Subregular subalgebras and invariant generalized complex structures on Lie groups
aa r X i v : . [ m a t h . AG ] J a n Subregular subalgebras and invariant generalized complexstructures on Lie groups
Evgeny MayanskiyOctober 17, 2018
Abstract
We introduce the notion of a subregular subalgebra, which we believe is useful for classification ofsubalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structureon a Lie group. As an illustration of the study of invariant generalized complex structures, we computethem all for the real forms of G . Let g be a finite-dimensional complex Lie algebra, k ≥ Definition 1.1.
A subalgebra s ⊂ g is called subregular in codimension k if s is normalized by a codimension k subalgebra of a Cartan subalgebra of g .If k ≥ , then s ⊂ g is called subregular strictly in codimension k if s ⊂ g is subregular in codimension k ,but is not subregular in codimension k − . Note that any subalgebra s ⊂ g is subregular (strictly) in codimension k for some k ∈ { , , . . . , rank( g ) } .Regular subalgebras, as defined in [2], [4], are precisely those which are subregular in codimension 0.This notion may be useful for an explicit classification of subalgebras of Lie algebras as in [10]. In thisnote, we demonstrate how it can be applied to construction of invariant generalized complex structures onLie groups. Invariant generalized complex structures on homogeneous spaces were studied in [11] and [1]. In particular,Alekseevsky, David and Milburn classified invariant generalized complex structures on Lie groups in termsof the so-called admissible pairs. We will review a part of their classification.Throughout this section, G denotes a finite-dimensional connected real Lie group, g the (real) Liealgebra of G , g = g ⊗ R C its complexification and τ : g → g the corresponding antiinvolution. If g issemisimple and h ⊂ g is a Cartan subalgebra, then h = h ⊗ R C ⊂ g denotes its complexification andΦ ⊂ h ∗ the root system of g with respect to the Cartan subalgebra h . Definition 2.1 (Alekseevsky-David [1], Milburn [11]) . A g -admissible pair is a pair ( s , ω ) , where s ⊂ g isa complex subalgebra and ω ∈ V s ∗ is a closed -form such that: • s + τ ( s ) = g , and • Im( ω | g ∩ s ) is non-degenerate. Theorem 2.2 (Akelseevsky-David [1], Milburn [11]) . There is a one-to-one correspondence between theinvariant generalized complex structures on G and the g -admissible pairs ( s , ω ) . Definition 2.3 (Akelseevsky-David [1]) . An invariant generalized complex structure on G is called regularif the associated subalgebra s ⊂ g is normalized by a Cartan subalgebra of g . The following theorem strengthens [1], Theorem 15 and completes the classification of invariant general-ized complex structures on finite-dimensional compact connected real semisimple Lie groups.
Theorem 2.4. If G is a finite-dimensional compact connected real semisimple Lie group, then any invariantgeneralized complex structure on G is regular.Proof. Let s ⊂ g be the complex subalgebra associated by Theorem 2.2 to an invariant generalized complexstructure on G . Let N ( s ) ⊂ g be its normalizer.By [9], Theorem 13, N ( s ) ∩ g generates a closed subgroup of G . The same argument as in [1], Theo-rem 15, using [14], implies that N ( s ) is normalized by a Cartan subalgebra h ⊂ g , i.e. N ( s ) = L ⊕ M α ∈ R C X α , where R ⊂ Φ is a closed subset, X α , α ∈ Φ, are root vectors of g with respect to the Cartan subalgebra h = h ⊗ R C , and L ⊂ h is the solution set of a system of equations of the form α − β = 0, α, β ∈ Φ.Since τ is the conjugation with respect to a compact real form g of g and τ ( h ) = h , τ | h ( R ) = − Id h ( R ) , τ ( C X α ) = C X − α , where h ( R ) is the real span in h of the coroots of g with respect to h [6].Since N ( s ) + τ ( N ( s )) = g , L + τ ( L ) = h , which is possible only if L = h . Hence h ⊂ N ( s ) normalizes s .In general, not all invariant generalized complex structures on real semisimple Lie groups are regular.Let G be a finite-dimensional connected real Lie group, k ≥ Definition 2.5.
An invariant generalized complex structure J on G is called subregular in codimension k if the associated subalgebra s ⊂ g is normalized by a codimension k subalgebra of a Cartan subalgebra of g .If k ≥ , then J is called subregular strictly in codimension k if J is subregular in codimension k , but isnot subregular in codimension k − . We illustrate this notion with an example of a non-regular invariant generalized complex structure on SO (2 n − , n ≥ SO (2 n − , Let G = SO (2 n − , n ≥
3. Then g = so (2 n − ,
1) is a noncompact real form of g = so n ( C ). Weinterpret g as the Lie algebra of 2 n × n skew symmetric complex matrices. Then τ : g → g , A J · ¯ A · J, J = diag( 1 1 · · · | {z } n − ( −
1) ) , is the conjugation with respect to g , where bar denotes the usual complex conjugation.2et E ij , 1 ≤ i, j ≤ n , be a 2 n × n matrix with 1 in the ( i, j ) th place and 0 elsewhere. Following [6],define H k = √− · ( E k − , k − E k, k − ) , ≤ k ≤ n,G + jk = E j − , k − − E k − , j − + E j, k − E k, j + √− · ( E j − , k − E j, k − − E k, j − + E k − , j ) ,G + kj = − G + jk , ≤ j < k ≤ n,G − jk = E j − , k − − E k − , j − − E j, k + E k, j + √− · ( E j − , k + E j, k − − E k, j − − E k − , j ) ,G − kj = − G − jk , ≤ j < k ≤ n. Then h = n L k =1 C H k is a Cartan subalgebra of g . Let ǫ k ∈ h ∗ , 1 ≤ k ≤ n , be such that ǫ k ( H j ) = 1 if j = k and 0 otherwise. ThenΦ = { ǫ j − ǫ k | ≤ j = k ≤ n } ∪ {± ( ǫ j + ǫ k ) | ≤ j < k ≤ n } is the root system of ( g , h ). Let us choose the following root vectors: X jk = X ǫ j − ǫ k = G + jk , ≤ j = k ≤ n,Y jk = X ǫ j + ǫ k = G − kj , ≤ j < k ≤ n,Z jk = X − ( ǫ j + ǫ k ) = G − jk , ≤ j < k ≤ n. Note that [ Y jk , Z jk ] = 4 · ( H j + H k ) , [ X jk , X kj ] = 4 · ( H j − H k ) , ≤ j < k ≤ n. Let h ⊂ h be a hyperplane cut out by the equation ǫ n − − ǫ n = 0, L ( h a vector subspace containing H n − + H n , and H ∈ h \ L . Define s = L ⊕ C ( H + X n − ,n ) ⊕ M ≤ j The subalgebra s ⊂ g is subregular strictly in codimension .Proof. By construction, s is normalized by a codimension 1 subalgebra h ⊂ h . At the same time, s is notregular, because, for a suitable l ∈ L , l + H + X n − ,n lies in the radical of s but its nilpotent component X n − ,n does not.Note that s + τ ( s ) = g if and only if L + C H + τ ( L + C H ) = h .To illustrate the general idea, let us assume for simplicity that H = H , L = n − L k =2 C H k ⊕ C ( H n − + H n ).Then s ∩ g = { √− · b · ( H + X n − ,n − Z n − ,n ) + n − X j =2 √− · b j · H j | b j ∈ R } is a real abelian Lie algebra of dimension n − 2. If n is even, s ∩ g carries a symplectic form ω , which maybe any non-degenerate 2-form on the real vector space s ∩ g = R √− H + X n − ,n − Z n − ,n ) ⊕ n − L j =2 R √− H j ∼ = R n − . One can extend √− ω to a closed 2-form ω ∈ V s ∗ . Assume that ǫ j , 1 ≤ j ≤ n , vanish on the rootvectors of g . This proves 3 heorem 3.2. Let n ≥ be even, s ⊂ g the complex subalgebra defined above, H = H , L = n − L k =2 C H k ⊕ C ( H n − + H n ) , and ω ∈ V s ∗ a closed -form such that ω | s ∩ g = √− · n − X j =1 ǫ j − ∧ ǫ j . Then ( s , ω ) is a g -admissible pair and defines a non-regular invariant generalized complex structure on SO (2 n − , . G G denotes a connected real Lie group whose Lie algebra g is a real form of the complexsimple Lie algebra g of type G , i.e. g is either the compact real form G c or the normal real form G n of g = G . Let τ : g → g be the conjugation with respect to g .Recall that G n has 4 conjugacy classes of Cartan subalgebras l : the maximally noncompact, the max-imally compact, the one with a single short real root and the one with a single long real root [13]. Theconjugation τ n : g → g with respect to G n acts on the root system of ( g , l ⊗ R C ) as Id, − Id, a reflectionthrough a short root and a reflection through a long root respectively.Let ( s , ω ) be a g -admissible pair corresponding to an invariant generalized complex structure on G .The subalgebras of the complex simple Lie algebra of type G were classified in [10]. We will use the notationof [4] and [10]. Since s + τ ( s ) = g , dim( s ) ≥ dim( G ) / s ⊂ g is regular. Lemma 4.1. The subalgebra s ⊂ g is normalized by a Cartan subalgebra of g and is not isomorphic to sl ( C ) .Proof. By [4], up to conjugacy either s = g or s = A or s ⊂ G [ β ] or s ⊂ G [ α ].Suppose s = A . Since s is semisimple, the 2-dimensional subalgebra s ∩ τ ( s ) contains semisimple andnilpotent components of its elements. Since H ( s , C ) = 0, ω is exact. Thus, if s ∩ τ ( s ) is abelian, ω | s ∩ g = 0,a contradiction. Hence s ∩ τ ( s ) is not abelian, and so every element of s ∩ τ ( s ) is either semisimple ornilpotent. Then we can choose a basis x , x of s ∩ τ ( s ) such that [ x , x ] = 2 · x , where x is semisimpleand x is nilpotent. Since τ ( x ) ∈ C x , we may assume that τ ( x ) = x , τ ( x ) = x .The proof of the Jacobson-Morozov theorem in [3] goes through and provides x ∈ s such that x , x , x span an sl ( C ) subalgebra of s . Since τ ( x ) = x , we obtain a contradiction.Suppose s ⊂ G [ β ] or s ⊂ G [ α ]. By [10], Table 1, s either is solvable and contains a Cartan subalgebraof g or is normalized by a Borel subalgebra of g or is the subalgebra s = h ⊕ C Y β ⊕ C Y − β ⊕ C Y α + β ⊕ C Y α + β ⊕ C Y α +2 β , where h ⊂ g is a Cartan subalgebra, Φ = {± α, ± β, ± ( α + β ) , ± (2 α + β ) , ± (3 α + β ) , ± (3 α + 2 β ) } is the rootsystem of ( g , h ), Y γ , γ ∈ Φ, are root vectors.Note that any Borel subalgebra b ⊂ g contains a Cartan subalgebra h ⊂ g [15]. Let h = h ⊗ R C .If s ⊂ b contains a Cartan subalgebra of g , then h is maximally compact and h ∩ s = 0. This impliesthat either h normalizes s or h ∩ s ⊂ s ∩ τ ( s ) = 0, a contradiction.Suppose s = s . Let b ⊂ G [ α ] be the Borel subalgebra of ( g , h ) containing Y α and Y β , n = [ b , b ]. Since[[ n , n ] , n ] ⊂ s , we may write s = h ⊕ C x ⊕ C x ⊕ C X α + β ⊕ C X α + β ⊕ C X α +2 β , = a · X α + a · X α + β + X β , x = X − β + b · X α + b · X α + β , where X γ , γ ∈ Φ, are root vectors of ( g , h ), n contains X α and X β .We may assume that h = C x ⊕ C x , where x = z + X ρ + X γ ≻ ρ u γ · X γ , x = z + X γ ≻ ρ v γ · X γ , z , z ∈ h , ρ ( z ) = 0 , for some ρ ∈ { α, β, α + β } .Since s ∋ [ x , x ] = α ( z ) · a · X α + β ( z ) · X β + x , x ∈ [ n , n ], and ( α − β )( z ) = 0, a = 0. If ρ = α ,then also s ∋ ( α + β )( z ) · a · X α + β + β ( z ) · X β , and so a = 0 in this case.If x = X β , then [ x , x ] = H β + b · X α + β . Hence ρ = α , and so [ x , x ] ∈ s implies that b = 0. Then s = C x ′ ⊕ C x ′ ⊕ C x ⊕ C X β ⊕ C X α + β ⊕ C X α + β ⊕ C X α +2 β , where x ′ = z ′ + u · X α + β , x ′ = z ′ + v · X α + β , z ′ , z ′ ∈ h . We may assume that u = 1, v = 0, and so( α + β )( z ′ ) = 0.In this case, τ acts on the roots either as − Id or as a reflection through β . Hence either z ′ or X β iscontained in s ∩ τ ( s ) = 0, a contradiction.Hence we may assume that x is not proportional to a root vector, and so ρ = α . In this case, s contains x ′ = z + X α + u · X α + β and x ′ = z + v · X α + β .If v = 0, we may assume that v = 1 and u = 0. Since [ x ′ , x ′ ] ∈ s , ( α + β )( z ) = 0.Since [ x ′ , x ] ∈ s , − β ( z ) · X − β + α ( z ) b · X α ∈ s , and so b = 0.Since [ x ′ , x ] ∈ s , − β ( z ) · X − β − X α ∈ s , and so b = 1 /β ( z ).Since [ x ′ , x ] ∈ s , β ( z ) · X β − X α + β ∈ s , and so a = − /β ( z ). Hence s = C x ′ ⊕ C x ′ ⊕ C x ′ ⊕ C x ′ ⊕ C X α + β ⊕ C X α + β ⊕ C X α +2 β , where x ′ = r · H α + β − r · X − β , x ′ = r · H α +2 β − r · X β , x ′ = X α + β + r · X β , x ′ = X α + r · X − β .Since τ acts on the roots either as − Id or as a reflection through β , h ⊕ C X β ⊕ C X − β is spanned by x ′ , x ′ , τ ( x ′ ), τ ( x ′ ). We can choose the root vectors such that τ ( X γ ) = ± X τ ( γ ) , γ ∈ Φ.If τ acts on the roots as − Id, then τ ( x ′ ) = − r · H α + β ∓ r · X β , τ ( x ′ ) = − r · H α +2 β ∓ r · X − β . Hence C X β ⊕ C X − β is spanned by a single element ( r /r ) · X − β ± ( r /r ) · X β , a contradiction.If τ acts on the roots as a reflection through β , then τ ( x ′ ) = ± r · H α +2 β + r · X − β , τ ( x ′ ) = ± r · H α + β + r · X β . Hence h is spanned by a single element ( r /r ) · H α + β ± ( r /r ) · H α +2 β , a contradiction.If v = 0, then u = 0. Since [ x ′ , x ] ∈ s , X − β − b · X α + β ∈ s , and so b = b = 0. Since s ∩ τ ( s ) = 0and α ( z ) = 0, g = G n and τ = τ n acts on the roots as a reflection through β . Hence X − β ∈ s ∩ τ ( s ), acontradiction. 5 orollary 4.2. The subalgebra s ⊂ g is normalized by a maximally compact Cartan subalgebra h ⊂ g .Moreover, either s = L ⊕ n or s = b , where b ⊂ g is a Borel subalgebra of ( g , h ) , L ⊂ h = h ⊗ R C and n = [ b , b ] .Proof. Let h ⊂ g be a Cartan subalgebra normalizing s , h = h ⊗ R C , X γ , γ ∈ Φ, root vectors of ( g , h ).By Lemma 4.1 and [10], Table 1, the subalgebra s ⊂ g is one of the following: L ⊕ n , b = h ⊕ n , C H α ⊕ C X − α ⊕ n , C H β ⊕ C X − β ⊕ n ,G [ β ] = C X − α ⊕ b , G [ α ] = C X − β ⊕ b , g , where L ⊂ h , n = L γ ∈ Φ + C X γ and Φ is suitably ordered so that Φ + = { α, β, α + β, α + β, α + β, α + 2 β } ⊂ Φis the subset of positive roots.By [1], Lemma 7, only two of these subalgebras can form a g -admissible pair: s = L ⊕ n or s = b . In both cases, τ acts on the roots as − Id.Let φ : G → G be the universal complexification [8], i.e. G is the connected complex simple Lie groupof type G , with Lie algebra g , ker( φ ) is the center of G , and the differential of φ is the embedding g ⊂ g .Let B ⊂ G be a fixed Borel subgroup, with Lie subalgebra b ⊂ g containing a maximally compact Cartansubalgebra h ⊂ g . By [15], Theorem 5 . H = B ∩ G is connected and so is generated by h = b ∩ g .Consider the flag manifold G/B parametrizing the Borel subalgebras of g . Let N ∗ be the holomorphic ho-mogeneous vector bundle over G/B corresponding to the isotropy representation B → GL (Hom C ([ b , b ] , C ))coming from the adjoint action of B on b .Let B ⊂ G/B be the union of the open orbits of G . By [15], Theorem 4 . B parametrizes the Borelsubalgebras containing a maximally compact Cartan subalgebra of g . If G is compact, then B = G/B .Otherwise, B consists of exactly three open orbits of G on G/B , corresponding to the three Weyl chambersof G contained in a Weyl chamber of A + ˜ A , [15], Theorem 4 . I = B × GL (2 , R ) /GL (1 , C ) , Σ = B × Σ , where Σ = { σ ∈ ^ ( C ) ∗ | Im( σ | R ) is symplectic } , be the trivial bundles over B parametrizing the complex structures and certain extensions of symplecticstructures on the fibers of B × h → B respectively, h identified with R . Remark 4.3. As sets, GL (2 , R ) /GL (1 , C ) ∼ = { z ∈ C | Im( z ) = 0 } ∼ = Σ . Now we state the main result of this section. Theorem 4.4. Any invariant generalized complex structure on G , a real Lie group of type G and realdimension , is regular. The set of invariant generalized complex structures on G is parametrized by thedisjoint union C ∪ S , where C = I × B N ∗ ∼ = N ∗ | B × GL (2 , R ) /GL (1 , C ) and S = Σ × B N ∗ ∼ = N ∗ | B × Σ . roof. We use the notation of Corollary 4.2.Suppose s = L ⊕ n , dim( L ) = 1. Then s + τ ( s ) = g if and only if L ⊂ h is the holomorphic subspaceof a complex structure on h . Since s ∩ τ ( s ) = 0, any closed 2-form ω ∈ V s ∗ gives a g -admissible pair ( s , ω ).The Chevalley-Eilenberg resolution gives H ( s , C ) = 0. Hence ω = d ξ for a uniquely determined linearmap ξ : [ b , b ] → C .Thus, the g -admissible pairs ( s , ω ) with s = L ⊕ n are parametrized by the triples ( b , ξ, λ ), where b ⊂ g is a Borel subalgebra containing a maximally compact Cartan subalgebra of g , ξ ∈ Hom C ([ b , b ] , C ) and λ ∈ GL (2 , R ) /GL (1 , C ) is a complex structure on the real vector space h = b ∩ g ∼ = R . Cf. [12].Suppose s = b . In this case, H ( s , C ) = C · ω , where 0 = ω ∈ V h ∗ is extended by zero to a 2-form on s . Since b ∩ g = h , a 2-form ω ∈ V b ∗ gives a g -admissible pair ( b , ω ) if and only if ω = c · ω + d ξ for auniquely determined linear map ξ : [ b , b ] → C , where Im( c · ω | h ) is non-degenerate.Thus, the g -admissible pairs ( s , ω ) with s = b are parametrized by the triples ( b , ξ, σ ), where b ⊂ g is aBorel subalgebra containing a maximally compact Cartan subalgebra of g , ξ ∈ Hom C ([ b , b ] , C ) and σ ∈ Σ .As we recalled above, B consists of one or three orbits of G [15]. Corollary 4.5. The set of invariant generalized complex structures on G , up to conjugacy by G , isparametrized by r copies of the disjoint union N ∗ × GL (2 , R ) /GL (1 , C ) ∪ N ∗ × Σ , where N ∗ = Hom C ([ b , b ] , C ) /H , r = 1 if G is compact and otherwise. The following remark is an immediate consequence of Milburn’s study of invariant generalized complexstructures on homogeneous spaces [11]. Remark 4.6. There is no SO (2 n + 1) -invariant generalized complex structure on the n -dimensional sphere S n = SO (2 n + 1) /SO (2 n ) , n ≥ , and no G -invariant generalized complex structure on S = G c /SU (3) .The SO (3) -invariant generalized complex structures on S = SO (3) /SO (2) are two biholomorphic complexstructures CP and CP , and a family of invariant generalized complex structures with holomorphic subbun-dles of the form L ( so ( C ) , ω c ) , c ∈ C , Im( c ) = 0 , which are B-transforms of the symplectic structures (up tosymplectomorphism) on S . Notation is from [5], [11], ω c ∈ V so ( C ) ∗ is defined by ω c ( X, Y ) = Trace( c − c · [ X, Y ]) for × skew symmetric complex matrices X, Y ∈ so ( C ) . See also [7]. Acknowledgement The author is grateful to Beijing International Center for Mathematical Research, the Simons Foundationand Peking University for support, excellent working conditions and encouraging atmosphere. References [1] D. V. Alekseevsky and L. David, Invariant generalized complex structures on Lie groups , Proceedingsof the London Mathematical Society. Third Series (2012), no. 4, 703–729.72] N. G. Chebotarev, A theorem of the theory of semi-simple Lie groups , Matematicheskii Sbornik N.S. (1942), no. 3, 239–244.[3] D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras , Van NostrandReinhold Mathematics Series, Van Nostrand Reinhold, New York, 1993.[4] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras , Matematicheskii Sbornik N.S. 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