Algebraic hulls of solvable groups and exponential iterated integrals on solvmanifolds
aa r X i v : . [ m a t h . G T ] M a y ALGEBRAIC HULLS OF SOLVABLE GROUPS ANDEXPONENTIAL ITERATED INTEGRALS ON SOLVMANIFOLDS
HISASHI KASUYA
Abstract.
We represent the coordinate ring of algebraic hulls (which are gen-eralizations of the Malcev completions of nilpotent groups for solvable groups)of solvmanifolds G/ Γ by using Miller’s exponential iterated integrals (whichare extensions of Chen’s iterated integrals) of invariant differential forms. Introduction
Let G be a simply connected Lie group and g the Lie algebra of G . We considerthe space V g ∗ C of C -valued left G -invariant differential forms on G . We assume that G has a lattice (i. e. cocompact discrete subgroup) Γ. We consider the compact ho-mogeneous space G/ Γ and V g ∗ C as a subcomplex of the de Rham complex A ∗ C ( G/ Γ)of G/ Γ. Suppose G is nilpotent. Then we have the unique unipotent algebraic group U Γ called the Malcev completion of Γ such that there is a injection Γ → U Γ with theZariski-dense image. We can represent the coordinate ring of U Γ by using Chen’siterated integrals on G/ Γ (see [2]). Since the inclusion V g ∗ C ⊂ A ∗ C ( G/ Γ) induces acohomology isomorphism by Nomizu’s theorem [11], V g ∗ C is the Sullivan minimalmodel of A ∗ C ( G/ Γ) (see [4]). This implies H ( ¯ B ( V g ∗ C )) ∼ = H ( ¯ B ( A ∗ C ( G/ Γ))) where¯ B ( V g ∗ C ) and ¯ B ( A ∗ C ( G/ Γ)) are the reduced bar constructions of V g ∗ C and A ∗ C ( G/ Γ)respectively (see [3]). Hence we can represent the coordinate ring of U Γ by usingChen’s iterated integrals of left-invariant forms.Suppose G is solvable. Then Chen’s iterated integrals on G/ Γ does not givesufficient information on the fundamental group of G/ Γ. For example, let G = R ⋉ φ R such that φ ( t ) = (cid:18) e t e − t (cid:19) . Then G has a lattice Γ and the inclusion V g ∗ C ⊂ A ∗ C ( G/ Γ) induces a cohomology isomorphism (see [5]). Since we have H ( G/ Γ , C ) = H ( V g ∗ C ) = C , by Chen’s results, iterated integrals represent thecoordinate ring of a additive algebraic group G ad = C (see [7]). But since Γ issolvable and not abelian, we can’t embed Γ in G ad .In [10], as an extension of the Malcev completion, Mostow constructed a Zariski-dense embedding of Γ in an algebraic group H Γ called algebraic hull of Γ. In[7], Miller gave extensions of Chen’s iterated integrals called exponential iteratedintegrals. In this paper we represent the coordinate ring of H Γ by using Miller’sexponential iterated integrals of left-invariant differential forms on G/ Γ.2.
Relative completions and algebraic hulls
Let G be a discrete group (resp. a Lie group). We call a map ρ : G → GL n ( C ) arepresentation, if ρ is a homomorphism of groups (resp. Lie groups). In this paper Key words and phrases. exponential iterated integral, algebraic hull, solvmanifold. we denote by T n ( C ) the group of n × n upper triangular matrix and denote by U n ( C ) the group of n × n upper triangular unipotent matrix.2.1. algebraic groups and pro-algebraic groups. In this paper an algebraicgroup means an affine algebraic variety G over C with a group structure such thatthe multiplication and inverse are morphisms of varieties. All algebraic groups ariseas Zariski-closed subgroups of GL n ( C ). A pro-algebraic group is an inverse limit ofalgebraic groups. If a pro-algebraic group is an inverse limit of unipotent algebraicgroups, it is called pro-unipotent. Let G be a pro-algebraic group. We denote by U ( G ) the maximal pro-unipotent normal subgroup called the pro-unipotent radical.If U ( G ) = e , G is called reductive. We denote by C [ G ] the coordinate ring of G .The group structure on G induces a Hopf algebra structure on C [ G ]. It is knownthat we have the anti-equivalence between algebras and affine varieties induces ananti-equivalence between pro-algebraic groups and reduced Hopf algebras. Theorem 2.1. ([9],[6])
Let G be a pro-algebraic group. Then the exact sequence / / U ( G ) / / G / / G / U ( G ) / / splits. Let G be a discrete group or Lie group. We denote by A ( G ) the inverse limit ofall representations φ : G → G with Zariski-dense images. We call the pro-unipotentradical U ( A ( G )) of A ( G ) the unipotent hull of G and denote it U G .2.2. Relative completion.
Let ρ : G → S be a representation of G to a diagonalalgebraic group S with the Zariski-dense image. Let φ : G → G be a representationof G to an algebraic group G with the Zariski-dense image. We call φ a ρ -relativerepresentation if we have the commutative diagram1 / / U ( G ) / / G / / S / / G φ O O ρ ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) . If S is contained in an algebraic torus, for any ρ -relative representation φ : G → G there exists a faithful representation G ֒ → T n ( C ) such that G ∩ U n ( C ) = U ( G )(see [7]).We denote by G ρ ( G ) the inverse limit of ρ -relative representations φ i : G → G i .We call G ρ ( G ) the ρ -relative completion of G . If S is trivial, G ρ ( G ) is the classicalMalcev (or unipotent) completion.2.3. Algebraic hulls.
We define the algebraic hulls of polycyclic groups (resp. Liegroups) which constructed in [10].A group Γ is polycyclic if it admits a sequenceΓ = Γ ⊃ Γ ⊃ · · · ⊃ Γ k = { e } of subgroups such that each Γ i is normal in Γ i − and Γ i − / Γ i is cyclic. For apolycyclic group Γ, we denote rank Γ = P i = ki =1 rank Γ i − / Γ i . Let G be a simply con-nected solvable Lie group and Γ be a lattice of G . Then Γ is torsion-free polycyclicand dim G = rank Γ.Let G be a simply connected solvable Lie group or torsion-free polycyclic group.Consider the algebraic completion A ( G ). Then it is known that the unipotent hull LGEBRAIC HULLS AND EXPONENTIAL ITERATED INTEGRALS 3 U G = U ( A ( G )) is finite dimensional (see [10]). By Theorem 2.1, we have a splitting A ( G ) = ( A ( G ) / U G ) ⋉ φ U G . Let K be the kernel of the action φ : ( A ( G ) / U G ) → Aut( U G ). Then K is the maximal reductive normal subgroup of A ( G ) (see [10]).Denote H G = A ( G ) /K and call it the algebraic hull of G . Theorem 2.2. ([10],[13])
Let G be a simply connected solvable Lie group (resp.torsion-free polycyclic group). Then G → H G is injective and H G is a finite di-mensional algebraic group such that: (1) dim U ( H G ) = dim G (resp. = rank G ). (2) The centralizer of U ( H G ) in H G is contained in U ( H G ) .Conversely if an algebraic group H with an injective homomorphism ψ : G → H with the Zariski-dense image satisfies the properties (1) and (2) , then H is isomor-phic to H G . Direct constructions of algebraic hulls.
The idea of this subsection isbased on classical works of semi-simple splitting (see [14], [12] and the referencesgiven there). Let g be a solvable Lie algebra, and n = { X ∈ g | ad X is nilpotent } . n is the maximal nilpotent ideal of g and called the nilradical of g . Then wehave [ g , g ] ⊂ n . Let D ( g ) be the Lie algebra of derivations of g . By the Jordandecomposition, we have ad X = ad sX + ad nX such that ad sX is a semi-simpleoperator and ad nX is a nilpotent operator. Since we have d X , n X ∈ D ( g ), we havethe map ad s : g → D ( g ). Since ad is trigonalizable (Lie’s theorem), this map ishomomorphism with the kernel n . Let ¯ g = Im ad s ⋉ g . and ¯ n = { X − ad sX ∈ ¯ g | X ∈ g } . Proposition 2.3. ¯ n is a nilpotent ideal of ¯ g and we have a decomposition ¯ g =Im ad s ⋉ ¯ n .Proof. By ad X − ad sX = ad X − ad sX on g , ad X − ad sX is a nilpotent operator andhence ¯ n consists of nilpotent elements. By Lie’s theorem, we have a basis X , . . . , X l , X l +1 . . . , X n of g ⊗ C such that ad is represented by upper triangular matrices. Since the nilradical n is an ideal, n ⊗ C is ad-invariant subspace of g ⊗ C . We choose X , . . . , X l a basisof n ⊗ C . By [ g , g ] ⊂ n , we have ad X ( g ⊗ C ) ⊂ n ⊗ C = h X , . . . , X l i , and hence adrepresented asad X = a ( X ) . . . . . . a l ( X ). . . ... a ll ( X ) . . . a lm ( X )0 . . .
0. . . ...0 . Thus we have ad sX ( X i ) = a ( X ) X i for 1 ≤ i ≤ l and ad sX ( X i ) = 0 for l + 1 ≤ i ≤ n . By this we have[ X i + ad sY , X j + ad sZ ] ∈ h X , . . . , X l i = n ⊗ C for any 1 ≤ i, j ≤ n , Y, Z ∈ g . This implies [¯ g , ¯ g ] ⊂ n . By n ⊂ ¯ n , ¯ n is an ideal of ¯ g and we have ¯ g = { ad sX + Y − ad sY | X, Y ∈ g } = Im ad s ⋉ ¯ n . (cid:3) HISASHI KASUYA
By this proposition, we have the inclusion i : g → D (¯ n ) ⋉ ¯ n given by i ( X ) =ad sX + X − ad sX for X ∈ g .Let G be a simply connected solvable Lie group and g be the Lie algebra of G .For the adjoint representation Ad : G → Aut( g ), we consider the semi-simple partAd s : G → Aut( g ) as similar to the Lie algebra case. Denote by T the universalcovering of Ad s ( G ). Let ¯ N be the simply connected Lie group which correspondsto ¯ n . Then by Proposition 2.3, we have T ⋉ G = T ⋉ ¯ N . By the proof of thisproposition, the action T → Aut(¯ n ) is the extension of the action of Im ad s . Hencewe have Ad s ( G ) ⋉ G = Ad s ( G ) ⋉ ¯ N .A simply connected nilpotent Lie group is considered as the real points of aunipotent R -algebraic group (see [12, p. 43]) by the exponential map. We have theunipotent R -algebraic group ¯N with ¯N ( R ) = ¯ N . We identify Aut a ( ¯N ) with Aut( n C )and Aut a ( ¯N ) has the R -algebraic group structure with Aut a ( ¯N )( R ) = Aut( N ). Sowe have the R -algebraic group Aut a ( ¯N ) ⋉ ¯N . Then by Ad s ( G ) ⋉ G = Ad s ( G ) ⋉ ¯ N ⊂ Aut a ( ¯N ) ⋉ ¯N , we consider the Zariski-closure G of G in Aut a ( ¯N ) ⋉ ¯N . Since Ad s ( G )is a group of diagonal automorphisms, we have U ( G ) = ¯ N . By dim G = dim ¯ N ,we can easily check that G satisfies the properties (1), (2) in Theorem 2.2 andhence it is the algebraic hull H G of G . Hence the inclusion i : g → D (¯ n ) ⋉ ¯ n induces the algebraic hull I : G → H G of G . Since i : g → D (¯ n ) ⋉ ¯ n is given by i ( X ) = ad sX + X − ad sX ∈ D (¯ n ) ⋉ ¯ n , the composition G → H G → H G / U ( H G )is induced by the Lie algebra homomorphism ad s : g → D ( g ) by U ( G ) = ¯ N . Thuswe have the following lemma. Lemma 2.4.
The algebraic hull G → H G is an Ad s -relative representation. Algebraic hulls and relative completions of solvable groups.Theorem 2.5.
Let G be a simply connected Lie group. Then the algebraic hull H G is isomorphic to the Ad s -relative completion G Ad s ( G ) of G .Proof. Consider a commutative diagram H ′ Φ / / H G G O O = = ④④④④④④④④ for some Ad s -relative representation G → H ′ . Since G → H ′ and G → H G have Zariski-dense images, Φ : H ′ → H G is surjective and the restriction Φ : U ( H ′ ) → U ( H G ) is also surjective. By U ( H G ) = U G , Φ : U ( H ′ ) → U ( H G ) isan isomorphism. Since G → H ′ and G → H G are Ad s -relative representations,Φ induces the isomorphism H ′ / U ( H ′ ) → H G / U ( H G ). Hence Φ : H ′ → H G isan isomorphism. By the definition of Ad s -relative completion of G , we have thetheorem. (cid:3) Theorem 2.6.
Let G be a simply connected solvable Lie group and Γ a latticeof G . Then the algebraic hull H Γ of Γ is isomorphic to Ad s | Γ -relative completion G Ad s | Γ (Γ) of Γ .Proof. For the algebraic hull ψ : G → H G of G , we consider the Zariski-closure of ψ (Γ) in H G . Then by dim G = rank Γ we can easily check that this algebraic groupsatisfies (1) and (2) in Theorem 2.2 and hence it is the algebraic hull H Γ of Γ. By LGEBRAIC HULLS AND EXPONENTIAL ITERATED INTEGRALS 5 the above theorem, Γ → H Γ is a Ad s | Γ -relative representation. As similar to theabove proof, we have the theorem. (cid:3) Exponential iterated integral on solvmanifolds
In this section we consider Miller’s exponential iterated integrals. Let M be a C ∞ -manifold and Ω x M be a space of piecewise smooth loops λ : [0 , → M with λ (0) = x . For 1-forms ω , . . . , ω n ∈ A ∗ C ( M ), the iterated integral R ω ω · · · ω n :Ω x M → C is defined by Z λ ω ω · · · ω n = Z ≤ t ≤ t ≤··≤ t n ≤ F ( t ) F ( t ) · · · F ( t n ) dt dt · · · dt n λ ∈ Ω x M where F i ( t ) dt = λ ∗ ω i ∈ A ([0 , δ , δ , · · · , δ n , ω , ω , · · · , ω n − n ∈ A C ( M ) Miller defined the exponential iterated integral R e δ ω e δ ω ··· e δ n − ω n − n e δ n :Ω x M → C as Z λ e δ ω e δ ω · · · e δ n − ω n − n e δ n = X m ,m , ··· m n ≥ Z λ δ . . . δ | {z } m terms ω δ . . . δ | {z } m terms · · · ω n − n δ n . . . δ n | {z } m n terms . Then this infinite sum converges (see [7]). Let L ⊂ A C ( M ) be a finitely generated Z -module of 1-forms such that dL = 0. We denote E L ( M, x ) the C -vector space offunctions on Ω x M generated by { Z e δ ω · · · ω n − n e δ n | δ , · · · δ n ∈ L, ω , ω , · · · , ω n − n ∈ A ∗ C ( M ) } . If I ∈ E L ( M, x ) is constant on homotopy classes of loops λ : [0 , → M relative to { , } , we call I a closed exponential iterated integral. Let H ( E L ( M, x )) denotethe subspace of closed exponential iterated integrals. Take a Z -basis { δ , δ , . . . , δ n } of L . Then we have the diagonal representation ρ : π ( M, x ) → D n ( C ) such that ρ ( λ ) = diag( R λ e δ , . . . , R λ e δ n ) for λ ∈ π ( M, x ). Consider the ρ -relative completion G ρ ( π ( M, x )) of the fundamental group of M . Miller showed the following theorem. Theorem 3.1. ([7, Theorem 6.1])
The space H ( E L ( M, x )) is a Hopf algebra andwe have a Hopf algebra isomorphism H ( E L ( M, x )) ∼ = C [ G ρ ( π ( M, x ))] . Remark 3.1.
For any ρ -relative representation φ : π ( M, x ) → G , Miller showedthat φ is the monodromy of a flat connection ω on M × C n whose connectionform is an upper triangular matrix. Then the monodromy of ω is given by I + P ∞ i =1 R ωω · · · ω | {z } i terms and its matrix entries are exponential iterated integrals. In theproof of Theorem 6.1 of [7] , Miller showed that these matrix entries generate thecoordinate ring C [ G ] . Consider a simply connected solvable Lie group G with a lattice Γ. Take adiagonalization of the semi-simple part ad s of the adjoint representation ad on g .Write ad s = diag( δ , . . . , δ n ) where δ , . . . , δ n are characters of g . By δ , . . . , δ n ∈ Hom( g , C ), we regard δ , . . . , δ n as left-invariant closed 1-forms. Let L be the Z -module generated by δ , . . . , δ n . Consider the algebraic hull H Γ of Γ. Since we have π ( G/ Γ , x ) ∼ = Γ, by Theorem 2.6, we have: HISASHI KASUYA
Corollary 3.2.
We have a Hopf algebra isomorphism H ( E L ( G/ Γ , x )) ∼ = C [ H Γ ] . Let E L ( g ∗ C ) denote the subvector space of E L ( G/ Γ , x ) generated by { Z e δ ω · · · ω n − n e δ n | δ , · · · δ n ∈ L ω , ω , · · · , ω n − n ∈ g ∗ C } . Studying the proof of [7, Lemma 5.1], we can see that E L ( g ∗ C ) is closed under themultiplication. We define the subring H ( E L ( g ∗ C )) = E L ( g ∗ C ) ∩ H ( E L ( G/ Γ , x ))of H ( E L ( G/ Γ , x )). Theorem 3.3.
We have H ( E L ( g ∗ C )) = H ( E L ( G/ Γ , x )) .Proof. Consider the algebraic hull ψ : G → H G of G . Since ψ : G → H G is Ad s -relative, we can assume H G ⊂ T r ( C ) and U r ( C ) ∩ H G = U ( H G ) as in Section 2.2.Let ψ ∗ : g → t r ( C ) be the derivative of ψ where t r ( C ) is the Lie algebra of T r ( C ).We write ψ ∗ = ω ω · · · ω r . . . . . . .... . . . . . ω r − r ω rr as we consider ψ ∗ ∈ Hom( g , C ) ⊗ T r ( C ). Then we have( dψ ∗ − ψ ∗ ∧ ψ ∗ )( X, Y ) = ψ ∗ ([ X, Y ]) − [ ψ ∗ ( X ) , ψ ∗ ( Y )] = 0for X, Y ∈ g . Hence we have the flat connection d − ψ ∗ on the vector bundle G × C r . Consider the parallel transport T = I + P ∞ i =1 R ψ ∗ · · · ψ ∗ of this connection.Let P e G be the space of the paths γ : [0 , → G with γ (0) = e where e is theidentity element of G . We consider the spaces P e G/ ∼ of homotopy classes of γ ∈ P e G relative to { , } . Since G is simply connected, we have P e G/ ∼ = G . It iseasily seen that the parallel transport T on P e G/ ∼ = G is a homomorphism whosederivative is equal to ψ ∗ . Hence we can identify the parallel transport T on P e G/ ∼ with the representation ψ . Since ψ is Ad s -relative and the diagonal entries of T are R e ω , . . . , R e ω rr , we have ω , . . . , ω rr ∈ L . By the proof of Theorem 2.6, theinjection φ : Γ → H Γ is the restriction of ψ on Γ. Thus the representation φ is themonodromy I + P ∞ i =1 R ψ ∗ · · · ψ ∗ of the left-invariant flat connection d − ψ ∗ on thevector bundle G/ Γ × C r . By Remark 3.1, the ring C [ H Γ ] is generated by matrixentries of I + P ∞ i =1 R ψ ∗ · · · ψ ∗ . Hence the theorem follows from Corollary 3.2. (cid:3) An Example and a remark
Let N be a simply connected nilpotent Lie group and n the Lie algebra of N .We suppose that G has a lattice Γ. Then we can represent the coordinate ring ofthe Malcev completion of Γ by using Chen’s iterated integral of left-invariant formson N . In this paper we give another representation of the Malcev completion ofthe fundamental group of some nilmanifold. LGEBRAIC HULLS AND EXPONENTIAL ITERATED INTEGRALS 7
Consider the solvable Lie group G = R ⋉ µ C such that µ ( t ) = (cid:18) e iπt te iπt e iπt (cid:19) .We have the lattice Γ = 2 Z ⋉ ( Z + i Z ). We consider the inclusion V g ∗ C ⊂ A ∗ ( G/ Γ).The map H ∗ ( V g ∗ C ) → H ∗ ( G/ Γ , C ) induced by this inclusion is injective (see [13]).By ( V g ∗ C ) = C and ( V g ∗ C ) ∩ dA ( G/ Γ) = 0, we have an isomorphism H ( B ( ^ g ∗ C , x )) ∼ = H ( ¯ B ( ^ g ∗ C ))where H ( B ( V g ∗ C , x )) is the space of closed Chen’s iterated integrals of the left-invariant forms on the based loop space Ω x G/ Γ and H ( ¯ B ( V g ∗ C )) is the reducedbar construction (see [2]). Since we have H ( V g ∗ C ) ∼ = C , we have an isomorphism H ( B ( V g ∗ C , x )) ∼ = C [ G ad ].On the other hand, let L be the sub Z -module of g ∗ C generated by { iπdt } . Thenby Corollary 3.2 and Theorem 3.3, we have an isomorphism H ( E L ( g ∗ C )) ∼ = C [ H Γ ] . Since we have µ (2 t ) = (cid:18) t (cid:19) for t ∈ Z , Γ is nilpotent. Hence H Γ is the Malcevcompletion of Γ. Since two compact solvmanifolds having the same fundamentalgroup are diffeomorphic (see [8] or [13]), G/ Γ is diffeomorphic to a nilmanifold. Bythese arguments we notice:
Remark 4.1.
By closed Chen’s iterated integrals of the -forms g ∗ C on G/ Γ , we cannot represent the coordinate ring of Malcev completion of the fundamental group ofthe nilmanifold G/ Γ . But the closed L -exponential iterated integrals of g ∗ C enableus to represent it. Acknowledgements.
The author would like to express his gratitude to Toshitake Kohno for help-ful suggestions and stimulating discussions. This research is supported by JSPSResearch Fellowships for Young Scientists.
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