Small commutators in compact semisimple Lie groups and Lie algebras
aa r X i v : . [ m a t h . G R ] M a y SMALL COMMUTATORS IN COMPACT SEMISIMPLELIE GROUPS AND LIE ALGEBRAS
ALESSANDRO D’ANDREA AND ANDREA MAFFEI C ONTENTS
1. Introduction 12. Openness of the commutator map in a compact semisimple Lie algebra 23. Openness of the commutator map in a Lie group 4References 51. I
NTRODUCTION
Let G be a connected semisimple (Lie or algebraic) group. Then G equals its derived subgroupand it is expected that, in many cases, every element of G is indeed a commutator. The problem ofunderstanding under what conditions this claim holds, or at least every element can be expressedas a product of a uniformly bounded quantity of commutators, has been investigated at length.The fact that every element in a semisimple compact Lie group is a commutator dates back toGotˆo [7], whereas counterexamples are easy to construct in non compact cases — for instance,in SL ( R ) , − id does not arise as a commutator. Later, Thompson [13] provided a classificationof all groups of the form SL n ( k ) , where k is an arbitrary field, containing elements that are notcommutators.Connected semisimple groups are treated in the complex case in [10], and in [11] over analgebraically closed field of any characteristic. More recently, ¯Dokovi´c showed [6], under mildtechnical assumptions, that in the real semisimple case every element is a product of at most twocommutators.Many variations on the topic have also been considered. To name just a few, Brown consideredthe analogous statement in the case of simple Lie algebras [4]; Borel studied instead maps G n → G induced by nontrivial group words in n letters, showing [2] that they yield dominant maps.In this paper, we establish a different property of the commutator map in semisimple compactLie groups and Lie algebras: its openness at the identity element. We say that a map f betweentwo topological spaces X and Y is open at a point x ∈ X if for all neighborhoods U of x theimage f ( U ) is a neighborhood of f ( x ) , so that our claim amounts to showing that elements thatare sufficiently small (i.e., close to the identity element) arise as commutators of prescribedlysmall elements. The usual proofs that in a compact semisimple Lie group every element is acommutator provide little information towards this statement as they proceed by expressing eachelement in the group as the commutator between an element lying in a torus and some expression,which is typically “far” from the identity, depending on a nontrivial Coxeter element chosen inthe associated Weyl group.In Section 2, we treat the infinitesimal case of compact semisimple Lie algebras: in this setting,the commutator map is a surjective bilinear map. It was a classical problem answered negativelyby Cohen and Horowitz to establish whether the surjectivity of a bilinear map implies its openessin zero [5, 8]. Date : September 17, 2018.
We show that the commutator map of compact semisimple Lie algebras can be inductivelyproved to be open by exploiting combinatorial properties of the corresponding root systems. Thebasis of induction corresponds to Lie algebras of type A , and needs to be done by hand.Our next step is to integrate to the group level the knowledge we have gathered for Lie alge-bras. Once more, this is not totally immediate: indeed, the commutator map for a Lie algebra isonly a second order approximation of the commutator map for the corresponding Lie group, andwe face a scarcity of tools for translating second order information from the infinitesimal levelto the local one.In Section 3, we solve this technical issue in two steps. First, we integrate half of the commu-tator map of the Lie algebra g to the map g × g ∋ ( x, y ) x − exp(ad y ) x ∈ g , and then use techniques from Rouvi`ere [12], related to the Kashiwara-Vergne method, to get tothe group level.We should stress that our strategy employs more than once the fact that all elements in acompact Lie group (resp. Lie algebra) lie in a torus, and therefore does not immediately extendto noncompact structures.We would like to thank Alessandro Berarducci for drawing our attention to this problem,which arises from his work on definable groups. Our openness statement is equivalent to theclaim that every element belonging to the infinitesimal neighbourhood of the identity (which isa perfect subgroup) in the non-standard version of a compact semisimple Lie group is a com-mutator. This issue was considered by Berarducci, Peterzil and Pillay — see in particular thecomments after [1, Proposition 2.14] — in connection to the question whether a finite centralextension of a group definable in an o -minimal structure M is interpretable in M . A positiveanswer to the latter question would also imply that a finite central extension of a compact Liegroup has an induced Lie structure making the extension a topological cover.2. O PENNESS OF THE COMMUTATOR MAP IN A COMPACT SEMISIMPLE L IE ALGEBRA
Throughout the paper, G will be a semisimple Lie group and g its Lie algebra. On g weconsider the Killing form κ g .In the following Lemma we characterize pairs of maximal toral subalgebras of su n which areorthogonal to each other with respect to the Killing form. If u , . . . , u n is an orthonormal basisof C n then we denote by t u the set of elements in su n which are diagonal with respect to thisbasis. Lemma 2.1.
Let u , . . . , u n and v , . . . , v n be two orthonormal bases of C n . Then t u is orthog-onal to t v if and only if (2.1) | u i · v h | = | u j · v k | , for all i, j, h, k .Proof. Define U ij ∈ su n by U ij ( u h ) = √− u i if h = i ; −√− u j if h = j ;0 otherwise . and similarly define V ij using the orthonormal basis v i . Then the operators U ij span t u and theoperators V ij span t v . Easy computations show thatTr (cid:0) U ij V hk (cid:1) = | u i · v k | + | u j · v h | − | u i · v h | − | u j · v k | . Hence t u is orthogonal to t v if and only if | u i · v k | + | u j · v h | = | u i · v h | + | u j · v k | MALL COMMUTATORS IN COMPACT SEMISIMPLE LIE GROUPS AND LIE ALGEBRAS 3 for all i, j, h, k . Hence if Equation 2.1 is verified, then the two subalgebras are orthogonal. Viceversa, assume they are orthogonal. Then, summing over h , we see that the above equalities imply | u i · v k | = | u j · v k | for all i, j and summing over j we get | u i · v h | = | u i · v k | proving the claim. (cid:3) Lemma 2.2.
Let G be compact, t be a maximal toral subalgebra of g . Then there exists amaximal toral subalgebra of g orthogonal to t .Proof. We first analyse the case g = su n . Set ζ = e πin , and let u , . . . , u n be an orthonormalbasis of C n such that t = t u . For j = 1 , . . . , n define v j = 1 √ n ( u + ζ j u + ζ j u + · · · + ζ ( n − j u n ) . Then v , . . . , v n is an orthonormal basis of C n and | u i · v j | = 1 / √ n for all i, j . Hence t u isorthogonal to t v . For g not isomorphic to su n we proceed by induction on the rank of g . If g isnot simple the claim follows immediately by induction, so we assume that g is simple.Let g C (resp. t C ) be the complexification of g (resp. t C ), denote by Φ the associated rootsystem and choose a simple basis ∆ ⊂ Φ . Let ω α , for α ∈ ∆ , be the corresponding fundamentalweights, and θ be the highest root of Φ . Since g is not of type A there exists a simple root α suchthat θ = ω α or θ = 2 ω α , see [3, Planches II-IX]. Let Ψ be the root system generated by ∆ \ { α } .We can choose a standard Chevalley basis, h α with α ∈ ∆ and x α with α ∈ Φ such thatelements k α = √− h α u α = x α − x − α and v α = √− x α + x − α ) are a basis of g , see [9, Theorem 6.11, Formula (6.12)]. Notice that the subspace orthogonal of t is the linear span of the elements u α and v α . Define h = h k β , u β , v β : β ∈ Ψ i . This is the semisimple part of the maximal Levi subalgebra associated to α . In particular theclaim is true for h . Let s be a maximal toral subalgebra orthogonal to the maximal toral subalge-bra of h given by t ∩ h .Notice also that we have [ u θ , h ] = [ u − θ , h ] = 0 . Hence, for dimension reasons, s ⊕ R u θ is a maximal toral subalgebra of g orthogonal to t . (cid:3) We can now prove the following fact.
Theorem 2.1.
Let G be compact. Then the commutator map comm g : g × g ∋ ( x, y ) [ x, y ] ∈ g is open at (0 , .Proof. We need to prove that if U is neighbourhood of then comm g ( U × U ) contains a neigh-bourhood of . Notice first that being G compact we can assume that U is G -stable under theadjoint action.Choose now a regular element x ∈ U (an element is said to be regular if its centralizer is atoral subalgebra) and let t be its centralizer. Let m be the orthogonal of t . Then ad x : m −→ m is a linear isomorphism. Hence there exists a G -stable neighbourhood V of G such that ad x ( m ∩ U ) ⊃ m ∩ V . A. D’ANDREA AND A. MAFFEI
Consider now ψ : G × ( m ∩ U ) given by ψ ( g, y ) = Ad g [ x, y ] . It is clear that the image of ψ is contained in comm g ( U × U ) and that the image of ψ contains G · ( V ∩ m ) . Finally fromthe previous Lemma we have that G · ( V ∩ m ) = V . Hence comm g ( U × U ) ⊃ V proving theTheorem. (cid:3)
3. O
PENNESS OF THE COMMUTATOR MAP IN A L IE GROUP
We will now show that the commutator map
Comm G : G × G ∋ ( X, Y ) XY X − Y − ∈ G is open at (id , id) as soon as the corresponding infinitesimal commutator map comm g : g × g → g is open at (0 , . Lemma 3.1.
The map comm g is open at (0 , if and only if the map C g : g × g ∋ ( x, y ) x − exp(ad y ) x ∈ g is so.Proof. The map φ : g × g ∋ ( x, y ) (cid:18) exp(ad y ) − y ( x ) , y (cid:19) ∈ g × g is smooth, and has invertible differential in (0 , , so it is a local diffeomorphism. However, thecomposition of comm g ◦ φ equals C g . (cid:3) In order to deal with openness of the commutator map in a group, we are going to use thefollowing variant [12, Remarque 4.2], related to the Kashiwara-Vergne method, of the Baker-Campbell-Hausdorff formula.
Proposition 3.1.
There exist, in a neighbourhood of (0 , ∈ g × g , analytical functions P, Q : g × g → G, P (0 ,
0) = Q (0 ,
0) = id , satisfying exp( a + b ) = exp( P ( a, b ) .a ) exp( Q ( a, b ) .b ) , for all a, b . Theorem 3.1.
The group commutator map
Comm G is open at (id , id) as soon as the infinitesimalcommutator map comm g is open at (0 , .Proof. Let us apply Proposition 3.1 to a = x, b = − exp(ad y ) x . Using the notation introducedabove, we set P = P ( a, b ) , Q = Q ( a, b ) and obtain exp( x − exp(ad y ) x ) = exp( P.x ) exp( − Q. exp(ad y ) x )= P exp( x ) P − ( Q exp( y )) exp( − x )( Q exp( y )) − = ABA − B − , where A = P exp( x ) P − , B = Q exp( y ) P − .Let U be a ( G -stable) neighbourhood of ∈ g on which exp restricts to a diffeomorphism.The map ψ : g × g → G defined by ( x, y ) Q ( x, exp(ad y ) x ) exp( y ) P ( x, exp(ad y ) x ) − is analytical, hence continuous. We may then find a neighbourhood U ′ of ∈ g , which weassume to be G -stable and contained in U , such that ψ ( U ′ × U ′ ) ⊂ exp U . If x, y lie in U ′ , then A and B = ψ ( x, y ) lie in exp U ; moreover, the composition exp ◦ C g maps ( x, y ) in ABA − B − and is open at (0 , .We thus conclude that all elements in a suitable neighbourhood of id ∈ G arise as commutatorsof elements from exp U . (cid:3) Corollary 3.1. If G is a compact semisimple Lie group, then Comm G is open at (id , id) . MALL COMMUTATORS IN COMPACT SEMISIMPLE LIE GROUPS AND LIE ALGEBRAS 5 R EFERENCES [1] Alessandro Berarducci, Ya’acov Peterzil and Anand Pillay,
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