Non-solvable Lie groups with negative Ricci curvature
aa r X i v : . [ m a t h . DG ] D ec NON-SOLVABLE LIE GROUPS WITH NEGATIVE RICCI CURVATURE
EMILIO A. LAURET AND CYNTHIA E. WILL
Abstract.
Until a couple of years ago, the only known examples of Lie groups admittingleft-invariant metrics with negative Ricci curvature were either solvable or semisimple.We use a general construction from a previous article of the second named author to pro-duce a large amount of examples with compact Levy factor. Given a compact semisimplereal Lie algebra u and a real representation π satisfying some technical properties, theconstruction returns a metric Lie algebra l ( u , π ) with negative Ricci operator. In thispaper, when u is assumed to be simple, we prove that l ( u , π ) admits a metric havingnegative Ricci curvature for all but finitely many finite-dimensional irreducible represen-tations of u ⊗ R C , regarded as a real representation of u . We also prove in the last sectiona more general result where the nilradical is not abelian, as it is in every l ( u , π ). Introduction
A natural question that has inspired a great deal of research is what can be said abouta differentiable manifold M that admits a metric with curvature of a certain sign. Inthe homogeneous case, while the scalar and sectional curvature behavior are settled, thequestion of when a homogeneous manifold admits a left-invariant metric with negativeRicci curvature seems far from being understood (see e.g. the introductions of [1] or [10]).In this paper we are interested in the case when M is a Lie group. We note that in thiscase one can study the problem at the Lie algebra level. There are three kinds of examplesin the literature of Lie groups admitting metrics with negative Ricci curvature.Dotti, Leite and Miatello proved in [3] that the only unimodular Lie groups that canadmit a left-invariant metric with Ric < , C ) , SL(3 , C ) , SO(5 , C ) , SO(7 , C ) , Sp(3 , C ) , Sp(4 , C ) , Sp(5 , C ) , ( G ) C , SL(2 , R ) , Sp(2 , R ) , Sp(3 , R ) , G (cf. remark after [3, Thm. 2.1]). It is known that SL(2 , R ) does not admit a Ricci negativemetric (see [8]) but the existence of such a metric on the other groups listed above is stillopen. It is worth mentioning that Jablonski and Petersen recently proved in [5] that suchsemisimple Lie group cannot have any compact factor.The second sort of examples are solvable Lie groups. This is the most developed case(see [2], [10], [9], [1]) probably due to the relationship with Einstein solvmanifolds. Recallthat any non-flat Einstein solvmanifold is an example of a Lie group with a metric ofnegative Ricci curvature. Date : May 29, 2019.2010
Mathematics Subject Classification.
Primary 53C30, Secondary 53C21, 17B22.
Key words and phrases.
Negative Ricci curvature, Lie groups, Riemannian metrics.This research was partially supported by Grants from CONICET, FONCYT and SeCyT (UniversidadNacional de C´ordoba). The first named author was also supported by the Alexander von HumboldtFoundation (return fellowship).
Let s be a solvable Lie algebra admitting an inner product with negative Ricci operator.Besides the fact that s cannot be unimodular (see [2]), Nikolayevsky and Nikonorov [10]gave the only necessary condition known so far, namely, there exists Y ∈ s such thattr ad Y > Y | z ( n ) have positive real part, where n is the nilradicalof s and z ( n ) is the center of n .The second named author constructed in [11] the first examples of Lie groups, whichare not solvable nor semisimple, admitting a left-invariant metric with negative Riccicurvature. This construction was extended to a more general setting in [12] that we nextdescribe.Let u be a compact semisimple Lie algebra. Given ( π, V ) a finite-dimensional realrepresentation of u , let l ( u , π ) be the Lie algebra defined by(1.2) l ( u , π ) := ( R Z ⊕ u ) ⋉ V determined by ad Z | u = 0 and ad Z | V = Id . It was proven in [12, Thm. 3.3] that, if ( π, V ) satisfies some technical conditions, theLie algebra l ( u , π ) admits an inner product with negative definite Ricci operator. SeeTheorem 3.1 for the precise statement.Furthermore, the article [12] provides explicit examples of the above construction byusing representations of the classical compact real Lie algebras ( su ( n ), so ( n ) and sp ( n ) for n ≥
2) realized as a vector space of homogeneous polynomials.The main goal of this article is to find systematic ways to construct metric Lie algebraswith negative Ricci operator from Theorem 3.1. From now on we will consider complexrepresentations of the complexified Lie algebra u C := u ⊗ R C , viewed as real representationsof u by restriction and by restricting scalars. More precisely, a finite-dimensional complexrepresentation π : u C → gl ( V ) induces the real representation π | u : u → gl ( V ), where V isregarded as a real vector space. This representation will still be denoted by ( π, V ).We develop three essentially different approaches to decide whether such a representa-tion satisfies the hypotheses in Theorem 3.1. The corresponding representations are givenin Theorems 3.4, 3.7, and 3.8. The first approach is called the Weyl chamber approach because it assumes the existence of a weight of the representation inside a Weyl chamber.This allows us to prove the main result of the article. Theorem 1.1.
Let u be a compact real Lie algebra such that its complexified Lie algebra u C is simple. For all but finitely many finite-dimensional irreducible complex representations ( π, V ) of u C , the real Lie algebra l ( u , π ) given by (1.2) admits a metric with negative Riccicurvature. Furthermore, for u an arbitrary compact real semisimple Lie algebra, Theorem 3.5ensures that l ( u , π ) admits an inner product with negative Ricci curvature for infinitelymany finite-dimensional irreducible complex representations ( π, V ) of u C .Although the Weyl chamber approach (Theorem 3.4) provides a great amount of ex-amples, it does not work very well for small representations. We then show two moreconstructions, the Weyl group orbit approach and the zero weight approach . The first onewas inspired by the constructions in [12], where it was used implicitly.In Section 4 we show many explicit examples. In particular, when u C has rank atmost two, we classify all irreducible representations π of u C where Theorem 3.1 can beapplied. We obtain that the above holds for every such π with the only exceptions of the7-dimensional irreducible representation of the complex simple Lie algebra of type G , andthe fundamental representations when the type is A or B = C .With all these examples in mind, it is clear that the set of Lie groups admitting left-invariant metrics of negative Ricci curvature is much bigger than it seemed and the clas-sification problem has become more complicated. EGATIVE RICCI CURVATURE 3
We show in the last section that one can get examples with non-abelian nilradical. Tobe able to do that we need to show a slightly more general version of [12, Thm. 3.3] (seeTheorem 5.1). More precisely, we consider a Lie algebra ( R Z ⊕ u ) ⋉ n , where u is a compactsemisimple Lie algebra acting on n by derivations and [ Z, u ] = 0. Note that in order toget such a Lie algebra, ad Z must be a derivation of n and therefore it can never act as amultiple of the identity unless n is abelian.It is known that there is no topological obstruction on a differential manifold to theexistence of a complete Riemannian metric with negative Ricci curvature (see [7]). Nev-ertheless, the situation changes when dealing with left-invariant metrics on Lie groups.First, recall that if K is a maximal compact subgroup of a Lie group G , then all thenontrivial topology of G is in K , in the sense that as a differentiable manifold, G is theproduct K × R n . Therefore, from the semisimple examples in [3], it follows that almostall compact simple Lie groups may appear as maximal compact subgroups of a Lie groupwith negative Ricci curvature with the exceptions (1.1) namely,(1.3) SO(2) , SU(2) , SU(3) , Sp(2) , Sp(3) , Sp(4) , Sp(5) , SO(7) , G c . Here, G c denotes the simply connected compact Lie group with the Lie algebra of typeG . In [11] and [12] were covered all the groups in the above list with the only exceptionof G c , which is now attained by Theorem 1.1 (see also Proposition 4.1). Therefore, asin the general case, there are (almost) no topological obstructions for the existence of aleft-invariant metric of negative Ricci curvature on a Lie group.The paper is organized as follows. In Section 2 we recall all the facts about compactLie groups and their representations that will be used in the sequel. Section 3 containsthe three approaches to ensure that l ( u , π ) admits a metric of negative Ricci curvature, aswell as the proof of the main theorem. Several explicit examples are shown in Section 4.In the last section, we construct Lie algebras with the non-abelian nilradical admitting aninner product of negative Ricci curvature. Acknowledgments.
The authors wish to thank Jorge R. Lauret and the referees forseveral helpful comments on a preliminary version of this paper.2.
Preliminaries
Throughout the paper, all Lie algebras as well as their representations are assumed finitedimensional.
In this section we first recall well-known facts on compact real forms and representationsof a complex semisimple Lie algebra via root systems. Very good general references are[4, § § Root system.
Let g be a complex semisimple Lie algebra and B its Killing form.We fix a Cartan subalgebra h of g , and let ∆ denote the corresponding system of rootsand W the Weyl group. One has the root decomposition(2.1) g = h ⊕ M α ∈ ∆ g α , where g α = { X ∈ g : [ H, X ] = α ( H ) for all H ∈ h } is one dimensional for all α ∈ ∆.For α ∈ ∆, we denote by H α the corresponding coroot, that is, the only element in h such that B ( H, H α ) = α ( H ) for all H ∈ h . We denote by h R the R -linear span of all H α for α ∈ ∆, which is a real form of h . We will consider the inner product on h ∗ R determinedby h α, β i := B ( H α , H β ). EMILIO A. LAURET AND CYNTHIA E. WILL
For each α ∈ ∆, it is possible to choose X α ∈ g α such that, for all α, β ∈ ∆,(2.2) [ X α , X − α ] = H α , [ X α , X β ] = N α,β X α + β if α + β ∈ ∆ , [ X α , X β ] = 0 if α + β = 0 and α + β / ∈ ∆ , where N α,β = − N − α, − β ∈ R (see [6, Thm 6.6]). It turns out that(2.3) u := X α ∈ ∆ R i H α + X α ∈ ∆ R ( X α − X − α ) + X α ∈ ∆ R i ( X α + X − α )is a compact real form of g , that is, a real Lie algebra whose complexified Lie algebra is g ,which has negative definite Killing form (see [6, Thm. 6.11]).We denote H α = i H α , X α = ( X α − X − α ) and Y α = i ( X α + X − α ) for each α ∈ ∆. For α, β ∈ ∆, one has that(2.4) [ H α , X β ] = c α,β Y β , [ X α , X β ] = N α,β X α + β − N − α,β X − α + β if β = ± α, [ H α , Y β ] = − c α,β X β , [ Y α , Y β ] = − N α,β X α + β − N − α,β X − α + β if β = ± α, [ X α , Y α ] = 2 H α , [ X α , Y β ] = N α,β Y α + β − N − α,β Y − α + β if β = ± α, where c α,β are real numbers and N α,β = 0 if α + β / ∈ ∆.2.2. Weights of representations.
Let ( π, V ) be a complex representation of g , that is,a C -linear map π : g → gl ( V ) satisfying that π ([ X, Y ]) = π ( X ) π ( Y ) − π ( Y ) π ( X ) for all X, Y ∈ g . We recall our convention that all representations are assumed finite-dimensional.One has the weight decomposition(2.5) V = M µ V ( µ ) , where µ ∈ h ∗ and(2.6) V ( µ ) = { v ∈ V : π ( H ) v = µ ( H ) v for all H ∈ h } . Those µ ∈ h ∗ satisfying V ( µ ) = 0 are called the weights of π and belong to the weightlattice P ( g ) := { µ ∈ h ∗ : h µ, α i / h α, α i ∈ Z for all α ∈ ∆ } ⊂ h ∗ R . Furthermore,(2.7) π ( g α ) V ( µ ) ⊂ V ( µ + α )for α ∈ ∆ and µ ∈ P ( g ). The dimension of V ( µ ) is called the weight multiplicity of µ in π . One has dim V ( w · µ ) = dim V ( µ ) for all w ∈ W .We now pick a positive system ∆ + of ∆. Let Π be the set of simple roots. An element µ ∈ h ∗ R is called dominant if h µ, α i ≥ α ∈ ∆ + . Each connected component of thecomplement of the set of hyperplanes { µ ∈ h ∗ R : h µ, α i = 0 } for α ∈ ∆ is called a Weylchamber . The cone of dominant elements coincides with the closure of the fundamentalWeyl chamber { µ ∈ h ∗ R : h µ, α i > α ∈ ∆ + } .Let P + ( g ) denote the set of dominant elements in the weight lattice P ( g ). We writeΠ = { α , . . . , α n } ( n is the rank of g ). The fundamental weights ω , . . . , ω n are given by2 h ω i , α j i / h α j , α j i = δ i,j and satisfy P ( g ) = L ni =1 Z ω i and P + ( g ) = L ni =1 Z ≥ ω i .The next result will be useful in the sequel (see for instance [4, Prop. 3.2.11]). Lemma 2.1.
Let µ ∈ P ( g ) dominant and ν a weight of V . If ν − µ can be written as asum of positive roots, then µ is a weight of V . The Highest Weight Theorem parameterizes irreducible complex representations of g with dominant elements in P ( g ). For λ ∈ P ( g ) dominant, we write ( π λ , V λ ) the corre-sponding irreducible complex representation of g with highest weight λ . EGATIVE RICCI CURVATURE 5
We conclude the preliminaries section by showing the existence of a weight of a rep-resentation in a Weyl chamber, for all but finitely many irreducible representations of acomplex simple Lie algebra. Although this result may be present in the mathematical lit-erature, we include a case-by-case proof because the authors could not find any reference.The reader will find two shorter and uniform proofs in the answers by Sam Hopkins andDavid E Speyer of the MathOverflow question [13].
Lemma 2.2.
Let g be a complex simple Lie algebra. For all but finitely many complexirreducible representations ( π, V ) of g , we have that ( π, V ) contains a weight in a Weylchamber.Proof. By the Highest Weight Theorem, every irreducible representation of g is in corre-spondence with an element in { P nj =1 a j ω j : a j ∈ Z ≥ for all j } . Since P r := { P nj =1 a j ω j : a j ∈ Z and 0 ≤ a j ≤ r for all j } is finite for every r ≥
0, it is sufficient to prove that, for r sufficiently large, every irreducible representation of g with highest weight λ ∈ P + ( g ) r P r has a weight in the fundamental Weyl chamber.To do that, we will use Lemma 2.1. More precisely, for such λ , we will show that there are β , . . . , β l ∈ ∆ + such that λ − ( β + · · · + β l ) is in the fundamentalWeyl chamber. Clearly, it is sufficient to show that there is an integer r big enough satisfying that for each ≤ i ≤ n , there are β , . . . , β l ∈ ∆ + such that rω i − ( β + · · · + β l ) is dominant and its ω j -coefficient is positive for all ≤ j ≤ n . The checking process of the above condition involves only computations in the corre-sponding irreducible root system. We will do it case by case.We start considering complex simple Lie algebras of exceptional type. We will use thedata available in [6, § C.2], where the fundamental weights ω , . . . , ω n are written in termsof the simple roots α , . . . , α n . Type G : In this case, n = 2, ω = 2 α + α , and ω = 3 α + 2 α . It followsimmediately that rω − α − α = ( r − ω + ω and rω − α − α = ( r − ω + ω ,thus the required condition holds with r = 3. Type F : In this case, n = 4, and the fundamental weights are ω = 2 α + 3 α +2 α + α , ω = 3 α + 6 α + 4 α + 2 α , ω = 4 α + 8 α + 6 α + 3 α , and ω =2 α + 4 α + 3 α + 2 α . It follows that rω − α − α − α = ( r − ω + (2 ω − α ) + (3 ω − α )+ (2 ω − α − α − α )= ( r − ω + ω + ω + ω ,rω − α − α − α − α = ( r − ω + ω + ω + ω ,rω − α − α − α − α = ( r − ω + ω + ω + ω ,rω − α − α − α − α = ( r − ω + ω + ω + ω . Hence, the required condition holds with r = 8. Type E n : In these cases, n = 6 , , ≤ i ≤ n , ω i = P nj =1 a j α j with a , . . . , a n positive integers. Although it is pretty involved to give the explicitcalculations as in the previous two cases, it is clear that for a positive integer r sufficiently large, one has rω i = ( r − s ) ω i + n X j =1 b j ω j + µ EMILIO A. LAURET AND CYNTHIA E. WILL for some integer s ≤ r , b , . . . , b n positive integers, and µ a sum of simple roots.Hence, the condition is valid.We next consider the classical Lie algebras. The root system data can be found in [6, § C.1]. We give all the details for type D n since it is the one that presents more difficulties.The rest of the types are given in a brief way because the method is analogous. Type D n : In this case, g = so (2 n, C ) for any n ≥ h R = Span R { ε , . . . , ε n } ,∆ + = { ε i ± ε j : 1 ≤ i < j ≤ n } , ω i = ε + · · · + ε i for 1 ≤ i ≤ n − ω n − = ( ε + · · · + ε n − − ε n ), and ω n = ( ε + · · · + ε n ). We have that rω − ( ε − ε ) = ( r − ω + ε = ( r − ω + ω , (2.8) rω i − ( ε i − ε i +1 ) = ( r − ω i + ( ε + · · · + ε i ) − ( ε i − ε i +1 )(2.9) = ω i − + ( r − ω i + ω i +1 for 2 ≤ i ≤ n − ,rω n − − ( ε n − − ε n − ) = ( r − ω n − + ( ε + · · · + ε n − ) + ε n − (2.10) = ω n − + ( r − ω n − + ( ε + · · · + ε n − )= ω n − + ( r − ω n − + ω n − + ω n , Furthermore, rω n − − ε n − − ε n ) − ( ε n − − ε n − ) = ( r − ω n − + 3 ω n − − ( ε n − − ε n − )(2.11) = ω n − + ω n − + ( r − ω n − + ω n rω n − ε n − + ε n ) − ( ε n − − ε n − ) = ( r − ω n + 2 ω n − − ( ε n − − ε n − )(2.12) = ω n − + ω n − + ( r − ω n . The above identities tell us that, for any 1 ≤ i ≤ n , one can subtract to rω i asum of positive roots obtaining a dominant element with positive ω i − and ω i +1 -coefficients (it is understood that there are no ω i -coefficient for i = 0 , n + 1).Proceeding in this way several times, one obtains a dominant element in P ( g )with positive ω j -coefficient for all j , for r large enough. This proves the requiredcondition. Type B n : g = so (2 n + 1 , C ) for any n ≥ h R = Span R { ε , . . . , ε n } , ∆ + = { ε i ± ε j :1 ≤ i < j ≤ n } ∪ { ε i : 1 ≤ i ≤ n } , ω i = ε + · · · + ε i for 1 ≤ i ≤ n −
1, and ω n = ( ε + · · · + ε n ).One can check that (2.8) holds, as well as (2.9) for 2 ≤ i ≤ n −
2. Furthermore, rω n − − ( ε n − − ε n ) = ω n − + ( r − ω n − + 2 ω n , and rω n − ε n = ω n − + ( r − ω n . Type C n : g = sp ( n, C ) for any n ≥ h R = Span R { ε , . . . , ε n } , ∆ + = { ε i ± ε j : 1 ≤ i < j ≤ n } ∪ { ε i : 1 ≤ i ≤ n } , ω i = ε + · · · + ε i for 1 ≤ i ≤ n .One can check that (2.8) holds, as well as (2.9) for 2 ≤ i ≤ n −
1. Furthermore, rω n − ε n = 2 ω n − + ( r − ω n . Type A n : g = sl ( n + 1 , C ) for any n ≥ h R = { P n +1 i =1 a i ε i : a i ∈ R ∀ i, P n +1 i =1 a i =0 } , ∆ + = { ε i − ε j : 1 ≤ i < j ≤ n + 1 } , ω i = ε + · · · + ε i − in +1 ( ε + · · · + ε n +1 )for 1 ≤ i ≤ n .One can check, for any 1 ≤ i ≤ n , that rω i − ( ε i − ε i +1 ) = ( r − ω i − + 2( ε + · · · + ε i ) − in +1 ( ε + · · · + ε n +1 ) − ε i + ε i +1 = ( r − ω i − + ( ε + · · · + ε i − ) − i − n +1 ( ε + · · · + ε n +1 )+ ( ε + · · · + ε i +1 ) − i +1 n +1 ( ε + · · · + ε n +1 )= ω i − + ( r − ω i + ω i +1 , where ω = ω n +1 = 0. EGATIVE RICCI CURVATURE 7
We conclude that the required condition holds for every complex simple Lie algebra,which completes the proof. (cid:3) Existence of Ricci negative metric Lie algebras
In this section we introduce three different approaches to use Theorem 3.1. Furthermore,it also contains the proof of the main theorem.In what follows, u denotes a compact semisimple real Lie algebra and, since a com-pact real form of a semisimple complex Lie algebra is unique up to isomorphism (see [6,Cor. 6.20]), there is a root system ∆ of the complex Lie algebra u C := u ⊗ R C and elements X α ∈ ( u C ) α for each α ∈ ∆ such that the compact real form given by (2.3) coincides with u . Furthermore, we fix ∆ + a positive system for ∆ and Π = { α , . . . , α n } an ordered setof simple roots.In the sequel, we will use the objects introduced in Section 2 without further comments,for instance, the weight lattice P ( u C ), its dominant elements P + ( u C ), weights of a complexrepresentation, the Weyl group W , Weyl chambers, the fundamental weights ω , . . . , ω n ,the (unique up to equivalence) irreducible representation ( π λ , V λ ) with the highest weight λ ∈ P + ( u C ), etc.3.1. Ricci negative inner products.
We first recall the main theorem in [12], whichwill be the main tool in the sequel.To a given real representation ( π, V ) of a real Lie algebra u we associate the real Liealgebra l ( u , π ) := ( R Z ⊕ u ) ⋉ V , where R Z ⊕ u is a central extension of u (i.e. ad Z | u = 0)and Z acts as the identity on V (i.e. ad Z | V = Id). More precisely, the brackets in l ( u , π )are determined by[ Z, X ] = 0 , [ Z, v ] = v, [ X, v ] = π ( X ) v, [ v, w ] = 0 , (3.1)for all X ∈ u and v, w ∈ V . Theorem 3.1. [12, Thm. 3.3]
Let ( V, π ) be a real representation of u . We assume thedecomposition V = V ⊕ V and the existence of an inner product h· , ·i on V satisfying thefollowing properties: (1) V and V are i h R -invariant. (2) π ( X α )( V ) ⊂ V and π ( Y α )( V ) ⊂ V for every α ∈ ∆ + . (3) V is orthogonal to V . (4) π ( H ) is a skew-symmetric operator of V for every H ∈ i h R . (5) π ( X α ) | V and π ( Y α ) | V are not trivial for every α ∈ ∆ + . (6) tr π ( Y ) t | V π ( X ) | V = 0 whenever X = Y are elements of { X α , Y α , α ∈ ∆ + } .Then, the real Lie algebra l ( u , π ) given by (3.1) (i.e. l ( u , π ) = ( R Z ⊕ u ) ⋉ V where ad Z | u = 0 and ad Z | V = Id ) admits an inner product with negative Ricci curvature. The idea of the proof is to first prove that l ( u , π ) degenerates into a solvable Lie algebra l ∞ to finally show that this limit admits an inner product with Ric <
0. By continuity, sodoes the starting Lie algebra.3.2.
Common generalities.
Let ( π, V ) be a complex representation of u C . We can re-gard V as a real vector space by restricting scalars, and then the restriction of π to u becomes a real representation of u . To simplify the notation, the resulting real represen-tation π | u : u → gl ( V ) is again denoted by ( π, V ).Let U be the simply connected Lie group with Lie algebra u . We denote again by π the associated homomorphism of groups U → GL( V ). Since U is compact, there is acomplex inner product hh· , ·ii on the complex vector space V such that π is unitary. More EMILIO A. LAURET AND CYNTHIA E. WILL precisely, for v, w ∈ V , one has that hh π ( a ) v, π ( a ) w ii = hh v, w ii for all a ∈ U , which gives hh π ( X ) v, w ii = −hh v, π ( X ) w ii for all X ∈ u . We set(3.2) h v, w i = Re( hh v, w ii ) for v, w ∈ V. This is clearly a real inner product on V . Moreover, h π ( X ) v, w i = −h v, π ( X ) w i for all X ∈ u and v, w ∈ V , that is, π ( X ) acts as a skew-symmetric operator on V for all X ∈ u .From now on, we will associate such real inner product on V to any complex representationof u C without further comments.It follows immediately from the definition (2.6) that a weight space is invariant by h R .Thus, property (1) in Theorem 3.1 suggests us to decompose V = V ⊕ V , by taking V and V subspaces given by sums of distinct weight spaces.The simple observations from the two last paragraphs guarantee us four properties fromTheorem 3.1. Proposition 3.2.
Let ( π, V ) be a complex representation of u C and let S be a subset ofweights of π . We set V = M µ ∈S V ( µ ) , V = M µ/ ∈S V ( µ ) . Then, there exists a real inner product on V = V ⊕ V such that the induced real represen-tation of u on V satisfies properties (1) , (3) , (4) , and (6) from Theorem 3.1. Moreover,if in addition S has exactly one element, then (2) also holds.Proof. Property (1) follows from (2.6). We already mentioned that π ( X ) : V → V isskew-symmetric on V with respect to h· , ·i for all X ∈ u , thus property (4) holds.Let µ and ν be distinct weights of π and let v ∈ V ( µ ) and w ∈ V ( ν ). For any H ∈ h ,one has that(3.3) µ ( H ) hh v, w ii = hh π ( H ) v, w ii = −hh v, π ( H ) w ii = − ν ( H ) hh v, w ii , which implies h v, w i = Re( hh v, w ii ) = 0 by taking H ∈ h such that µ ( H ) = ν ( H ). Thisshows that different weight spaces are orthogonal with respect to h· , ·i ; in particular (3)holds.We next prove the validity of property (6). Let { v , . . . , v d } be an orthogonal C -basisof ( V, hh· , ·ii ) given by weight vectors, such that { v , . . . , v c } is a C -basis of V ( c ≤ d ). Itturns out that { v , i v , . . . , v d , i v d } is an R -basis of V , as well as { v , i v , . . . , v c , i v c } is an R -basis of V We note for
X, Y ∈ u that(3.4) tr π ( Y ) t | V π ( X ) | V = c X j =1 (cid:0) h π ( Y ) t | V π ( X ) | V v j , v j i + h π ( Y ) t | V π ( X ) | V i v j , i v j i (cid:1) = c X j =1 (cid:0) h π ( X ) v j , π ( Y ) v j i + h π ( X ) i v j , π ( Y ) i v j i (cid:1) . Since π ( X α ) V ( µ ) ⊂ V ( µ − α ) ⊕ V ( µ + α ), π ( Y α ) V ( µ ) ⊂ V ( µ − α ) ⊕ V ( µ + α ) by (2.7),and h V ( µ ) , V ( ν ) i = 0 for µ = ν , it follows that tr π ( X β ) t | V π ( X α ) | V , tr π ( Y β ) t | V π ( X α ) | V ,tr π ( X β ) t | V π ( Y α ) | V , and tr π ( Y β ) t | V π ( Y α ) | V are all equal to zero for α = β in ∆ + .It remains to show that tr π ( Y α ) t | V π ( X α ) | V = 0 for all α ∈ ∆ + . This follows from thefacts that π ( X α ) preserves the (orthogonal) subspaces(3.5) Span R { v , . . . , v c } and Span R { i v , . . . , i v c } , while π ( Y α ) switches them. We conclude that (6) holds. EGATIVE RICCI CURVATURE 9
We now assume that S has exactly one element, say µ . Then, the validity of (2)follows since, for any α ∈ ∆ + , (2.7) yields that π ( X α ) V and π ( Y α ) V are included into V ( µ + α ) ⊕ V ( µ − α ), which is included in V because µ ± α / ∈ S = { µ } . (cid:3) The next lemma will be very useful to check property (5).
Lemma 3.3.
Let ( π, V ) be a complex representation of u C , let µ be a weight of π , andlet α ∈ ∆ . If the integer h α, µ i is non-zero, then π ( X α ) | V ( µ − kα ) and π ( Y α ) | V ( µ − kα ) arenon-trivial for all k between and the integer h α, µ i .Proof. It is well known that a α := Span C { H α , X α , X − α } is isomorphic to sl (2 , C ). Let v ∈ V ( µ ) with v = 0.We first assume that the integer m := h α, µ i is positive. From the well-known repre-sentation theory of sl (2 , C ) (see for instance [6, § I.9]), since(3.6) π ( H α ) v = µ ( H α ) v = h µ, α i v = mv, we deduce that the irreducible a α -submodule of V containing v has dimension at least m + 1, and moreover, π ( X − α ) k v = 0 for all 0 ≤ k ≤ m . Consequently, π ( X α ) | V ( µ − kα ) and π ( Y α ) | V ( µ − kα ) are non-trivial for all 0 ≤ k ≤ m .We now assume that m is negative. Analogously as above, π ( X α ) k v = 0 for all m ≤ k ≤
0, thus π ( X α ) | V ( µ − kα ) and π ( Y α ) | V ( µ − kα ) are non-trivial. (cid:3) Weyl chamber approach.
The following result is the first approach to use Theo-rem 3.1. The main result of the paper, Theorem 1.1, is proved below as a consequence ofthis approach.
Theorem 3.4.
Let ( π, V ) be a complex representation of u C . We assume that there is aweight µ of π lying in a Weyl chamber. By setting V = V ( µ ) and V = L µ = µ V ( µ ) ,all the properties in Theorem 3.1 hold. Consequently, the real Lie algebra l ( u , π ) admitsan inner product with negative Ricci curvature.Proof. The decomposition V = V ⊕ V chosen is as in Proposition 3.2 with S = { µ } ,thus it only remains to show property (5). This follows immediately from Lemma 3.3 since h µ , α i 6 = 0 for all α ∈ ∆ + because µ is not orthogonal to any root from being inside aWeyl chamber. (cid:3) We are now in position to prove the main theorem.
Proof of Theorem 1.1.
By applying Theorem 3.1, Theorem 3.4 tell us that for every com-plex representation ( π, V ) of u C containing a weight in a chamber Weyl, l ( u , π ) admits aninner product with negative Ricci curvature. Then, Lemma 2.2 ensures that this prop-erty holds for all but finitely many irreducible representations of u C , which completes theproof. (cid:3) Theorem 3.5.
Let u be a compact semisimple real Lie algebra. The real Lie algebra l ( u , π ) given by (1.2) admits an inner product with negative Ricci curvature for infinitely manyfinite-dimensional irreducible complex representations ( π, V ) of u C .Proof. There are u , . . . , u m real compact Lie algebras such that u ≃ m M j =1 u ( j ) u C ≃ m M j =1 u ( j ) C , and u ( j ) C is simple for all j . There is also a root system ∆ = S mj =1 ∆ ( j ) of u C such that∆ ( j ) is a root system of u ( j ) C with associated real compact form u ( j ) . Every irreducible representation of u C is of the form ( π, V (1) ⊗ · · · ⊗ V ( m ) ), where( π ( j ) , V ( j ) ) is an irreducible representation of u ( j ) C for all j and(3.7) π ( X , . . . , X m ) v (1) ⊗ · · · ⊗ v ( m ) = m X j =1 v (1) ⊗ · · · ⊗ π ( j ) ( X j ) v j ⊗ · · · ⊗ v ( m ) for all X j ∈ u ( j ) C and v ( j ) ∈ V ( j ) for all j .By Lemma 2.2, there are infinitely many irreducible representations ( π, V (1) ⊗· · ·⊗ V ( m ) )of u C such that, for every j , V ( j ) contains a weight η ( j ) in a Weyl chamber associated to∆ ( j ) . It follows that η := η (1) + · · · + η ( m ) is in a Weyl chamber of ∆. In fact, if α ∈ ∆, then α ∈ ∆ ( j ) for some j , thus h α, η i = h α, η ( j ) i 6 = 0. The proof follows from Theorem 3.4. (cid:3) Remark . For g a complex semisimple non-simple Lie algebra, Lemma 2.2 is not longertrue. We assume for simplicity that g = g ⊕ g . We fix V an irreducible complexrepresentation of g having no weight in a Weyl chamber. Thus, for any irreduciblecomplex representation W of g , the irreducible complex representation V ⊗ W of g doesnot contain any weight in a Weyl chamber, and of course, there are infinitely many suchrepresentations.3.4. Other approaches.
The next result is our second approach to use Theorem 3.1.We call it the
Weyl group orbit approach because S is the Weyl orbit of a weight of therepresentations satisfying certain conditions. Theorem 3.7.
Let ( π, V ) be a complex representation of u C . We assume there is a non-zero weight µ of π such that µ + α / ∈ W · µ for all α ∈ ∆ . By setting V = M w ∈ W V ( w · µ ) , V = M µ/ ∈ W · µ V ( µ ) , all the properties in Theorem 3.1 hold. Consequently, the real Lie algebra l ( u , π ) admitsan inner product with negative Ricci curvature.Proof. The decomposition V = V ⊕ V coincides with the one from Proposition 3.2 bytaking S = W · µ , the Weyl orbit of µ . It follows that properties (1), (3), (4), and (6)hold.For α ∈ ∆ + and w ∈ W , π ( X α ) and π ( Y α ) map V ( w · µ ) to V ( w · µ + α ) ⊕ V ( w · µ − α )by (2.7). If V ( w · µ ± α ) ⊂ V , then w · µ ± α ∈ W · µ , thus µ ± w − · α ∈ W · µ , whichcontradicts the assumption since W (∆) ⊂ ∆. Hence, π ( X α ) and π ( Y α ) map V ( w · µ )into V for all α ∈ ∆ + and w ∈ W , thus property (2) holds.It only remains to establish the validity of (5). We fix α ∈ ∆ + . Let w ∈ W satisfyingthat h α, w · µ i 6 = 0. Such element exists because α = 0 and Span C ( W · µ ) = h ∗ (since µ = 0). We assume that the integer m := h α, w · µ i is positive; the negative case isanalogous.Let v ∈ V ( w · µ ) with v = 0. We have that π ( H α ) v = ( w · µ )( H α ) v = h α, w · µ i v = mv . Similarly as in the proof of Lemma 3.3, the representation theory of sl (2 , C ) impliesthat π ( X − α ) k v = 0 for all 0 ≤ k ≤ m . In particular, π ( X − α ) v = 0, hence π ( X α ) | V and π ( Y α ) | V are non-trivial. This completes the proof. (cid:3) The Weyl group orbit approach has been implicitly used in [12]. In fact, the proof of[12, Thm. 1.1] decomposes any polynomial representation π of a classical Lie algebra asin Theorem 3.7 with µ the highest weight of π .We now state the zero weight approach which picks S = { } . EGATIVE RICCI CURVATURE 11
Theorem 3.8.
Let ( π, V ) be a complex representation of u C . We assume that and allthe roots are weights of π . By setting V = V (0) and V = L µ =0 V ( µ ) , all the propertiesin Theorem 3.1 hold. Consequently, the real Lie algebra l ( u , π ) admits an inner productwith negative Ricci curvature.Proof. The decomposition V = V ⊕ V is as in Proposition 3.2 with S = { } , thus it onlyremains to check that property (5) holds.Let α ∈ ∆ + . By assumption, α is a weight of π , that is, V ( α ) = 0. Since h α, α i is apositive integer, Lemma 3.3 ensures that π ( X α ) | V (0) and π ( Y α ) | V (0) are non-trivial. (cid:3) Remark . It is clear that the adjoint representation of u C satisfies the assumptions ofTheorem 3.8. 4. Explicit examples
This section contains many explicit examples of metric Lie algebras with negative Riccioperator constructed from Theorems 3.4, 3.7 and 3.8.We recall that the irreducible complex representations of u C are in correspondencewith elements in P + ( u ) = L ni =1 Z ≥ ω i , where ω , . . . , ω n are the fundamental weightscorresponding to the simple root system Π = { α , . . . , α n } . In the sequel, when we considera particular complex simple Lie algebra, we will use the positive root system chosen in [6, § C.1–2].If λ = P ni =1 a i ω i ∈ P + ( u C ) satisfies a i > i , then λ lies in the fundamentalWeyl chamber C + . It follows immediately that l ( u , π λ ) admits an inner product havingnegative Ricci curvature from the Weyl chamber approach (Theorem 3.4). It remains toanalyze those dominant integral weights lying in the faces of C + , namely ( n X i =1 a i ω i : a i ∈ N ∀ i ∈ I, a i = 0 ∀ i / ∈ I ) for each non-empty and proper subset I of { , . . . , n } .When u = su (2), every non-trivial dominant integral weight is in the fundamental Weylchamber. Consequently, l ( su (2) , π ) admits an inner product with negative Ricci curvaturefor all non-trivial complex irreducible representation π of u C . Indeed, this fact was provedin [11, § u C is two, the dominant integral weights λ such that one of our approaches applies and therefore l ( u , π λ ) admits an inner productwith negative Ricci operator. Proposition 4.1.
Let u be a real Lie algebra such that u C is simple of rank two. Then, l ( u , π λ ) admits an inner product with negative Ricci curvature for every λ ∈ P + ( u C ) exceptpossibly { , ω , ω } for types A and B = C , and { , ω } for type G .Proof. For short, we say that a complex representation π of u C has Ricci negative curvatureif l ( u , π λ ) admits an inner product with negative Ricci operator.Every dominant integral weight is of the form aω + bω for some non-negative integers a, b . From the above discussion, the cases ab ≥ aω and aω for a ∈ N .We first assume that u C is of type G . Let α and α denote the simple roots, thus∆ + = { α , α , α + α , α + α , α + α , α + 2 α } and the fundamental weights are ω = 2 α + α and ω = 3 α + 2 α .From the Weyl chamber approach, π aω (resp. π aω ) has negative Ricci operator if a ≥ a ≥
2) since λ − α ∈ P λ ( u C ) ∩ C + ( λ − α − α ∈ P λ ( u C ) ∩ C + ). The case λ = ω follows from the zero weight approach (Theorem 3.8) since P λ ( u C ) = ∆ ∪ { } . The case λ = 2 ω follows from the Weyl orbit approach (Theorem 3.7) since λ + α / ∈ W · λ for all α ∈ ∆.The rest of the cases follows from the more general result in Lemma 4.2 below. (cid:3) Lemma 4.2.
Let u be a compact real simple Lie algebra such that u C is of classical typeand rank n . Then l ( u , π aω p ) admits an inner product with negative Ricci curvature forevery a ≥ and ≤ p ≤ n .Proof. Let u C be the simple complex Lie algebra of type C n for some n ≥
2. There is a C -basis { ε , . . . , ε n } of h ∗ such that ∆ + = { ε i ± ε j : 1 ≤ i < j ≤ n } ∪ { ε i : 1 ≤ i ≤ n } , ω p = ε + · · · + ε p for any 1 ≤ p ≤ n , W ≃ S n × {± } n and the element ( σ, { t i } ni =1 ) acts by P ni =1 a i ε i P ni =1 t i a σ ( i ) ε i . Thus W · aω p = {± aω i : 1 ≤ i ≤ n } . It follows immediatelythat aω p + α / ∈ W · aω p for all α ∈ ∆. Thus, the Lemma follows by Theorem 3.7.The rest of the cases are very similar. (cid:3) Remark . One can easily check that the non-trivial exceptions in Proposition 4.1 (i.e. π ω , π ω for types A and B , and π ω for type G ) do not follow from any of the threeapproaches. In case it is possible, it would be interesting to find in any of the correspondingLie algebras an inner product with negative Ricci curvature. We note that beyond thesolvable case there is no necessary condition in the literature for a Lie group to admit ametric with negative Ricci curvature. This makes the problem to prove that a Lie algebradoes not admit such a metric, a challenging one.5. General nilradical
Finally, we can use the same idea as in [12, Thm. 5.4] to get examples with a non-abelian nilradical. Explicitly, we will consider the more general setting of a Lie algebra g = ( R Z ⊕ u ) ⋉ n where u is a compact semisimple Lie algebra acting on a nilpotent Liealgebra n by derivations and as always, Z commutes with u . Note that in order to get aLie algebra, ad Z must be a derivation of n and therefore could not act as a multiple ofthe identity unless n is abelian.First we will prove a more general version of Theorem 3.1. Theorem 5.1.
Let u be a compact semisimple Lie algebra, and let ( π, V ) be a finitedimensional real representation of u . We assume V decomposes in u -submodules as V = W ⊕ · · · ⊕ W k such that, for some index i , there exists a decomposition W i = V ′ ⊕ V ′ anda real inner product on W i satisfying properties (1)–(6) from Theorem 3.1. Then, for allpositive real numbers c , . . . , c k , the Lie algebra ( R Z ⊕ u ) ⋉ V determined by ad Z | u = 0 and ad Z | Wi = c i Id W i admits a Ricci negative inner product.Proof. Assume W i = V ′ ⊕ V ′ , h· , ·i i satisfies (1)–(6) from Theorem 3.1. It is not hard tocheck that all these hypotheses remain valid for V = V ⊕ V and h· , ·i where V = V ′ , V = V ′ ⊕ P j = i W j and h· , ·i is obtained by extending h· , ·i i to V in such a way that V = W ⊕ · · · ⊕ W k is an orthogonal decomposition and such that every element of u acts as a skew-symmetric operator of P j = i W j (see (3.2)). Then, Theorem 3.1 proves thetheorem for the case c = · · · = c k = 1. Moreover, the case c = · · · = c k also followsfrom Theorem 3.1 since positive rescaling at the element Z in the definition (1.2) of l ( u , π )returns isomorphic Lie algebras.For the general case, we can follow the same proof as in [12, Thm. 3.3] with minorchanges coming from the fact that we have a different mean curvature vector on e l eventhough it still is a multiple of Z . We next sketch this proof by following the notation in[12]. EGATIVE RICCI CURVATURE 13
Let e l denote the Lie algebra ( R Z ⊕ u ) ⋉ V determined by ad Z | u = 0 and ad Z | Wi = c i Id W i .We first note that the same degeneration given in [12, Lem. 3.1] shows that e l degeneratesinto the solvable Lie algebra e l ∞ = ( R Z ⊕ u ⊕ V, µ ) which only differs from l ∞ by theaction of Z . The next step is to show that e l ∞ admits an inner product with negative Riccioperator. Note that the mean curvature vector of e l ∞ is H = c Z, where c = tr(ad µ Z ) = k X i =1 c i dim W i . We proceed by noting that − c ad Z is negative definite on V and diagonalizes in any basisformed as a union of basis of W j , so one can find an appropriated ρ . Finally note thatbecause of this diagonalization property, we can perturb the inner product on V and stillget a block diagonal Ricci operator as in [12]. (cid:3) Let g be a non-solvable real Lie algebra. From its Levi decomposition, we know thereis a semisimple Lie subalgebra u of g (called the Levi subalgebra of g ) such that g ≃ u ⋉ s ,where s is the radical of g . In the next theorem, we will assume that u is compact andthe nilradical n of s has codimension one. It is not hard to see that one can always geta complement a of n in s such that [ u , a ] = 0 (see [12] after Theorem 5.2). Thus, for anon-zero element Z ∈ a , we have that g ≃ ( R Z ⊕ u ) ⋉ n , with [ Z, u ] = 0. Theorem 5.2.
Let g = ( R Z ⊕ u ) ⋉ n be a real Lie algebra where u is a compact semisimpleLie algebra, n is a nilpotent Lie algebra and Z is a non-zero element commuting with u .Let n = n ⊕ · · · ⊕ n k be the decomposition of n as an u -module in irreducible components.If any of the n j admits a decomposition as in Theorem 3.1 and ad Z | n acts as a positivemultiple of the identity in every n i , then g admits an inner product with negative Riccicurvature.Proof. We know that ad Z | n i = c i Id n i for some c i >
0. We set a = R Z .It is shown in [12, Theorem 5.4 or (31)] that g = ( R Z ⊕ u ⊕ n , [ , ]) degenerates into aLie algebra l = ( R Z ⊕ u ⊕ n , [ , ] ) where the action of R Z ⊕ u on n is the same as in g but n is an abelian ideal of l . In fact, for each t > ψ t ∈ gl ( g ) such that ψ t | a ⊕ u = Id , ψ t | n = t Id . Using that n is an ideal of g , it is easy to check that [ , ] t := ψ t · [ , ] is given by(5.1) [ X , X ] t = [ X , X ] for X , X ∈ a ⊕ u , [ X , X ] t = t [ X , X ] for X , X ∈ n , [ X , X ] t = [ X , X ] for X ∈ a ⊕ u , X ∈ n . Hence, lim t →∞ [ , ] t =: [ , ] is well defined and is given by(5.2) [ X , X ] = [ X , X ] for X ∈ a ⊕ u , X ∈ g , [ X , X ] = 0 for X i ∈ n . Note that since [ n , n ] = 0, the Lie algebra ( R Z ⊕ u ⊕ n , [ , ] ) is an l ( u , π ) as in (1.2). Now,by Theorem 5.1, it admits an inner product with negative Ricci curvature and thereforeso does g . (cid:3) We note that when ad Z | n is a positive definite operator with respect to some innerproduct on n or equivalently it is diagonalizable with positive eigenvalues, if it preservesan irreducible submodule, then it necessarily acts as a multiple of the identity. In fact, inthis case there always exist a decomposition of n in irreducible submodules such that ad Z acts as a positive multiple of the identity in every term. Hence, with the same idea of theabove proof we obtain the following theorem. Theorem 5.3.
Let g = ( R Z ⊕ u ) ⋉ n be a real Lie algebra where u is a compact semisimpleLie algebra, n is a nilpotent Lie algebra and Z is a non-zero element commuting with u .Assume that ad Z | n is diagonalizable with positive eigenvalues and n = n (1) ⊕ · · · ⊕ n ( k ) isthe corresponding decomposition in eigenspaces. If any of the irreducible components of n ( i ) as an u -module admits a decomposition as in Theorem 3.1, then g admits an innerproduct with negative Ricci curvature.Proof. Since n = n (1) ⊕· · ·⊕ n ( k ) is the decomposition in eigenspaces of ad Z | n and [ Z, u ] = 0, u preserves each n ( i ) and one therefore gets n = n (1)1 ⊕ . . . n (1) j ⊕ · · · ⊕ n ( k )1 ⊕ · · · ⊕ n ( k ) j k , a decomposition of n in irreducible submodules and ad Z | n ( i ) j = c i Id n ( i ) j for some c i > (cid:3) References [1]
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Mathoverflow question(Version: 2019-05-07). https://mathoverflow.net/q/330929 . Instituto de Matem´atica (INMABB), Departamento de Matem´atica, Universidad Nacionaldel Sur (UNS)-CONICET, Bah´ıa Blanca, Argentina.
E-mail address : [email protected] Universidad Nacional de C´ordoba, FaMAF and CIEM, 5000 C´ordoba, Argentina.
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