cycles the simple roots that are complex.
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By the Vogan diagram of the triple Ç ; È0 ; 1+ , we mean the Dynkin diagram of 1+ with the 2-element orbits under σ labeled by an arrow and with the 1-element orbits painted or not, depending on whether the corresponding imaginary simple root is noncompact or compact. 3.7.1. Equivalence of Vogan Diagrams We define the equivalence of Vogan diagrams to be the equivalence relation generated by the following two operations: 1.
Application of an automorphism of the Dynkin diagram.
2. Change in the positive system by reflection in a simple, noncompact root, i.e., by a vertex which is colored in the Vogan diagram. As a consequence of reflection by a simple, noncompact root α, the rule for single and triple lines is that we leave α colored and its immediate neighbor is changed to the opposite color. The rule for double lines is that if α is the smaller root, then there is no change in the color of its immediate neighbor, but we leave α colored. If α is the bigger root, then we leave α colored and its immediate neighbor is changed to the opposite color. If two Vogan diagrams are not equivalent to each other, we call them non-equivalent. 4. EXAMPLES 1
4.1. Real Forms of A1 = ÓÌ2; t; t −1 ⊕ c ⊕ d Here, the generalized Cartan matrix is Dynkin diagram is
2 −2
−2 2
;
and the corresponding
. The generators are e1 = f1 =
0
1
0
0
0
0
1
0
! = e;
e2 = t
! = f;
f2 = t
−1
0
0
1
0
! = tf;
0
1
0
0
! = t −1 e:
1
And the Cartan subalgebra of A1 is h ⊕ c ⊕ d;
where h =
1
0
0
−1
! :
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The Cartan semi-involution ω0s of A1 is ω0s x t n e 7→ −t −n f t n f 7→ −t −n e t n h 7→ −t −n h it n e 7→ it −n f it n f 7→ it −n e it n h 7→ it −n h c 7→ −c d 7→ −d ic 7→ ic id 7→ id: 4.2. Almost Compact Real Forms of ÓÌ2; t; t −1 ⊕ c ⊕ d Using the involutions given by Kobayashi [7], we get the following four almost compact real forms of ÓÌ2; t; t −1 ⊕ c ⊕ d. 1. Involution of the first kind φ x e 7→ −e; f 7→ −f; tf 7→ −tf; t −1 e 7→ −t −1 e; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e + f; ie − f ; ih; tf + t −1 e; itf − t −1 e; th − t −1 h; ith + t −1 h ⊕ ic ⊕ id: 2. The involution of the first kind φ x e 7→ e; f 7→ f; tf 7→ −tf; t −1 e 7→ −t −1 e; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e − f; ie + f ; ih; tf + t −1 e; itf − t −1 e; th + t −1 h; ith − t −1 h ⊕ ic ⊕ id: 3. The involution of the first kind φ x e 7→ tf; f 7→ t −1 e; tf 7→ e; t e 7→ f; c 7→ c; d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e − t −1 e; ie + t −1 e; f − tf; if + tf ; h; th + t −1 h; ith − t −1 h −1
⊕ ic ⊕ id:
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4. The compact real form, corresponding to Cartan semi-involution ω0s , is given by n t e − t −n f; it n e + t −n f ; t n f − t −n e; it n f + t −n e; t n h − t −n h; it n h + t −n hy n ∈ ⊕ ic ⊕ id: 4.3. Vogan Diagrams Corresponding to Almost Compact Forms (1), (2) and (4) of ÓÌ2; t; t −1 ⊕ c ⊕ d Here, φ is the complexified Cartan involution for the real form. So its fixed point set will give us Ë0 . The maximally compact Cartan subalgebra È0 = ih + ic + id = Ô0 . We define simple roots as follows: α1 h = 2; α1 c = 0; α1 d = 0: The corresponding root vector is e. α2 h = −2; α2 c = 0; α2 d = 1: The corresponding root vector is tf . So the Vogan diagram corresponding to (1) is the following, because e and tf are in Ð: . The Vogan diagram corresponding to (2) is the following, because e is in Ë and tf is in Ð: . The Vogan diagram corresponding to (4) is the following, because e and tf are in Ë: .
4.4. Vogan Diagrams Corresponding to Almost Compact Forms (3) of ÓÌ2; t; t −1 ⊕ c ⊕ d The maximally compact Cartan subalgebra is È0 = h + ic + id = Ô0 ⊕ Á0 , where Ô0 = ic + id and Á0 = h. We define simple roots as follows: α1 h = 2; α2 h = −2;
α1 c = 0; α2 c = 0;
α1 d = 1: α2 d = 1:
So the Vogan diagram is the following, because α1 and α2 are complex roots: .
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4.5. Real Forms of A2 = ÓÌ3; t; t −1 ⊕ c ⊕ d Here, the generalized Cartan matrix and corresponding Dynkin diagram are .
The generators are 0 1 0 e1 = 0 0 0 ; 0
0
0 f1 = 1 0
0 0
0 0;
0
0
0
0 0 0 e2 = 0 0 1 ; 0 0 0 0 0 0 f2 = 0 0 0 ; 0 1 0
0 0 0 e3 = t 0 0 0 ; 1 0 0 0 0 1 f3 = t −1 0 0 0 : 0 0 0
1
The Cartan subalgebra of A2 is h1 + h2 ⊕ c ⊕ d; where
1 0 0 h1 = 0 −1 0 ; 0 0 0
0 0 0 0; h2 = 0 1 0 0 −1 1
−1 0 0 h3 = 0 0 0 : 0 0 1
The Cartan semi-involution ω0s of A2 is the following: ω0s x t n e1 7→ −t −n f1 t n e2 7→ −t −n f2 t n e3 7→ −t −n f3 t n f1 7→ −t −n e1 t n f2 7→ −t −n e2 t n f3 7→ −t −n e3 t n h1 7→ −t −n h1 t n h2 7→ −t −n h2 t n h3 7→ −t −n h3
real forms of affine lie algebras
223
it n e1 7→ it −n f1 it n e2 7→ it −n f2 it n e3 7→ it −n f3 it n f1 7→ it −n e1 it n f2 7→ it −n e2 it n f3 7→ it −n e3 it n h1 7→ it −n h1 it n h2 7→ it −n h2 it n h3 7→ it −n h3 c 7→ −c d 7→ −d ic 7→ ic id 7→ id: 4.6. Almost Compact Real Forms of ÓÌ3; t; t −1 ⊕ c ⊕ d Using the involutions given by Kobayashi [7], we get the following five almost compact real forms of ÓÌ3; t; t −1 ⊕ c ⊕ d. 1. Involution of the first kind φ x e1 7→ e1 , f1 7→ f1 , e2 7→ −e2 , f2 7→ −f2 , e3 7→ −e3 , f3 7→ −f3 , c 7→ c, d 7→ d. The corresponding semiinvolution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n − f1 t −n , ie1 t n + f1 t −n , f1 t n − e1 t −n , if1 t n + e1 t −n , e2 t n + f2 t −n , ie2 t n − f2 t −n , f2 t n + e2 t −n , if2 t n − e2 t −n , e3 t n + f3 t −n , ie3 t n − f3 t −n y n ∈ ⊕ ic ⊕ id. 2. Involution of the first kind φ x e1 7→ e1 , f1 7→ f1 , e2 7→ e2 , f2 7→ f2 , e3 7→ −e3 , f3 7→ −f3 , c 7→ c, d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n + −1n+1 f1 t −n , ie1 t n − −1n+1 f1 t −n , e2 t n + −1n+1 f2 t −n , ie2 t n − −1n+1 f2 t −n , e3 t n − −1n+1 f3 t −n , ie3 t n + −1n+1 f3 t −n ; n ∈ ⊕ic ⊕ id. 3. The compact real form, corresponding to the Cartan semiinvolution ω0s ; is given by e1 t n − f1 t −n , ie1 t n + f1 t −n , h1 t n − h1 t −n , ih1 t n + h1 t −n , e2 t n − f2 t −n , ie2 t n + f2 t −n , h2 t n − h2 t −n , ih2 t n + h2 t −n , e3 t n − f3 t −n , ie3 t n + f3 t −n , h3 t n − h3 t −n , ih3 t n + h3 t −n ; n ∈ ⊕ ic ⊕ id.
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4. Involution of the first kind φ x e1 7→ −f1 , f1 7→ −e1 , e2 7→ −f2 , f2 7→ −e2 , e3 7→ −t 2 f3 , f3 7→ −t −2 e3 , c 7→ c, d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n + e1 t −n , ie1 t n − e1 t −n , f1 t n + f1 t −n , if1 t n − f1 t −n , e2 t n + e2 t −n , ie2 t n − e2 t −n , f2 t n + f2 t −n , if2 t n − f2 t −n , e3 t n + e3 t −n+2 , ie3 t n − e3 t −n+2 , f3 t n + f3 t −n+2 ; if3 t n − f3 t −n+2 ;n ∈ ⊕ ic ⊕ id. 5. Involution of the first kind φ x e1 7→ −f1 , f1 7→ −e1 , e2 7→ −f2 , f2 7→ −e2 , e3 7→ t 2 f3 , f3 7→ t −2 e3 , c 7→ c, 4d 7→ d. The corresponding semi-involution of the second kind is ω0s ◦ φ. The corresponding almost compact real form is generated by e1 t n − −1n+1 e1 t −n , ie1 t n + −1n+1 e1 t −n , f1 t n − −1n+1 f1 t −n , if1 t n + −1n+1 f1 t −n , e2 t n − −1n+1 e2 t −n , ie2 t n + −1n+1 e2 t −n , f2 t n − −1n+1 f2 t −n , if2 t n + −1n+1 f2 t −n , e3 t n + −1n+1 e3 t −n+2 , ie3 t n − −1n+1 e3 t −n+2 , f3 t n + −1n+1 f3 t −n+2 , if3 t n − −1n+1 f3 t −n+2 ; n ∈ ⊕ ic ⊕ id. 4.7. Vogan Diagrams Corresponding to Almost Compact Real Forms (1), (2) and (3) of ÓÌ3; t; t −1 ⊕ c ⊕ d Here, φ is the complexified Cartan involution for the real form. So its fixed point set will give us Ë0 . The maximally compact Cartan subalgebra È0 = ih1 + ih2 + ic + id = Ô0 . We define simple roots as follows: α1 h1 = 2; α1 h2 = −1; α1 c = 0; α1 d = 0; with corresponding root vector e1 . α2 h1 = −1; α2 h2 = 2; α2 c = 0; α2 d = 0; with corresponding root vector e2 . α3 h1 = −1; α3 h2 = −1; α3 c = 0; α3 d = 1; with corresponding root vector e3 . So the Vogan diagram corresponding to (1) is the following, because e1 is in Ë and e2 , e3 are in Ð:
.
The Vogan diagram corresponding to (2) is the following, because e1 , e2 are in Ë and e3 is in Ð:
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225
.
The Vogan diagram corresponding to (3) is the following, because e1 , e2 , e3 are in Ë:
.
4.8. Vogan Diagrams Corresponding to Almost Compact Real Forms (4) and (5) of ÓÌ3; t; t −1 ⊕ c ⊕ d The maximally compact Cartan subalgebra is
1
0
È0 = 0 0
0
0
0 + −i
1 0
−2
0
i
0
0
0 + ic + id:
0
0
We see from φ’s that
0
Ô0 = −i 0
i
0
0
0 + ic + id;
0
0
1
Á0 = 0 0
and
0 1 0
We define simple roots as follows:
1
α1 0 0
0 1 0
0
0 =3; −2 α1 c = 0;
0
i
0
α1 −i
0
0 =1;
0
0
0
α1 d = 0:
0
0 :
−2
226 The corresponding 1 α2 0 0
punita batra root vector is itf3 + e2 : 0 0 0 1 0 α2 =−3; −i 0 −2 0 α2 c = 0;
The corresponding 1 α3 0 0
0
0
0 = 1;
0
0
α2 d = 0:
root vector is it −1 e3 +f2 : 0 0 0 −i 1 0 = 0; α 3 0 0 −2 α3 c = 0;
i
i 0 0
0 0 =−2; 0
α3 d = 1:
The corresponding root vector is e1 t + f1 t − ih1 t. So the Vogan diagram for (4) is the following, because e1 t + f1 t − ih1 t is in Ð:
.
The Vogan diagram for (5) is the following, because e1 t + f1 t − ih1 t is in Ë:
.
5. MAIN RESULTS Using [1, Theorem 4.6], we prove the following corollary. Corollary 5.1. If two Cartan semi-involutions θ and θ0 are conjugate by an element τ of IntÇ, then corresponding compact real forms Õ0 and Õ00 are also conjugate by τ.
real forms of affine lie algebras
227
Proof. Each compact real form has an associated conjugation θ that determines it. From the construction of Õ0 = Ë0 ⊕ iÐ0 and Õ00 = Ë00 ⊕ iÐ00 , we see that this conjugation is a Cartan involution of Ç . So from the assumption of the corollary θ and θ0 are conjugate by an element of IntÇ . Since IntÇ = IntÇ, the corollary follows. f
Notation. Aut Ç is the group of linear and semi-linear automorphisms of Ç. Let Ç be an affine Kac–Moody Lie algebra. Let φ be a semi-linear automorphism of Ç. Let ·; · be the invariant, bilinear form defined as in Section 2.4. Then φX; φY = X; Y for X; Y ∈ Ç if and only if φ is in f Aut Ç [1, Proposition 3.4, Part R2]. Remark 2. It is easy to check that φX; φY = X; Y for X; Y ∈ Ç, for all Cartan semi-involutions φ of Ç. Theorem 5.2. Suppose two almost compact real forms of a nontwisted affine Kac–Moody Lie algebra Ç have equivalent Vogan diagrams. Then these two real forms are isomorphic. Remark 3. We may assume that the Vogan diagrams actually match. Proof. Suppose Ç1 and Ç2 are two almost compact real forms of Ç. As they are both real forms of Ç, they have the same complexification Ç. Let Õ0 = Ë0 ⊕ iÐ0 and Õ00 = Ë00 ⊕ iÐ00 be the compact real forms of Ç corresponding to Ç1 and Ç2 , respectively. Let θ and θ0 be the respective corresponding Cartan semi-involutions. By [1, Theorem 4.6], θ and θ0 are conjugate by IntÇ. So the corresponding compact real forms Õ0 and Õ00 are also conjugate by Corollary 5.1. So there exists x ∈ IntÇ such that xÕ00 = Õ0 . The real form xÇ2 of Ç is isomorphic to Ç2 and has Cartan decomposition xË00 ⊕ xÐ00 . Since xË00 ⊕ ixÐ00 = xÕ00 = Õ0 , there is no loss of generality in assuming that Õ00 = Õ0 . Then θÕ0 = Õ0
and
θ0 Õ0 = Õ0 :
(5.1)
Case 1. Let Á0 and Á00 be 0. Let È1 = Ô0 and È2 = Ô00 be, respectively, compact Cartan subalgebras of Ç1 and Ç2 . As Õ00 = Õ0 , Ô0 and Ô00 ⊆ Õ0 . So Ô0 and Ô00 are conjugate by an interior automorphism by [1, Proposition 4.9c]; i.e., there exists k ∈ IntÕ0 such that kÔ00 = Ô0 . Again replacing Ç2 by kÇ2 and arguing as above we may assume that Ô00 = Ô0 . Then the complexification of È1 and È2 is the same, denoted by È. Now that the complexifications Ç and È have been aligned, the root systems are the same. Let 1+ and 1+0 be the positive root systems given in the respective triples. Then there exists an automorphism of the first kind k0 such that k0 1+0 = 1+ . (This is the point where we are making use of the
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hypothesis that both real forms are almost compact.) Replacing Ç2 by k0 Ç2 and arguing as above we may assume that 1+0 = 1+ from the outset. Using the conjugacy of compact real forms of Ç, we construct X X X t n eα − fα t −n + it n eα + fα t −n + it n hα + hα t −n Õ0 = n; α
+
X n; α
n; α
n
t hα − hα t
−n
n; α
+ ic + id:
We define Yn; α = t n eα and Y−n; −α = t −n fα . Since the automorphisms of 1+0 defined by θ and θ0 are the same, it follows that θH = θ0 H
∀H ∈ È:
(5.2)
Since Á0 and Á00 are 0, all of the roots are imaginary. So if α is unpainted, then θYn; α = Yn; α = θ0 Yn; α ;
where n is fixed:
(5.3)
And if α is painted, then θYn; α = −Yn; α = θ0 Yn; α ; Since Ç=È⊕
M α>0
Çα ⊕
where n is fixed: M α<0
(5.4)
Çα ;
it follows that È and Yn; α and Yn; −α for α simple imaginary will generate Ç. So θ = θ0 and θid = θ0 id = id, and Ë = Ë 0 and Ð = Ð0 . So Ç1 = Ç2 = Õ0 ∩ Ë ⊕ iÕ0 ∩ Ð: Case 2. Let Á0 and Á00 be nonzero. The imaginary simple roots are handled as in Case 1. Suppose α ∈ 1 is a complex simple root. We write θYn; α = an; α Yθn;α . We have θ Õ0 ∩ spanYn; α ; Y−n; −α ⊆ Õ0 ∩ span Yθn; α ; Yθ−n; −α : From the compact real form Õ0 , θ Yn; α − Y−n; −α + iYn; α + Y−n; −α ⊆ Yθn; α − Yθ−n; −α + i Yθn; α + Yθ−n; −α : If z = x + yi where x; y are real, then xYn; α − Y−n; −α + yiYn; α + ¯ −n; −α : Thus the expression θzYn; α − zY ¯ −n; −α = Y−n; −α = zYn; α − zY
real forms of affine lie algebras
229
¯ −n; −α Yθ−n; −α must be of the form wYθn;α − wY ¯ θ−n;−α : zan; α Yθn; α − za So a¯ n; α = a−n; −α : Using Remark 2 we get an; α a−n; −α = an; α Yθn;α ; a−n; −α Yθ−n;−α = θYn; α ; θY−n; −α = Yn; α ; Y−n; −α = 1: So we get an; α = 1:
(5.5)
Again Yn; α = θ2 Yn; α = θan; α Yθn;α = an; α aθn;α Yn; α . Therefore an; α aθn;α = 1:
(5.6) 1/2
For each pair of complex simple roots α and θα, choose square roots an; α 1/2 and aθn;α so that 1/2
1/2 an; α aθn;α = 1:
This is possible by (5.6). Similarly θ0 Yn; α = bn; α Yθn;α with bn; α = 1; 1/2
(5.7)
1/2
and define bn; α and bθn;α for α and θα simple so that 1/2
1/2 bn; α bθn;α = 1:
(5.8)
So by (5.5) and (5.7), we can define H and H 0 in Õ0 ∩ È by the conditions that αH = αH 0 = 0 for α imaginary simple, while for α and θα complex simple, we define 1/2 exp 21 αH = an; α;
exp
1 2
exp exp
1 2
1/2
θαH = aθn;α ; 1 2
1/2 αH 0 = bn; α; 1/2
θαH 0 = bθn;α :
Let us verify the identity θ0 ◦ Ad exp 21 H − H 0 = Ad exp 21 H − H 0 ◦ θ:
(5.9)
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If α is imaginary simple, then (5.9) above holds on Yn; α : If α is complex simple, then 0 θ0 ◦ Ad exp 21 H − H 0 Yn; α = θ0 e1/2αH−H Yn; α 1/2 −1/2 = an; α bn; α bn; α Yθn; α −1/2 1/2 = an; α bn; α θYn; α 1/2
−1/2
= aθn;α bθn;α θYn; α
= Ad exp 21 H − H 0 ◦ θYn; α : Applying (5.9) to Ë and then to Ð, we see that Ad exp 21 H − H 0 Ë ⊆ Ë 0 ;
(5.10)
Ad exp 21 H − H 0 Ð ⊆ Ð0 :
(5.11)
Then equality must hold in (5.10) and (5.11). Since the element Ad exp 21 H − H 0 carries Õ0 to itself, it must carry Ë0 = Õ0 ∩ Ë to Ë00 = Õ0 ∩ Ë 0 ; and Ð0 = Õ0 ∩ Ð to Ð00 = Õ0 ∩ Ð0 . Hence it must carry Ç1 = Ë0 ⊕ Ð0 to Ç2 = Ë00 ⊕ Ð00 : 6. APPENDIX 1
6.1. Non-equivalent Vogan Diagrams of An for n ≥ 2 1
The largest root in An is e1 − en+1 . So αn+1 in An acts like −e1 − en+1 . 1 So the completed Dynkin diagram for An is
.
When there is a compact Cartan subalgebra, there are the following Vogan diagrams:
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231
.
Here 1; 2; : : : ; n − 1; n, and 00 give one non-equivalent Vogan diagram. Again 10 and n0 give one non-equivalent Vogan diagram, 20 and n − 10 give one non-equivalent Vogan diagram, and so on. So if n is even, then the number of non-equivalent Vogan diagrams is 1 + 1 + n2 . And if n is odd, then the number of non-equivalent Vogan diagrams is + 1. 1 + 1 + n−1 2 When there are complex simple roots, then if n = 2rr ≥ 1 is even, we get the following two non-equivalent Vogan diagrams:
.
So if n = 2r, we get 2 + r + 2 = r + 4 non-equivalent Vogan diagrams. If n = 2r − 1 r ≥ 2 is odd, then we get the following five non-equivalent Vogan diagrams:
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.
So if n = 2r − 1, we get 2 + r − 1 + 1 + 5 = r + 7 non-equivalent Vogan 1 diagrams. Using [1], the number of almost compact real forms in A2r−1 is 1 also r + 7. And the number of almost compact real forms in A2r is also r + 4. 1
6.2. Non-equivalent Vogan Diagrams of Bn
1
The largest root in Bn is e1 + e2 . So αn+1 in Bn acts like −e1 + e2 . So 1 the completed Dynkin diagram for Bn is
.
There is one non-trivial automorphism, the flip of left two roots:
.
When there is a non-trivial automorphism, then none or one of the α2 ; α3 ; : : : ; αn ’s is painted. So we get n non-equivalent Vogan diagrams.
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When there is a compact Cartan subalgebra, then there are the following Vogan diagrams:
.
We write 0 or 1 of the n roots α1 ; α2 ; : : : ; αn is painted. We get diagrams 0, 1; : : : ; n: This gives n + 1 diagrams. For the diagram l0 , take l and paint αn+1 . We add 10 in the list of non-equivalent Vogan diagrams. Thus our list now contains n + n + 1 + 1 = 2n + 2 diagrams. By inductive argument we can show that 00 ; 20 ; : : : ; n0 ; are redundant. Using [1], 1 the number of almost compact real forms in Bn is also 2n + 2. 1
6.3. Non-equivalent Vogan Diagrams of Cn
1
The largest root in Cn is 2e1 . So the completed Dynkin diagram for Cn is . There is one non-trivial automorphism, the reflection about the middle root. This gives the following two diagrams if n is even:
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.
And the following single diagram if n is odd: . When there is a compact Cartan subalgebra, then there are the following Vogan diagrams:
. Here 00 and n are the same. All of 10 through n − 10 reduces to 00 . Also 1 and n − 1 are the same, 2 and n − 2 are the same, and so on. We get 0; 1; : : : ; n2 ; n and n0 if n is even, and ; n and n0 if n is odd. 0; 1; : : : ; n−1 2 From here n = 2r − 1 gives r + 2, and n = 2r gives r + 3 non-equivalent Vogan diagrams. So for n = 2r − 1, we get a total of r + 3 non-equivalent Vogan diagrams. For n = 2r, we get total r + 5 non-equivalent Vogan diagrams. Using [1], 1 the number of almost compact real forms in C2r−1 is also r + 3. And the 1 number of almost compact real forms in C2r is also r + 5. 1
6.4. Non-equivalent Vogan Diagrams of D4
1
The largest root in D4 is e1 + e2 . So α5 in D4 acts like −e1 + e2 . So 1 the completed Dynkin diagram for D4 is
.
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When there are complex simple roots, we have the following five nonequivalent Vogan diagrams:
.
When there is a compact Cartan subalgebra, we have the following Vogan diagrams:
.
Here 00 and 10 are the same. And 00 and 2 are the same. So we have four non-equivalent Vogan diagrams in this case. So we have a total 1 of nine non-equivalent Vogan diagrams in D4 . Using [1], the number of 1 almost compact real forms in D4 is also nine.
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punita batra ACKNOWLEDGMENTS
This work was done at SUNY at Stony Brook, NY under the guidance of Prof. A. W. Knapp. I would like to thank Prof. A. W. Knapp for his help and useful suggestions in preparing this paper. I would also like to thank Prof. G. Rousseau for a private communication about Theorem 3.7.
REFERENCES 1. J. Bausch and G. Rousseau, “Algebres de Kac–Moody affines (Automorphismes et Formes Reelles),” Institut Elie Cartan, Nancy, 1989. 2. V. Back, N. Bardy, H. Ben Messaoud, and G. Rousseau, Formes presque-d´eploy´ees des alg`ebres de Kac–Moody: Classification et racines relatives, J. Algebra 171 (1995), 43–96. 3. V. G. Kac, “Infinite Dimensional Lie Algebras,” 3rd Ed., Cambridge Univ. Press, London/New York, 1990. 4. V. G. Kac and D. H. Peterson, Infinite flag varieties and conjugacy theorems, Proc. Natl. Acad Sci. U.S.A. 80 (1983), 1778–1782. 5. A. W. Knapp, “Lie Groups Beyond an Introduction,” Progress in Mathematics, Vol. 140, Birkh¨auser, Boston, 1996. 6. A. W. Knapp, A quick proof of the classification of simple real Lie algebras, Proc. Amer. Math. Soc. 124, No. 10 (1996), 3257–3259. 1 7. Z. Kobayashi, Automorphisms of finite order of the affine Lie algebra Al , Tsukuba J. Math. 10, No. 2 (1986), 269–283. 8. F. Levstein, A classification of involutive automorphisms of an affine Kac–Moody algebra, J. Algebra 114 (1988), 489–518. 9. G. Rousseau and H. Ben Messaoud, Private communication. 10. G. Rousseau, Almost split K-forms of Kac–Moody algebras, in “Infinite dimensional Lie Algebras and Groups,” Marseille, 1988 (V. G. Kac, ed.), pp. 70–85, Advanced Series in Mathematical Physics, Vol. 7, World Scientific, Singapore, 1989.