Quantum subgroups of a simple quantum group at roots of 1
aa r X i v : . [ m a t h . QA ] S e p QUANTUM SUBGROUPS OF A SIMPLE QUANTUMGROUP AT ROOTS OF 1
NICOL ´AS ANDRUSKIEWITSCH AND GAST ´ON ANDR´ES GARC´IA
Abstract.
Let G be a connected, simply connected, simple complexalgebraic group and let ǫ be a primitive ℓ -th root of 1, ℓ odd and 3 ∤ ℓ if G is of type G . We determine all Hopf algebra quotients of the quantizedcoordinate algebra O ǫ ( G ). Introduction and preliminaries
Introduction.
The purpose of this paper is to determine all quantumsubgroups of a quantum group at a root of one, or in equivalent terms, todetermine all Hopf algebra quotients of a quantized coordinate algebra at aroot of one (over the complex numbers). This problem was first consideredby P. Podle´s [P95] for quantum SU (2) and SO (3). The characterizationof all finite-dimensional Hopf algebra quotients of the quantized coordinatealgebra O q ( SL N ) was obtained by Eric M¨uller [M00]. M¨uller’s approachis via explicit computations with matrix coefficients; this strategy does notapply to more general simple groups.The present work can be viewed as a continuation of the long traditionof studying subgroups of a simple algebraic group. In fact, our main theo-rem assumes the knowledge of such subgroups, see Definition 1.1. Besidesits intrinsical mathematical interest, our result would have implications inquantum harmonic analysis– see for example [L02]– and in the study of mod-ule categories over the tensor category of comodules over the Hopf algebra O ǫ ( G )– in the sense of [EO04].An outcome of our main theorem is the construction of many new exam-ples of finite-dimensional Hopf algebras. At the present time, all examplesof finite-dimensional Hopf algebras, we are aware of, are: • group algebras of finite groups, • small quantum groups introduced by Lusztig [L90a, L90b], and vari-ations thereof [AS], Date : June 22, 2007.2000
Mathematics Subject Classification.
Keywords: quantum groups, quantized enveloping algebras, quantized coordinate algebras.Results in this paper are part of the Ph.D. thesis of G. A. G., written under the adviseof N. A. The work was partially supported by CONICET, ANPCyT, Secyt (UNC) andTWAS. • other pointed Hopf algebras with abelian group arising from theNichols algebras discovered in [G˜n00, He], • a few examples of pointed Hopf algebras with non-abelian group[MS00, G˜n], • combinations of the preceding via some standard operations (duals,twisting, Hopf subalgebras and quotients, extensions).How to build examples of Hopf algebras via extensions of a group algebraby a dual group algebra is well understood– see for instance [Ma02]. Out ofthis, extensions can in principle be constructed by means of weak actions andcoactions, and pairs of compatible 2-cocycles. However, very few explicitexamples were presented in this way, to our knowledge no one in finitedimension, except for the trivial tensor product of two Hopf algebras. Ourexamples are indeed nontrivial extensions of finite quantum groups by finitegroups, but it is not clear how they could be explicitly presented throughactions, coactions and cocycles. A natural subsequent question is when thenew examples of Hopf algebras are isomorphic with each other; this will beaddressed in (the forthcoming new version of) [AG].Furthermore, a result of S¸tefan [St99, Thm. 1.5] says that a non-semi-simple finite-dimensional Hopf algebra generated by a simple 4-dimensionalcoalgebra stable by the antipode is a quotient of the quantized coordinatealgebra of SL (2) at a root of one. It is tempting to suggest that finite-dimensional quotients of more general quantized coordinate algebras mightplay a prominent role in the classification of Hopf algebras.We notice that a different problem is sometimes referred to with a similarname: this is the classification of indecomposable module categories overfusion categories arising in conformal field theory, e. g. from the represen-tation theory of finite quantum groups at roots of one. See [O02, KiO02].There is no evident relation between these two problems.1.2. Statement of the main result.
Let g be the Lie algebra of G , h ⊆ g a fixed Cartan subalgebra, Π = { α , . . . , α n } a basis of the root systemΦ = Φ( g , h ) of g with respect to h and n = rk g . Definition 1.1. A subgroup datum is a collection D = ( I + , I − , N, Γ , σ, δ )where • I + ⊆ Π and I − ⊆ − Π. Let Ψ ± = { α ∈ Φ : Supp α ⊆ I ± } , l ± = P α ∈ Ψ ± g α and l = l + ⊕ h ⊕ l − ; l is an algebraic Lie subalgebraof g . Let L be the connected Lie subgroup of G with Lie( L ) = l . • N is a subgroup of d T I c , see Remark 2.12 below. • Γ is an algebraic group. • σ : Γ → L is an injective homomorphism of algebraic groups. • δ : N → b Γ is a group homomorphism.
UANTUM SUBGROUPS 3
If Γ is finite, we call D a finite subgroup datum . We parameterize withinjective group homomorphisms rather than group inclusions for a betterdescription of the isomorphism classes [AG]. An equivalence relation amongsubgroup data is defined in Subsection 2.4.Our main result is the following. Theorem 1.
There is a bijection between ( a ) Hopf algebra quotients q : O ǫ ( G ) → A . ( b ) Subgroup data up to equivalence.
In Section 2, we carry out the construction of a quotient A D of O ǫ ( G )starting from a subgroup datum D , see Theorem 2.17. In Subsection 2.4,we study the lattice of quotients A D . In Section 3, we attach a subgroupdatum D to an arbitrary Hopf algebra quotient A and prove that A D ≃ A as quotients of O ǫ ( G ). This concludes the proof of the Theorem 1. As animmediate corollary of Theorem 1, we get Theorem 2.
There is a bijection between ( a ) Hopf algebra quotients q : O ǫ ( G ) → A such that dim A < ∞ . ( b ) Finite subgroup data up to equivalence.
Theorem 2 generalizes the main result of [M00].1.3.
Conventions.
Let C = ( a ij ) ≤ i,j ≤ n be the Cartan matrix of g andsuppose that g is generated by the elements { h i , e i , f i | ≤ i ≤ n } subjectto the Chevalley-Serre relations. Let Q = Z Φ = L ni =1 Z α i be the rootlattice, ̟ , . . . , ̟ n the fundamental weights, P = L ni =1 Z ̟ i the weightlattice and W the Weyl group. Let P + be the cone of dominant weights and Q + = P + ∩ P . Let ( − , − ) be the positive definite symmetric bilinear formon h ∗ induced by the Killing form of g . Let d i = ( α i ,α i )2 ∈ { , , } .For t, m ∈ N , q ∈ C and u ∈ Q ( q ) r { , ± } we denote:[ t ] u : = u t − u − t u − u − , [ t ] u ! := [ t ] u [ t − u · · · [1] u , (cid:20) mt (cid:21) u := [ m ] u ![ t ] u ![ m − t ] u ! , ( t ) u := u t − u − , ( t ) u ! := ( t ) u ( t − u · · · (1) u , (cid:18) mt (cid:19) u := ( m ) u !( t ) u !( m − t ) u ! . Definitions.
In this subsection we recall the definition of the quantizedcoordinate algebra of G . Let R = Q [ q, q − ], q an indeterminate. If p ℓ ( q ) ∈ R denotes the ℓ -th cyclotomic polynomial, then R/ [ p ℓ ( q ) R ] ≃ Q ( ǫ ). Definition 1.2.
The simply connected quantized enveloping algebra ˇ U q ( g )of g is the Q ( q )-algebra with generators { K λ | λ ∈ P } , E , . . . , E n and F , . . . , F n , satisfying the following relations for λ, µ ∈ P and 1 ≤ i, j ≤ n : N. ANDRUSKIEWITSCH AND G. A. GARC´IA K = 1 , K λ K µ = K λ + µ ,K λ E j K − λ = q ( λ,α j ) E j , K λ F j K − λ = q − ( λ,α j ) F j ,E i F j − F j E i = δ ij K α i − K − α i q i − q − i , − a ij X l =0 ( − l h − a ij l i q i E − a ij − li E j E li = 0 ( i = j ) , − a ij X l =0 ( − l h − a ij l i q i F − a ij − li F j F li = 0 ( i = j ) . Definition 1.3. [DL94, Section 3.4] Let q i = q d i , 1 ≤ i ≤ n . The algebraΓ( g ) is the R -subalgebra of ˇ U q ( g ) generated by the elements K − α i (1 ≤ i ≤ n ) , (cid:18) K α i ; 0 t (cid:19) := t Y s =1 (cid:18) K α i q − s +1 i − q si − (cid:19) ( t ≥ , ≤ i ≤ n ) ,E ( t ) i := E ti [ t ] q i ! ( t ≥ , ≤ i ≤ n ) ,F ( t ) i := F ti [ t ] q i ! ( t ≥ , ≤ i ≤ n ) . Let C be the strictly full subcategory of Γ( g )-mod whose objects are Γ( g )-modules M such that M is a free R -module of finite rank and the operators K α i and (cid:16) K αi ;0 t (cid:17) are diagonalizable with eigenvalues q mi and ( mt ) q i respec-tively, for some m ∈ N and for all 1 ≤ i ≤ n . Definition 1.4. [DL94, Section 4.1] Let R q [ G ] denote the R -submodule ofHom R (Γ( g ) , R ) spanned by the coordinate functions t ji of representations M from C : < g, t ji > = < g · m i , m j > , where ( m i ) is an R -basis of M , ( m j ) isthe dual basis of the dual module and g ∈ Γ( g ). Since the subcategory C isa tensor one, R q [ G ] is a Hopf algebra. Definition 1.5. [DL94, Section 6] The algebra R q [ G ] / [ p ℓ ( q ) R q [ G ]] is de-noted by O ǫ ( G ) Q ( ǫ ) and is called the quantized coordinate algebra of G over Q ( ǫ ) at the root of unity ǫ . In the same way as for O ǫ ( G ) Q ( ǫ ) , we can formthe Q ( ǫ )-Hopf algebra Γ ǫ ( g ) := Γ( g ) / [ p ℓ ( q )Γ( g )].We now relate the Hopf algebras O ǫ ( G ) Q ( ǫ ) and Γ ǫ ( g ). Definition 1.6. A Hopf pairing between two Hopf algebras U and H overa ring R is a bilinear form ( − , − ) : H × U → R such that, for all u, v ∈ U UANTUM SUBGROUPS 5 and f, h ∈ H ,( i ) ( h, uv ) = ( h (1) , u )( h (2) , v ); ( iii ) (1 , u ) = ε ( u );( ii ) ( f h, u ) = ( f, u (1) )( h, u (2) ); ( iv ) ( h,
1) = ε ( h ) . It follows that ( h, S ( u )) = ( S ( h ) , u ), for all u ∈ U , h ∈ H . Given a Hopfpairing, one has Hopf algebra maps U → H ◦ and H → U ◦ , where H ◦ and U ◦ are the Sweedler duals. The pairing is called perfect if these maps areinjections. Proposition 1.7. [DL94, 4.1 and 6.1]
There exists a perfect Hopf pairing R q [ G ] ⊗ R Γ( g ) → R , which induces a perfect Hopf pairing O ǫ ( G ) Q ( ǫ ) ⊗ Q ( ǫ ) Γ ǫ ( g ) → Q ( ǫ ) . In particular, O ǫ ( G ) Q ( ǫ ) ⊆ Γ ǫ ( g ) ◦ and Γ ǫ ( g ) ⊆ O ǫ ( G ) ◦ Q ( ǫ ) . (cid:3) If k is any field containing Q ( ǫ ), we denote O ǫ ( G ) k := O ǫ ( G ) Q ( ǫ ) ⊗ Q ( ǫ ) k .When k = C we simply write O ǫ ( G ) for O ǫ ( G ) C . The following two resultsimply by [Mo93, Prop. 3.4.3] that O ǫ ( G ) is a central extension of O ( G ) bya finite-dimensional Hopf algebra. Theorem 1.8. ( a ) [DL94, Prop. 6.4] O ǫ ( G ) contains a central Hopfsubalgebra isomorphic to the coordinate algebra O ( G ) of G . ( b ) [BG, III.7.11] O ǫ ( G ) is a free O ( G ) -module of rank ℓ dim G . (cid:3) We end this section by spelling out explicitly the quotient of O ǫ ( G ) by itscentral Hopf subalgebra O ( G ).Let O ǫ ( G ) = O ǫ ( G ) / [ O ( G ) + O ǫ ( G )] and denote by π : O ǫ ( G ) → O ǫ ( G )the quotient map. By Theorem 1.8 and [Mo93, Prop. 3.4.3], O ǫ ( G ) is aHopf algebra of dimension ℓ dim G which fits into the exact sequence1 → O ( G ) → O ǫ ( G ) → O ǫ ( G ) → . Let u ǫ ( g ) be the Frobenius-Lusztig kernel of g at ǫ ; that is, the Hopf subal-gebra of Γ ǫ ( g ) generated by the elements E i , F i and K α i for 1 ≤ i ≤ n . See[BG] for details. We denote by(1) T := { K α , . . . , K α n } = G ( u ǫ ( g ))the “finite torus” of group-like elements of u ǫ ( g ). Theorem 1.9. [BG, III.7.10] O ǫ ( G ) ≃ u ǫ ( g ) ∗ as Hopf algebras. (cid:3) Summarizing, the quantized coordinate algebra O ǫ ( G ) of G at ǫ fits intothe central exact sequence(2) 1 → O ( G ) ι −→ O ǫ ( G ) π −→ u ǫ ( g ) ∗ → . We shall need the following technical lemma.
Lemma 1.10.
There exists a surjective algebra map ϕ : Γ ǫ ( g ) → u ǫ ( g ) suchthat ϕ | u ǫ ( g ) = id . N. ANDRUSKIEWITSCH AND G. A. GARC´IA
Proof.
Since Γ ǫ ( g ) = Γ( g ) / [ p ℓ ( q )Γ( g )], we may define ϕ as a map from Γ( g )such that ϕ ( q ) = ǫ . Let ϕ be the unique algebra map which takes thefollowing values on the generators: ϕ ( E ( m ) i ) = ( E ( m ) i if 1 ≤ m < ℓ ϕ ( F ( m ) i ) = ( F ( m ) i if 1 ≤ m < ℓ ϕ ( (cid:0) K αi ;0 m (cid:1) ) = ((cid:0) K αi ;0 m (cid:1) if 1 ≤ m < ℓ ϕ ( K − α i ) = K ℓ − α i , ϕ ( q ) = ǫ, for all 1 ≤ i ≤ n . Since ϕ is the identity on the generators of u ǫ ( g ) and E ℓi = 0 = F ℓi , K ℓα i = 1 on u ǫ ( g ), it follows from a direct computation that ϕ satisfies the relations given in [DL94, Section 3.4], see [G07, 4.1.17] fordetails. Hence ϕ is a well-defined algebra map whose image is u ǫ ( g ). (cid:3) Hopf subalgebras of a pointed Hopf algebra.
We describe in thissubsection Hopf subalgebras of pointed Hopf algebras. Let U be a Hopfalgebra such that the coradical U is a Hopf subalgebra. Let ( U n ) n ≥ be thecoradical filtration of U , set U − = 0, gr U ( n ) = U n /U n − and let gr U = ⊕ n ≥ gr U ( n ) be the associated graded Hopf algebra. Let ι : U → gr U bethe canonical inclusion and let π : gr U → U be the homogeneous projection.Let R = (gr U ) co π be the diagram of U ; R is a graded braided Hopf algebra,that is, a Hopf algebra in the category U U YD of Yetter-Drinfeld modulesover U . Its coalgebra structure is given by ∆ R ( r ) = ϑ R ( r (1) ) ⊗ r (2) , for all r ∈ R , where ϑ R : gr U → R is the map defined by(3) ϑ R ( a ) = a (1) ιπ ( S a (2) ) , ∀ a ∈ gr U. It can be easily shown that ϑ R ( rh ) = rǫ ( h ), ϑ R ( hr ) = h · r for r ∈ R , h ∈ U .One has that gr U ≃ R U , R = ⊕ n ≥ R ( n ), R (0) ≃ C and R (1) = P ( R ).We say that R is a Nichols algebra if R is generated as algebra by R (1). See[AS02] for more details.To state the following result, we need to introduce some terminology. Let A be a Hopf algebra, M a Yetter-Drinfeld module over A and B a Hopfsubalgebra of A . We say that a vector subspace N of M is B -compatible if( a ) it is stable under the action of B , and( b ) it bears a B -comodule structure inducing the coaction of A .In inaccurate but descriptive words, “ N is a Yetter-Drinfeld submoduleover B ” (although M is not necessarily a Yetter-Drinfeld module over B ). Lemma 1.11.
Let Y be a Hopf subalgebra of U . Then the coradical Y is aHopf subalgebra and the diagram S of Y is a braided Hopf subalgebra of R . UANTUM SUBGROUPS 7 If R = B ( V ) is a Nichols algebra with dim V < ∞ , then S is also aNichols algebra. In this case, Hopf subalgebras of U are parameterized bypairs ( Y , W ) where Y is a Hopf subalgebra of U and W ⊂ V = R (1) is Y -compatible.Proof. The first claim follows since Y = Y ∩ U and the intersection oftwo Hopf subalgebras is a Hopf subalgebra. By [Mo93, Lemma 5.2.12],the coradical filtration of Y is given by Y n = Y ∩ U n ; thus we have aninjective homogeneous map of Hopf algebras γ : gr Y ֒ → gr U inducing thecommutative diagram gr Y (cid:31) (cid:127) γ / / π Y (cid:15) (cid:15) gr U π (cid:15) (cid:15) Y (cid:31) (cid:127) / / U . Thus S = { a ∈ gr Y : (id ⊗ π Y )∆( a ) = a ⊗ } is a subalgebra, and alsoa braided vector subspace, of R . Note that γϑ S = ϑ R γ , cf. (3); thus S is a subcoalgebra of R . Assume now that R ≃ B ( V ) is a Nichols algebrawith dim V < ∞ . Taking graded duals, we have a surjective map of gradedbraided Hopf algebras ℘ : B ( V ∗ ) → S gr dual . Since B ( V ∗ ) and S gr dual arepointed irreducible coalgebras, by [Sw69, Thm. 9.1.4], ℘ maps the corad-ical filtration of the first onto the coradical filtration of the second; hence P ( S gr dual ) = S gr dual (1) and a fortiori S is generated in degree 1, i. e. is aNichols algebra. Furthermore, Y is determined by Y and S (1), the last being Y -compatible. Conversely, if Y is a Hopf subalgebra of U and W ⊂ R (1)is Y -compatible, then choose ( y i ) i ∈ I in U such that the classes ( y i ) i ∈ I in U /U generate W Y of U generated by Y and( y i ) i ∈ I is a actually a Hopf subalgebra giving rise to the pair ( Y , W ). (cid:3) The lemma above also holds if V is a locally finite braided vector space.Let us now turn to Hopf subalgebras of pointed Hopf algebras. The notionof “compatibility” for groups reads as follows. Let G be a group and M aYetter-Drinfeld module over the group algebra C [ G ]. If F is a subgroup of G , a vector subspace N of M is F - compatible if( a ) it is stable under the action of F , and( b ) it is a C [ G ]-subcomodule and Supp N := { g ∈ G : N g = 0 } iscontained in F . Corollary 1.12.
Let U be a pointed Hopf algebra whose diagram R is aNichols algebra. Then Hopf subalgebras of U are parameterized by pairs ( F, W ) where F is a subgroup of G ( U ) and W ⊂ R (1) is F -compatible. (cid:3) The Corollary reads even nicer if G ( U ) is abelian and dim R (1) g = 1 forall g ∈ Supp R (1). Indeed, Hopf subalgebras of U are parameterized in thiscase by pairs ( F, J ) where F is a subgroup of G ( U ) and J ⊂ Supp R (1) iscontained in F . We recover in this way results from [CM96, M98]. N. ANDRUSKIEWITSCH AND G. A. GARC´IA
Corollary 1.13. [M98, Thm. 6.3]
The Hopf subalgebras of u ǫ ( g ) are pa-rameterized by triples (Σ , I + , I − ) , where Σ is a subgroup of T and I + ⊆ Π , I − ⊆ − Π such that K α i ∈ Σ if α i ∈ I + ∪ − I − . (cid:3) A five-lemma for extensions of Hopf algebras.
The following gen-eral lemma was kindly communicated to us by Akira Masuoka.
Lemma 1.14.
Let H be a bialgebra over an arbitrary commutative ring,and let A , A ′ be right H -Galois extensions over a common algebra B of H -coinvariants. Assume that A ′ is right B -faithfully flat. Then any H -comodule algebra map θ : A → A ′ that is identical on B is an isomorphism.Proof. Let β : A ⊗ B A → A ⊗ H , β ( x ⊗ y ) = xy (0) ⊗ y (1) and β ′ : A ′ ⊗ B A ′ → A ′ ⊗ H , β ′ ( x ′ ⊗ y ′ ) = x ′ y ′ (0) ⊗ y ′ (1) be the corresponding Galois maps, for x, y ∈ A , x ′ , y ′ ∈ A ′ . Using the A -module structure of A ′ given by θ , we canextend β to an isomorphism α : A ′ ⊗ B A ≃ A ′ ⊗ A A ⊗ B A id ⊗ β −−−→ A ′ ⊗ A A ⊗ H ≃ A ′ ⊗ H. Explicitly, α ( a ′ ⊗ a ) = a ′ θ ( a (0) ) ⊗ a (1) for all a ′ ∈ A ′ , a ∈ A . Then α fits intothe following commutative diagram A ′ ⊗ B A id ⊗ θ / / α ≃ % % LLLLLLLLLL A ′ ⊗ B A ′ β ≃ x x rrrrrrrrrr A ′ ⊗ H Hence id ⊗ θ is an isomorphism; since A ′ is right B -faithfully flat, θ is anisomorphism. (cid:3) The lemma applies to a commutative diagram of Hopf algebras(4) 1 / / B ι / / A π / / θ (cid:15) (cid:15) (cid:15) (cid:15) H / / / / B ι ′ / / A ′ π ′ / / H / / , where the rows are exact sequences of Hopf algebras, in the sense of [AD95]: A co π = B and ker π = B + A ; ditto for A ′ . If the top row is a cleft exact se-quence, then θ is an isomorphism [AD95, Lemma 3.2.19]. Masuoka’s Lemma1.14 implies another version of the five-lemma: If A and A ′ are H -Galoisover B , and A ′ is right B -faithfully flat, then θ is also an isomorphism. Corollary 1.15.
Assume in (4) that dim H is finite, A ′ is noetherian and B is central in A ′ . Then θ is an isomorphism.Proof. As the rows are exact, the corresponding Galois maps β and β ′ aresurjective; since dim H < ∞ , they are bijective [KT81, Thm. 1.7]. Thus A and A ′ are H -Galois over B . Now A ′ is B -faithfully flat by [S93, Thm.3.3]. (cid:3) UANTUM SUBGROUPS 9 Constructing quantum subgroups
In this section we construct quotients of the quantized coordinate algebra O ǫ ( G ). We do this in three steps.2.1. First step.
We construct in this subsection a quotient of O ǫ ( G ) as-sociated to a Hopf subalgebra of u ǫ ( g ); it corresponds to a connected Liesubgroup L of G . Let r : u ǫ ( g ) ∗ → H be a surjective Hopf algebra mor-phism. Then we have an injective Hopf algebra map t r : H ∗ → u ǫ ( g ) andby Corollary 1.13, the Hopf algebra H ∗ corresponds to a triple (Σ , I + , I − ).We shall eventually show that this triple is part of a subgroup datum as inDefinition 1.1.2.1.1. The Hopf subalgebra Γ ǫ ( l ) of Γ ǫ ( g ) . Definition 2.1.
For every triple (Σ , I + , I − ) define Γ( l ) to be the subalgebraof Γ( g ) generated by the elements K − α i (1 ≤ i ≤ n ) , (cid:18) K α i ; 0 m (cid:19) := m Y s =1 (cid:18) K α i q − s +1 i − q si − (cid:19) ( m ≥ , ≤ i ≤ n ) ,E ( m ) j := E mj [ m ] q j ! ( m ≥ , j ∈ I + ) ,F ( m ) k := F mk [ m ] q k ! ( m ≥ , k ∈ I − ) , where q i = q d i for 1 ≤ i ≤ n . Note that Γ( l ) does not depend on Σ.Choosing a reduced expression s i · · · s i N of the longest element of theWeyl group one can order totally the positive part Φ + of the root systemΦ with β = α i , β = s i α i , . . . , β N = s i · · · s i N − α i N . Then using thealgebra automorphisms T i introduced by Lusztig [L90b], one may definecorresponding root vectors E β k = T i · · · T i k − E i k and F β k = T i · · · T i k − F i k .Consider now the R -submodules of Γ( g ) given by J ℓ = R n Y β ≥ F ( n β ) β · n Y i =1 (cid:18) K α i ; 0 t i (cid:19) K Ent( t i / α i · Y α ≥ E ( m α ) α : ∃ n β , t i , m α ℓ ) o Γ ℓ = R n Y β ≥ F ( n β ) β · n Y i =1 (cid:18) K α i ; 0 t i (cid:19) K Ent( t i / α i · Y α ≥ E ( m α ) α : ∀ n β , t i , m α ≡ ℓ ) o Then, by [DL94, Thm. 6.3] there is a decomposition of free R -modulesΓ( g ) = J ℓ ⊗ Γ ℓ and Γ ℓ / [ p ℓ ( q )Γ ℓ ] ≃ U ( g ) Q ( ǫ ) . Let Q I ± = L i ∈ I ± Z α i anddefine the following R -submodules of Γ( l ): W ℓ = R n Y β ≥ F ( n β ) β · n Y i =1 (cid:18) K α i ; 0 t i (cid:19) K Ent( t i / α i · Y α ≥ E ( m α ) α : ∃ n β , t i , m α ℓ ) with β ∈ Q I − , α ∈ Q I + , ≤ i ≤ n o Θ ℓ = R n Y β ≥ F ( n β ) β · n Y i =1 (cid:18) K α i ; 0 t i (cid:19) K Ent( t i / α i · Y α ≥ E ( m α ) α : ∀ n β , t i , m α ≡ ℓ ) with β ∈ Q I − , α ∈ Q I + , ≤ i ≤ n o Using the decomposition of Γ( g ) as free R -module we get the following. Lemma 2.2.
There is a decomposition of free R -modules Γ( l ) = W ℓ ⊗ Θ ℓ .In particular, Γ( l ) is a direct summand of Γ( g ) .Proof. Clearly, Γ( l ) contains the free R -module W ℓ ⊗ Θ ℓ . Thus, it is enoughto show that Γ( l ) ⊆ W ℓ ⊗ Θ ℓ , but this follows directly from the fact thatΓ( l ) is generated as an algebra over R by the elements in Definition 2.1 andthese generators satisfy the relations given in [DL94, Sec. 3.4]. (cid:3) Let Γ ǫ ( l ) := Γ( l ) / [ p ℓ ( q )Γ( l )]. Then we have the following proposition. Proposition 2.3. ( a ) Γ ǫ ( l ) is a Hopf subalgebra of Γ ǫ ( g ) . ( b ) Γ ǫ ( g ) ≃ Γ( g ) ⊗ R R/ [ p ℓ ( q ) R ] and Γ ǫ ( l ) ≃ Γ( l ) ⊗ R R/ [ p ℓ ( q ) R ] .Proof. We prove only ( a ) since ( b ) is straightforward. By definition, theelements E j are ( K α j , F k ’s are (1 , K − α k )-primitives and the K α i ’s are group-like. Moreover, the antipode is given by S ( K α i ) = K − α i , S ( E j ) = − K − α j E j and S ( F k ) = − F k K α k with 1 ≤ i ≤ n , j ∈ I + and k ∈ I − . Hence, the subalgebra of Γ( l ) generated by these elements is a Hopfsubalgebra of Γ( g ) and Γ( l ) / [ p ℓ ( q )Γ( g ) ∩ Γ( l )] is a Hopf subalgebra of Γ ǫ ( g ).But by Lemma 2.2, we know that Γ( g ) = Γ( l ) ⊕ N for some R -submodule N . Then p ℓ ( q )Γ( g ) ∩ Γ( l ) = p ℓ ( q )(Γ( l ) ⊕ N ) ∩ Γ( l ) = p ℓ ( q )Γ( l ), which impliesthat Γ ǫ ( l ) = Γ( l ) / [ p ℓ ( q )Γ( g ) ∩ Γ( l )]. (cid:3) The regular Frobenius-Lusztig kernel u ǫ ( l ) . Let u ǫ ( l ) be the subalge-bra of Γ ǫ ( l ) generated by the elements { K α i , E j , F k : 1 ≤ i ≤ n, j ∈ I + , k ∈ I − } . Lemma 2.4. u ǫ ( l ) is a Hopf subalgebra of Γ ǫ ( l ) such that Γ ǫ ( l ) ∩ u ǫ ( g ) = u ǫ ( l ) and corresponds to the triple ( T , I + , I − ) , see (1) . UANTUM SUBGROUPS 11
Proof.
It is clear that u ǫ ( l ) is a Hopf subalgebra of Γ ǫ ( l ). Since the Frobenius-Lusztig kernel u ǫ ( g ) is the subalgebra of Γ ǫ ( g ) generated by the elements { K α i , E i , F i : 1 ≤ i ≤ n } , we have that u ǫ ( l ) ⊆ Γ ǫ ( l ) ∩ u ǫ ( g ). But fromLemma 2.2, it follows that every element of Γ ǫ ( l ) ∩ u ǫ ( g ) must be containedin u ǫ ( l ). The last assertion follows immediately from Corollary 1.13. (cid:3) Recall that the quantum Frobenius map Fr : Γ ǫ ( g ) → U ( g ) Q ( ǫ ) is definedon the generators of Γ ǫ ( g ) byFr( E ( m ) i ) = ( e ( m/ℓ ) i if ℓ | m F ( m ) i ) = ( f ( m/ℓ ) i if ℓ | m (cid:0) K αi ;0 m (cid:1) ) = ( ( h i ;0 m ) if ℓ | m K − α i ) = 1 , for all 1 ≤ i ≤ n, and one has an exact sequence of Hopf algebras– see [L90b], [DL94, Thm.6.3]: 1 → u ǫ ( g ) −→ Γ ǫ ( g ) Fr −→ U ( g ) Q ( ǫ ) → . If we define U ( l ) Q ( ǫ ) := Fr(Γ ǫ ( l )), then it follows that U ( l ) Q ( ǫ ) is a subalgebraof U ( g ) Q ( ǫ ) and the following diagram commutes(5) u ǫ ( g ) (cid:31) (cid:127) / / Γ ǫ ( g ) Fr / / / / U ( g ) Q ( ǫ ) u ǫ ( l ) (cid:31) (cid:127) / / ?(cid:31) O O Γ ǫ ( l ) Fr / / / / ?(cid:31) O O U ( l ) Q ( ǫ ) , ?(cid:31) O O where Fr is the restriction of Fr to Γ ǫ ( l ). Remarks . ( a ) Let l be the set of primitive elements P ( U ( l ) Q ( ǫ ) ) of U ( l ) Q ( ǫ ) . Then l is a Lie subalgebra of g , which is in fact regular in thesense of [D57]: it is the Lie subalgebra generated by the set { h i , e j , f k : 1 ≤ i ≤ n, j ∈ I + , k ∈ I − } . This agrees with Definition 1.1.( b ) Ker Fr is the two-sided ideal I of Γ ǫ ( l ) generated by the set n E ( m ) j , F ( m ) k , (cid:18) K α i ; 0 m (cid:19) , K α i − ≤ i ≤ n, j ∈ I + , k ∈ I − , m ≥ , ℓ ∤ m o , and coincides with W ℓ . Indeed, by [DL94, Thm. 6.3] we know that Ker Fr = J ℓ and coincides with the two-sided ideal generated by n E ( m ) i , F ( m ) i , (cid:18) K α i ; 0 m (cid:19) , K α i − ≤ i ≤ n, m ≥ , ℓ ∤ m o . But by Lemma 2.2, Ker Fr = Ker Fr ∩ Γ ǫ ( l ) = J ℓ ∩ Γ ǫ ( l ) = W ℓ and the lastone coincides with the ideal I . ( c ) Since by [DL94, Thm. 6.3], the morphism Γ ℓ / [ p ℓ ( q )Γ ℓ ] → U ( g ) Q ( ǫ ) induced by the quantum Frobenius map is bijective and by definition Θ ℓ ⊆ Γ ℓ and U ( l ) Q ( ǫ ) = Fr( U ( g ) Q ( ǫ ) ), it follows by Lemma 2.2 that Θ ℓ ∩ p ℓ ( q )Γ ℓ = p ℓ ( q )Θ ℓ and the morphism Θ ℓ / [ p ℓ ( q )Θ ℓ ] → U ( l ) Q ( ǫ ) is also bijective.The following proposition gives some properties of u ǫ ( l ). Proposition 2.6. ( a ) The following sequence of Hopf algebras is exact (6) 1 → u ǫ ( l ) j −→ Γ ǫ ( l ) Fr −→ U ( l ) Q ( ǫ ) → . ( b ) There is a surjective algebra map ψ : Γ ǫ ( l ) → u ǫ ( l ) such that ψ | u ǫ ( l ) = id .Proof. ( a ) We need only to prove that Ker Fr = u ǫ ( l ) + Γ ǫ ( l ) and co Fr Γ ǫ ( l ) = u ǫ ( l ). The first equality follows directly from Remark 2.5 ( b ), since thetwo-sided ideal generated by u ǫ ( l ) + coincides with I . The second equalityfollows from Lemma 2.4, because co Fr Γ ǫ ( g ) = u ǫ ( g ) by [A96, Lemma 3.4.1]and u ǫ ( l ) = u ǫ ( g ) ∩ Γ ǫ ( l ) = co Fr Γ ǫ ( g ) ∩ Γ ǫ ( l ) = co Fr Γ ǫ ( l ).( b ) By Lemma 1.10, there exists a surjective algebra map ϕ : Γ ǫ ( g ) → u ǫ ( g ) such that ϕ | u ǫ ( g ) = id. If we define ψ := ϕ | u ǫ ( l ) : Γ ǫ ( l ) → u ǫ ( g ), thenIm ψ ⊆ u ǫ ( l ) and ϕ | u ǫ ( l ) = id, from which follows that Im ψ = u ǫ ( l ). (cid:3) The quantized coordinate algebra O ǫ ( L ) . The inclusion Γ ǫ ( l ) ֒ → Γ ǫ ( g )determines by duality a Hopf algebra map Res : Γ ǫ ( g ) ◦ → Γ ǫ ( l ) ◦ . Since byProposition 1.7, we have that O ǫ ( G ) Q ( ǫ ) ⊆ Γ ǫ ( g ) ◦ , we may define O ǫ ( L ) Q ( ǫ ) := Res( O ǫ ( G ) Q ( ǫ ) ) . Moreover, as O ( G ) Q ( ǫ ) ⊆ O ǫ ( G ) Q ( ǫ ) , Res( O ( G ) Q ( ǫ ) ) is a central Hopf sub-algebra of O ǫ ( L ) Q ( ǫ ) and whence there exists an algebraic subgroup L of G such that Res( O ( G ) Q ( ǫ ) ) = O ( L ) Q ( ǫ ) . Next we show that L is connectedand the corresponding Lie subalgebra of g is no other than the Lie algebra l discussed in Remark 2.5 (a).Recall that a Lie subalgebra k ⊆ g is called algebraic if there exists analgebraic subgroup K ⊆ G such that k = Lie( K ). We say that k + is the algebraic hull of k if k + is an algebraic subalgebra of g such that k ⊆ k + andif a is an algebraic subalgebra of g that contains k , then k + ⊆ a . Proposition 2.7.
The algebraic group L is connected and Lie( L ) = l .Proof. Since O ( G ) Q ( ǫ ) ⊆ U ( g ) ◦ Q ( ǫ ) , dualizing diagram (5) we have O ( L ) Q ( ǫ ) =Res( O ( G ) Q ( ǫ ) ) ⊆ U ( l ) ◦ Q ( ǫ ) . But by [H81, XVI.3], U ( l ) ◦ Q ( ǫ ) and consequently O ( L ) Q ( ǫ ) are integral domains, implying that L is irreducible and thereforeconnected.To show Lie( L ) = l , we prove that Lie( L ) is the algebraic hull of l and l is an algebraic Lie algebra. Since Ker Res | O ǫ ( G ) Q ( ǫ ) = { f ∈ O ǫ ( G ) Q ( ǫ ) : f | Γ ǫ ( l ) = 0 } and the inclusion of O ( G ) Q ( ǫ ) in O ǫ ( G ) Q ( ǫ ) is given by the UANTUM SUBGROUPS 13 transpose of the quantum Frobenius map Fr (see page 11), it follows that O ( L ) Q ( ǫ ) ≃ O ( G ) Q ( ǫ ) /J , where J = { f ∈ O ( G ) Q ( ǫ ) : h f, Fr( x ) i = 0 , ∀ x ∈ Γ ǫ ( l ) } = { f ∈ O ( G ) Q ( ǫ ) : h f, x i = 0 , ∀ x ∈ U ( l ) Q ( ǫ ) } . In particular, 0 = h f, x i = x ( f ) for all x ∈ U ( l ) Q ( ǫ ) . Since by [FR05, Lemma6.9], Lie( L ) = { τ ∈ g : τ ( f ) = 0 , ∀ f ∈ J } , it is clear that l ⊆ Lie( L ). Nowlet K ⊆ G such that l ⊆ Lie( K ) =: k and denote by I the ideal of K ; then k = { τ ∈ g : τ ( I ) = 0 } . As l ⊆ k , τ ( I ) = 0 for all τ ∈ l . Since the pairing h , i is multiplicative, we have that I ⊆ J and whence L ⊆ K . Thus Lie( L ) ⊆ k for all algebraic Lie subalgebra k such that l ⊆ k , implying that Lie( L ) = l + .Now we show that l is algebraic, implying that l = l + = Lie( L ). Consider g as a G -module with the adjoint action and define G l = { x ∈ G : x · l = l } and g l = { τ ∈ g : [ τ, l ] ⊆ l } . Then by [FR05, Ex. 8.4.7], Lie( G l ) = g l . Thus,it is enough to show that l equals its normalizer in g .By construction, we know that l = l + ⊕ h ⊕ l − , where h is the Cartansubalgebra of g and l ± = L α ∈ Ψ ± g α , with Ψ ± = { α ∈ Φ : Supp( α ) ⊆ I ± } .Let x ∈ g l , then we may write x = P α ∈ Φ c α x α + x with x ∈ h . Thus, forall H ∈ h we have that [ H, x ] = P α ∈ Φ c α α ( H ) x α ∈ l . This implies that forall H ∈ h , c α α ( H ) = 0 for all α / ∈ Ψ = Ψ + ∪ Ψ − . Hence c α = 0 for all α / ∈ Ψand x ∈ l . (cid:3) Since O ( L ) Q ( ǫ ) is a central Hopf subalgebra of O ǫ ( L ) Q ( ǫ ) , the quotient O ǫ ( L ) Q ( ǫ ) := O ǫ ( L ) Q ( ǫ ) / [ O ( L ) + Q ( ǫ ) O ǫ ( L ) Q ( ǫ ) ]is a Hopf algebra which is finite-dimensional. The following propositionshows that, as expected, this algebra is isomorphic to u ǫ ( l ) ∗ , see 2.1.2. Proposition 2.8. ( a ) The following sequence of Hopf algebras is exact (7) 1 → O ( L ) Q ( ǫ ) ι L −→ O ǫ ( L ) Q ( ǫ ) π L −−→ O ǫ ( L ) Q ( ǫ ) → . ( b ) There exists a surjective Hopf algebra map P : u ǫ ( g ) ∗ → O ǫ ( L ) Q ( ǫ ) making the following diagram commutative: (8) 1 / / O ( G ) Q ( ǫ ) ι / / res (cid:15) (cid:15) (cid:15) (cid:15) O ǫ ( G ) Q ( ǫ ) π / / Res (cid:15) (cid:15) (cid:15) (cid:15) u ǫ ( g ) ∗ P (cid:15) (cid:15) (cid:15) (cid:15) / / / / O ( L ) Q ( ǫ ) ι L / / O ǫ ( L ) Q ( ǫ ) π L / / O ǫ ( L ) Q ( ǫ ) / / . ( c ) O ǫ ( L ) Q ( ǫ ) ≃ u ǫ ( l ) ∗ as Hopf algebras.Proof. ( a ) We need only to show that O ( L ) Q ( ǫ ) = co π L O ǫ ( L ) Q ( ǫ ) . Thealgebra O ǫ ( G ) Q ( ǫ ) is noetherian, by Theorem 1.8 ( b ). Therefore O ǫ ( L ) Q ( ǫ ) is also noetherian, since it is a quotient of O ǫ ( G ) Q ( ǫ ) . Then by [S93, Thm. O ǫ ( L ) Q ( ǫ ) is faithfully flat over O ( L ) Q ( ǫ ) and by [Mo93, Prop. 3.4.3] itfollows that O ( L ) Q ( ǫ ) = co π L O ǫ ( L ) Q ( ǫ ) = O ǫ ( L ) co π L Q ( ǫ ) .( b ) Since the sequence (2) is exact, we have Ker π = O ( G ) + Q ( ǫ ) O ǫ ( G ) Q ( ǫ ) and u ǫ ( g ) ∗ ≃ O ǫ ( G ) Q ( ǫ ) / [ O ( G ) + Q ( ǫ ) O ǫ ( G ) Q ( ǫ ) ]. But then, π L Res(Ker π ) = π L ( O ( L ) + Q ( ǫ ) O ǫ ( L ) Q ( ǫ ) ) = 0 and hence there exists a Hopf algebra map P : u ǫ ( g ) ∗ → O ǫ ( L ) Q ( ǫ ) which makes the diagram (8) commutative.( c ) Dualizing diagram (5) we obtain a commutative diagram(9) U ( g ) ◦ Q ( ǫ ) (cid:15) (cid:15) (cid:31) (cid:127) t Fr / / Γ ǫ ( g ) ◦ F / / Res (cid:15) (cid:15) u ǫ ( g ) ∗ p (cid:15) (cid:15) (cid:15) (cid:15) U ( l ) ◦ Q ( ǫ ) (cid:31) (cid:127) t Fr / / Γ ǫ ( l ) ◦ f / / u ǫ ( l ) ∗ . Since O ǫ ( L ) Q ( ǫ ) = Res( O ǫ ( G ) Q ( ǫ ) ), O ( L ) Q ( ǫ ) = Res( O ( G ) Q ( ǫ ) ) and O ( G ) Q ( ǫ ) ≃ U ( g ) ◦ Q ( ǫ ) , because g is simple, it follows that O ( L ) Q ( ǫ ) ⊆ t Fr( U ( l ) ◦ Q ( ǫ ) ). Inparticular, O ( L ) + Q ( ǫ ) ⊆ Ker f . Moreover, since F ( O ǫ ( G ) Q ( ǫ ) ) = π ( O ǫ ( G ) Q ( ǫ ) )= u ǫ ( g ) ∗ we have that u ǫ ( l ) ∗ = f Res( O ǫ ( G ) Q ( ǫ ) ) = f ( O ǫ ( L ) Q ( ǫ ) ). Hence,there exists a surjective Hopf algebra map β : O ǫ ( L ) Q ( ǫ ) → u ǫ ( l ) ∗ ; anddim O ǫ ( L ) Q ( ǫ ) ≥ dim u ǫ ( l ) ∗ .We show next that there exists a surjective morphism u ǫ ( l ) ∗ → O ǫ ( L ) Q ( ǫ ) implying that β is an isomorphism. Consider the map p : u ǫ ( g ) ∗ → u ǫ ( l ) ∗ as in (9) and let a ∈ Ker p . Since u ǫ ( g ) is finite-dimensional, the coordinatefunctions of the regular representation of u ǫ ( g ) span linearly u ǫ ( g ) ∗ and wemay assume that a is a coordinate function of a finite-dimensional represen-tation M of u ǫ ( g ). As p is just the map given by the restriction, we havethat a must be trivial on every basis of u ǫ ( l ), in particular the following: n Y β ≥ F n β β · n Y i =1 K t i α i · Y α ≥ E m α α : 0 ≤ n β , t i , m α < ℓ,β ∈ Q I − , ≤ i ≤ n, α ∈ Q I + o . On the other hand, we know by Lemma 1.10 that there exists a surjec-tive algebra map ϕ : Γ ǫ ( g ) → u ǫ ( g ) such that ϕ | u ǫ ( g ) = id. Hence, the u ǫ ( g )-module M admits a Γ ǫ ( g )-module structure via ϕ . Since M is finite-dimensional and K ℓα i acts as the identity for every 1 ≤ i ≤ n , it follows thateach operator K α i is diagonalizable with eigenvalues ǫ mi for some m ∈ N .This implies by definition that the coordinate function ϕ ∗ ( a ) of the Γ ǫ ( g )-module M must be contained in O ǫ ( G ) Q ( ǫ ) . Thus, using the definition of ϕ UANTUM SUBGROUPS 15 we have that Res ϕ ∗ ( a ) must annihilate the set W ℓ = Q ( ǫ ) n Y β ≥ F ( n β ) β · n Y i =1 (cid:18) K α i ; 0 t i (cid:19) K Ent( t i / α i · Y α ≥ E ( m α ) α : ∃ n β , t i , m α ℓ ) with β ∈ Q I − , ≤ i ≤ n, α ∈ Q I + o . Since by Lemma 2.2, Γ( l ) = W ℓ ⊗ Θ ℓ as free R -modules and by Remark 2.5,Ker Fr = W ℓ and the map Θ l / [ p ℓ ( q )Θ ℓ ] → U ( l ) Q ( ǫ ) induced by the restrictionof the quantum Frobenius map Fr is bijective. Then there exists b ∈ U ( l ) ◦ Q ( ǫ ) such that t Fr( b ) = Res( ϕ ∗ ( a )). Hence, P ( a ) = P ( π ( ϕ ∗ ( a ))) = π L (Res( ϕ ∗ ( a ))) = π L ( t Fr( b )) = ε ( b ) = ε ( a ) = 0 , and a ∈ Ker P . Thus Ker p ⊆ Ker P and there exists a surjective map u ǫ ( l ) ∗ → O ǫ ( L ) Q ( ǫ ) . (cid:3) Remark . By the Proposition above, we have the following commutativediagram of exact sequences of Hopf algebras(10) 1 / / O ( G ) Q ( ǫ ) ι / / res (cid:15) (cid:15) (cid:15) (cid:15) O ǫ ( G ) Q ( ǫ ) π / / Res (cid:15) (cid:15) (cid:15) (cid:15) u ǫ ( g ) ∗ p (cid:15) (cid:15) (cid:15) (cid:15) / / / / O ( L ) Q ( ǫ ) ι L / / O ǫ ( L ) Q ( ǫ ) π L / / u ǫ ( l ) ∗ / / Second Step.
We consider now the complex form of the algebras de-fined above. Denote the C -form of the Frobenius-Lusztig kernels just by u ǫ ( g ) and u ǫ ( l ).The following proposition tell us how to construct Hopf algebras from acentral exact sequence and a surjective Hopf algebra map. We perform it ina general setting and then we apply it to our situation. The characterizationof these algebras as pushouts will be crucial. Proposition 2.10.
Let A and K be Hopf algebras, B a central Hopf subal-gebra of A such that A is left or right faithfully flat over B and p : B → K asurjective Hopf algebra map. Then H = A/AB + is a Hopf algebra and A fitsinto the exact sequence → B ι −→ A π −→ H → . If we set J = Ker p ⊆ B ,then ( J ) = A J is a Hopf ideal of A and A/ ( J ) is the pushout given by thefollowing diagram: B (cid:31) (cid:127) ι / / p (cid:15) (cid:15) (cid:15) (cid:15) A q (cid:15) (cid:15) (cid:15) (cid:15) K (cid:31) (cid:127) j / / A/ ( J ) . Moreover, K can be identified with a central Hopf subalgebra of A/ ( J ) and A/ ( J ) fits into the exact sequence (11) 1 → K → A/ ( J ) → H → . Proof.
The first assertion follows directly from [Mo93, Prop. 3.4.3]. Since B is central in A , ( J ) is a two-sided ideal of A . Moreover, from the factthat ε and ∆ are algebra maps and S ( J ) ⊆ J , it follows that ( J ) is indeeda Hopf ideal. Identify K with B/ J . Then the map j : K → A/ ( J ) givenby j ( b + J ) = ι ( b ) + ( J ) defines a morphism of Hopf algebras because ι is a Hopf algebra map. Since A is faithfully flat over B , by [S92, Cor.1.8], B is a direct summand in A as a B -module, say A = B ⊕ M . Then( J ) ∩ B = J A ∩ B = ( J B ⊕ J M ) ∩ B = ( J ⊕ J M ) ∩ B = J . Thus, if j ( b + J ) = 0 then ι ( b ) ∈ ( J ) and this implies that b ∈ ( J ) ∩ B = J by theequality above. Hence, j is injective.Let us see now that A/ ( J ) is a pushout: let C be a Hopf algebra andsuppose that there exist Hopf algebra maps ϕ : K → C and ϕ : A → C such that ϕ p = ϕ ι . We have to show that there exists a unique Hopfalgebra map φ : A/ ( J ) → C such that φq = ϕ and φj = ϕ . B ι / / p (cid:15) (cid:15) A q (cid:15) (cid:15) ϕ (cid:21) (cid:21) K j / / ϕ , , A/ ( J ) ∃ ! φ FF " " FF C Since ϕ (( J )) = ϕ ( A J ) = ϕ ( A ) ϕ ( ι ( J )) = ϕ ( A ) ϕ ( p ( J )) = 0, thereexists a unique Hopf algebra map φ : A/ ( J ) → C such that φq = ϕ .Moreover, let x ∈ K and b ∈ B such that p ( b ) = x . Then φj ( x ) = φjp ( b )= φqι ( b ) = ϕ ι ( b ) = ϕ p ( b ) = ϕ ( x ), from which follows that φj = ϕ .Denote also by K the image of K under j . To see that K is central in A/ ( J ) we have to verify that j ( c )¯ a = ¯ aj ( c ) for all ¯ a ∈ A/ ( J ), c ∈ K . Since p is surjective, for all c ∈ K there exists b ∈ B such that p ( b ) = c and since q isan algebra map, it follows that ¯ aj ( c ) = q ( a ) j ( p ( b )) = q ( a ) q ( ι ( b )) = q ( aι ( b ))= q ( ι ( b ) a ) = q ( ι ( b )) q ( a ) = j ( c )¯ a , because B is central in A . In particular,the quotient e H = [ A/ ( J )] / [ K + ( A/ ( J ))] is a Hopf algebra. To see that A/ ( J ) is a central extension of K by e H , by [Mo93, Prop. 3.4.3] it is enoughto show that A/ ( J ) is flat over K and K is a direct summand of A/ ( J )as K -modules, since by [S92, Cor. 1.8] this implies that A/ ( J ) is faithfullyflat over K .First we show that A/ ( J ) is flat over K . Let M and M be two right K -modules and let f : M → M be an injective homomorphism. In particular,they admit a B -module structure via the map p : B → K , which we denoteby M i for i = 1 ,
2; thus f is an injective homomorphism of B -modules. Since UANTUM SUBGROUPS 17 A is faithfully flat over B , the homomorphism of A -modules f ⊗ id : M ⊗ B A → M ⊗ B A is also injective. As J is central in A , we have for i = 1 , M i ⊗ B A )( J ) = 0. Then the A -modules are also A/ ( J )-modulesand M i ⊗ B A ≃ M i ⊗ K A/ ( J ) as A/ ( J )-modules by the construction of M i . Hence the homomorphism of A/ ( J )-modules f ⊗ id : M ⊗ K A/ ( J ) → M ⊗ K A/ ( J ) is injective and A/ ( J ) is flat over K .As A = B ⊕ M as B -modules, we have that ( J ) = A J = J ⊕ M J , where M J is a B -submodule of M and J = B ∩ ( J ⊕ M J ). Hence A/ ( J ) =( B ⊕ M ) / ( J ⊕ M J ) = K ⊕ ( M/M J ) as K -modules, which implies that K is a direct summand of A/ ( J ).In conclusion, A/ ( J ) fits into an exact sequence of Hopf algebras1 → K j −→ A/ ( J ) r −→ e H → . Since the map Ψ : K + ( A/ ( J )) → ( B + A ) / ( J ) defined by Ψ( ba ) = ba is a k -linear isomorphism, it follows that e H = ( A/ ( J )) / [ K + ( A/ ( J ))] ≃ ( A/ ( J )) / [( B + A ) / ( J )] ≃ A/B + A = H and therefore A/ ( J ) fits into anexact sequence (11). (cid:3) Let Γ be an algebraic group and let σ : Γ → G an injective homomorphismof algebraic groups such that σ (Γ) ⊆ L . Then we have a surjective Hopfalgebra map t σ : O ( L ) → O (Γ). Applying the pushout construction givenin Proposition 2.10, we obtain a Hopf algebra A l ,σ which is part of an exactsequence of Hopf algebras and fits into the following commutative diagram(12) 1 / / O ( G ) ι / / res (cid:15) (cid:15) O ǫ ( G ) π / / Res (cid:15) (cid:15) u ǫ ( g ) ∗ / / p (cid:15) (cid:15) / / O ( L ) t σ (cid:15) (cid:15) ι L / / O ǫ ( L ) π L / / ν (cid:15) (cid:15) u ǫ ( l ) ∗ / / / / O (Γ) j / / A l ,σ ¯ π / / u ǫ ( l ) ∗ / / . Remark . Let 1 → K → A → H → β : A ⊗ K A → A ⊗ H , β ( x, y ) = xy (0) ⊗ y (1) denotes theGalois map, then β is surjective, since H ≃ A/K + A . If moreover H isfinite-dimensional, A is a finitely generated projective K -module, by [KT81,Thm. 1.7]. In particular, if dim K is finite, then dim A = dim K dim H isalso finite. In our case, if Γ is finite we obtain that dim A l ,σ = | Γ | dim u ǫ ( l ).2.3. Third Step.
In this subsection we make the third and last step of theconstruction. It consists essentially on taking a quotient by a Hopf idealgenerated by differences of central group-like elements of A l ,σ . The crucialpoint here is the description of H as a quotient of u ǫ ( l ) ∗ and the existenceof a coalgebra morphism ψ ∗ : u ǫ ( l ) ∗ → O ǫ ( L ). Recall that from the beginning of this section we fixed a surjective Hopfalgebra map r : u ǫ ( g ) ∗ → H and H ∗ is determined by the triple (Σ , I + , I − ).Since the Hopf subalgebra u ǫ ( l ) is determined by the triple ( T , I + , I − ) with T ⊇ Σ, we have that H ∗ ⊆ u ǫ ( l ) ⊆ u ǫ ( g ). Denote by v : u ǫ ( l ) ∗ → H thesurjective Hopf algebra map induced by this inclusion. Then H is a quotientof u ǫ ( l ) ∗ which fits into the following commutative diagram u ǫ ( g ) ∗ p / / / / r $ $ $ $ IIIIIIIII u ǫ ( l ) ∗ v (cid:15) (cid:15) (cid:15) (cid:15) H. Remark . Let I = I + ∪ − I − , I c = Π − I and T I = { K α i : i ∈ I } . Let s = | I c | . By Corollary 1.13, we know that T I ⊆ Σ ⊆ T = T I × T I c . If weset Ω = Σ ∩ T I c , it follows clearly that Σ ≃ T I × Ω.Thus, giving a subgroup Σ such that T I ⊆ Σ ⊆ T is the same as givinga subgroup Ω ⊆ T I c , and this is the same as giving a subgroup N ⊆ d T I c .Namely, N is the kernel of the group homomorphism ρ : d T I c → b Ω inducedby the inclusion. In particular, we have that | Σ | = | T I || Ω | = ℓ n − s | Ω | = ℓ n | N | . Definition 2.13.
For all 1 ≤ i ≤ n such that α i / ∈ I + or α i / ∈ I − we define D i ∈ G ( u ǫ ( l ) ∗ ) = Alg( u ǫ ( l ) , C ) on the generators of u ǫ ( l ) by D i ( E j ) = 0 ∀ j : α j ∈ I + , D i ( F k ) = 0 ∀ k : α k ∈ I − ,D i ( K α t ) = 1 ∀ t = i, ≤ t ≤ n, D i ( K α i ) = ǫ i , where ǫ i is a primitive ℓ -th root of 1. If α i / ∈ I + or α i / ∈ I − , then E i or F i is not a generator of u ǫ ( l ), respectively. Hence, D i is a well-defined algebramap, since it verifies all the defining relations of Γ ǫ ( g ) [DL94, Sec. 3.4], see[G07, 5.2.12] for details.Let I c = { α i , . . . , α i s } and let N ⊆ d T I c , correspond to Σ as in Remark2.12. We define for all z = ( z , . . . , z s ) ∈ d T I c the following group-like element D z := D z i · · · D z s i s . Recall that ( M ) denotes the two-sided ideal generated by a subset M ofan algebra R . Lemma 2.14. ( a ) If α i ∈ I c then D i is central in u ǫ ( l ) ∗ . In particular D z is central for all z ∈ d T I c . ( b ) H ≃ u ǫ ( l ) ∗ / ( D z − | z ∈ N ) .Proof. ( a ) We have to show that D i f = f D i for all f ∈ u ǫ ( l ) ∗ . First observethat D i coincide with the counit of u ǫ ( l ) in all elements of the basis whichdo not contain some positive power of K α i . By Lemma 2.2 we know that UANTUM SUBGROUPS 19 u ǫ ( l ) has a basis of the form n Y β ≥ F n β β · n Y i =1 K t i α i · Y α ≥ E m α α : 0 ≤ n β , t i , m α < ℓ, with β ∈ Q I − , α ∈ Q I + , ≤ i ≤ n o . Thus, using the defining relations of Γ ǫ ( g ) [DL94, Sec. 3.4], we may assumethat this basis is of the form K t i α i M with 0 ≤ t i < ℓ and M does not containany power of K α i . Then for every element of this basis we have D i f ( K t i α i M ) = D i ( K t i α i M (1) ) f ( K t i α i M (2) ) = D i ( K t i α i ) D i ( M (1) ) f ( K t i α i M (2) )= ǫ t i i ε ( M (1) ) f ( K t i α i M (2) ) = ǫ t i i f ( K t i α i M )= f D i ( K t i α i M ) , ( b ) By ( a ) we know that D z is a central group-like element of u ǫ ( l ) ∗ forall z ∈ N . Hence the quotient u ǫ ( l ) ∗ / ( D z − | z ∈ N ) is a Hopf algebra.On the other hand, following Corollary 1.13 we know that H ∗ is deter-mined by the triple (Σ , I + , I − ) and consequently H ∗ is included in u ǫ ( l ). Ifwe denote v : u ǫ ( l ) ∗ → H the surjective map induced by this inclusion, wehave that Ker v = { f ∈ u ǫ ( l ) ∗ : f ( h ) = 0 , ∀ h ∈ H ∗ } . But D z − ∈ Ker v for all z ∈ N , since D z ( ω ) = ρ ( z )( ω ) = 1 for all ω ∈ Ω. Hence there existsa surjective Hopf algebra map γ : u ǫ ( l ) ∗ / ( D z − | z ∈ N ) ։ H. Combining Corollary 1.13 with the PBW-basis of H and u ǫ ( l ) we have thatdim H = ℓ | I + | + | I − | | Σ | = ℓ | I + | + | I − | ℓ n − s | Ω | = ℓ | I + | + | I − | ℓ n − s | b Ω | = ℓ | I + | + | I − | ℓ n | N | = dim( u ǫ ( l ) ∗ / ( D z − | z ∈ N )) , which implies that γ is an isomorphism. (cid:3) Remark . The lemma above is very similar to a result used by E. M¨ullerin the case of type A n [M00, Sec. 4] for the classification of the finite-dimensional quotients of O ǫ ( SL N ). The new point of view here consists inregarding H as a quotient of the dual of u ǫ ( l ).Before going on with the construction we need the following technicallemma. Let X = { D z | z ∈ d T I c } be the set of central group-like elements of u ǫ ( l ) ∗ given by Lemma 2.14. Lemma 2.16.
There exists a subgroup Z := { ∂ z | z ∈ d T I c } of G ( A l ,σ ) isomorphic to X consisting of central elements.Proof. By Proposition 2.6 ( b ), we know that there exists an algebra map ψ : Γ ǫ ( l ) → u ǫ ( l ); it induces a coalgebra map ψ ∗ : u ǫ ( l ) ∗ → Γ ǫ ( l ) ◦ such that the following diagram commutesΓ ǫ ( g ) ◦ Res (cid:15) (cid:15) (cid:15) (cid:15) u ǫ ( g ) ∗ ϕ ∗ o o p (cid:15) (cid:15) (cid:15) (cid:15) Γ ǫ ( l ) ◦ u ǫ ( l ) ∗ . ψ ∗ o o Here, ϕ ∗ is the coalgebra map induced by the algebra map ϕ : Γ ǫ ( g ) → u ǫ ( l )given by Lemma 1.10, whose restriction to Γ ǫ ( l ) defines ψ . Furthermore, bythe proof of Proposition 2.6 ( c ), Im ϕ ∗ ⊆ O ǫ ( G ); since Res( O ǫ ( G )) = O ǫ ( L ),it follows that Im ψ ∗ ⊆ O ǫ ( L ). Consequently, we obtain a group of group-like elements Y = { d z = ψ ∗ ( D z ) | z ∈ d T I c } in O ǫ ( L ). Moreover, by Lemma2.2 and the definitions of ψ and the elements D i , the elements of Y arecentral.Since the map ν : O ǫ ( L ) → A l ,σ given by the pushout construction issurjective, the image of Y defines a group of central group-like elements in A l ,σ : Z = { ∂ z = ν ( d z ) | z ∈ d T I c } . Besides, | Z | = | Y | = | X | = ℓ s . Indeed, ¯ π ( Z ) = ¯ πν ( Y ) = π L ( Y ) = π L ψ ∗ ( X ) = X since the diagram (12) is commutative and π L ψ ∗ = id. Hence | ¯ π ( Z ) | = | X | , from which the assertion follows. (cid:3) We are now ready for our first main result.
Theorem 2.17.
Let D = ( I + , I − , N, Γ , σ, δ ) be a subgroup datum. Thenthere exists a Hopf algebra A D which is a quotient of O ǫ ( G ) and fits into theexact sequence → O (Γ) ˆ ι −→ A D ˆ π −→ H → . Concretely, A D is given by the quotient A l ,σ /J δ where J δ is the two-sidedideal generated by the set { ∂ z − δ ( z ) | z ∈ N } and the following diagram ofexact sequences of Hopf algebras is commutative (13) 1 / / O ( G ) ι / / res (cid:15) (cid:15) O ǫ ( G ) π / / Res (cid:15) (cid:15) u ǫ ( g ) ∗ / / p (cid:15) (cid:15) / / O ( L ) t σ (cid:15) (cid:15) ι L / / O ǫ ( L ) π L / / ν (cid:15) (cid:15) u ǫ ( l ) ∗ / / / / O (Γ) j / / A l ,σ ¯ π / / t (cid:15) (cid:15) u ǫ ( l ) ∗ / / v (cid:15) (cid:15) / / O (Γ) ˆ ι / / A D ˆ π / / H / / . UANTUM SUBGROUPS 21
Proof.
By Remark 2.12, N determines a subgroup Σ of T and the triple(Σ , I + , I − ) give rise to a surjective Hopf algebra map r : u ǫ ( g ) ∗ → H . Since σ : Γ → L ⊆ G is injective, by the first two steps developed before one canconstruct a Hopf algebra A l ,σ which is a quotient of O ǫ ( G ) and an extensionof O (Γ) by u ǫ ( l ) ∗ , where u ǫ ( l ) is the Hopf subalgebra of u ǫ ( g ) associated tothe triple ( T , I + , I − ). Moreover, by Lemma 2.14 ( b ), H is the quotient of u ǫ ( l ) ∗ by the two-sided ideal ( D z − | z ∈ N ). If δ : N → b Γ is a group map,then the elements δ ( z ) are central group-like elements in A l ,σ for all z ∈ N ,and the two-sided ideal J δ of A l ,σ generated by the set { ∂ z − δ ( z ) | z ∈ N } isa Hopf ideal. Hence, by [M00, Prop. 3.4 (c)] the following sequence is exact1 → O (Γ) / J → A l ,σ /J δ → u ǫ ( l ) ∗ / ¯ π ( J δ ) → , where J = J δ ∩ O (Γ). Since ¯ π ( ∂ z ) = D z and ¯ π ( δ ( z )) = 1 for all z ∈ N , wehave that ¯ π ( J δ ) is the two-sided ideal of u ǫ ( l ) ∗ given by ( D z − | z ∈ N ),which implies by Lemma 2.14 ( b ) that u ǫ ( l ) ∗ / ¯ π ( J δ ) = H . Hence, if wedenote A D := A l ,σ /J δ , we can re-write the exact sequence of above as(14) 1 → O (Γ) / J → A D → H → . To end the proof it is enough to see that J = J δ ∩ O (Γ) = 0. Clearly, J δ coincides with the two-sided ideal ( ∂ z δ ( z ) − − | z ∈ N ) of A l ,σ . Moreover,Υ := { ∂ z δ ( z ) − | z ∈ N } is a subgroup of central group-like elements of G ( A l ,σ ) and J δ = ( g − | g ∈ Υ) = A l ,σ C [Υ] + . Let ∂N = { ∂ z | z ∈ N } . Thenclearly the subalgebra B := O (Γ) C [ ∂N ] is a central Hopf subalgebra of A σ which contains C [Υ]. Further, B ≃ O (˜Γ) for some algebraic group ˜Γ andone has the following exact sequence of Hopf algebras1 → O (Γ) → O (˜Γ) → R → , where R = O (˜Γ) / O (˜Γ) O (Γ) + . But R ≃ ¯ π ( O (˜Γ)) = C [ N ], since¯ π ( O (˜Γ)) = [ O (˜Γ) + O (Γ) + A l ,σ ] / [ O (Γ) + A l ,σ ] ≃ O (˜Γ) / [ O (˜Γ) ∩ ( O (Γ) + A l ,σ )] ≃ O (˜Γ) / O (˜Γ) O (Γ) + . The last isomorphism follows from the fact that O (˜Γ) ∩ ( O (Γ) + A l ,σ ) = O (˜Γ) O (Γ) + . Indeed, since O (˜Γ) is a central Hopf subalgebra of the noe-therian algebra A l ,σ , by [S92, Thm. 3.3], O (˜Γ) is a direct summand of A l ,σ as O (˜Γ)-module, say A l ,σ = O (˜Γ) ⊕ M . Then O (Γ) + A l ,σ = O (Γ) + O (˜Γ) ⊕O (Γ) + M and the claim follows since O (˜Γ) ∩ O (Γ) + M = 0. Hence we havean exact sequence 1 → O (Γ) → O (˜Γ) ¯ π −→ C [ N ] → , which is cleft by the proof of Lemma 2.16, since ¯ π admits a coalgebra section.Moreover, this section on C [ N ] is by definition a bialgebra section, implyingthat O (˜Γ) ≃ O (Γ) ⊗ C [ ∂N ]. Let Λ = | Υ | P z ∈ N δ ( z ) ∂ − z be the integral of C [Υ] and denote by L Λ theendomorphism of O (˜Γ) given by left multiplication of Λ. Since O (˜Γ) ≃O (Γ) ⊗ C [ ∂N ] ≃ O (Γ) ⊗ C [Υ], it follows that Ker L Λ = O (Γ)( C [Υ]) + .But since A l ,σ = O (˜Γ) ⊕ M as O (˜Γ)-modules, we have that J δ ∩ O (˜Γ) = A l ,σ ( C [Υ]) + ∩ O (˜Γ) = O (˜Γ)( C [Υ]) + = O (Γ)( C [Υ]) + = Ker L Λ . Hence J δ ∩ O (Γ) = Ker L Λ ∩ O (Γ) = 0 for if x ∈ Ker L Λ ∩ O (Γ), then0 = Λ x = 1 | Υ | X z ∈ N ( δ ( z ) ⊗ ∂ − z )( x ⊗
1) = 1 | Υ | X z ∈ N δ ( z ) x ⊗ ∂ − z , which implies that δ ( z ) x = 0 for all z ∈ N , because the elements ∂ z arelinearly independent. Thus x = 0 since δ ( z ) is invertible for all z ∈ N . (cid:3) Remark . ( a ) If Γ is finite-dimensional, then O (Γ) = C Γ and by Remark2.11, dim A D = | Γ | dim H . In this case, D is a finite subgroup datum and thelast step of the proof of the theorem above follows easily by dimension argu-ments. Indeed, by [M00, Lemma 4.8], we have that dim A D = dim A l ,σ / | Υ | .Since A l ,σ and A D are extensions, it follows that(15)dim C Γ dim u ǫ ( l ) | Υ | = dim A D = dim( C Γ / J ) dim H = dim( C Γ / J ) dim u ǫ ( l ) | N | . Since ¯ π (Υ) = { D z | z ∈ N } and ¯ π ( ∂ z δ ( z ) − ) = D z = 1 if and only if z = 0, we have that | Υ | = | N | . Thus, from the equality (15) it follows that C Γ = C Γ / J .( b ) All exact sequences in the rows of diagram (13) are of the type B ֒ →A ։ H , where B is central in A and H is finite-dimensional. Thus, by [KT81,Thm. 1.7], B ⊂ A is an H -Galois extension and A is a finitely-generatedprojective B -module. Moreover, using Lemma 1.10 and Proposition 2.6 ( b ),one can see that the first three exact sequences are cleft.2.4. Relations between quantum subgroups.
Let U be any Hopf alge-bra and consider the category QU OT ( U ), whose objects are surjective Hopfalgebra maps q : U → A . If q : U → A and q ′ : U → A ′ are such maps,then an arrow q α / / q ′ in QU OT ( U ) is a Hopf algebra map α : A → A ′ such that αq = q ′ . In this language, a quotient of U is just an isomorphismclass of objects in QU OT ( U ); let [ q ] denote the class of the map q . Thereis a partial order in the set of quotients of U , given by [ q ] ≤ [ q ′ ] iff thereexists an arrow q α / / q ′ in QU OT ( U ). Notice that [ q ] ≤ [ q ′ ] and [ q ′ ] ≤ [ q ]implies [ q ] = [ q ′ ].Our aim is to describe the partial order in the set [ q D ], D a subgroupdatum, of quotients q D : O ǫ ( G ) ։ A D given by Theorem 2.17. Eventually,this will be the partial order in the set of all quotients of O ǫ ( G ). We beginby the following definition. By an abuse of notation we write [ A D ] = [ q D ]. Definition 2.19.
Let D = ( I + , I − , N, Γ , σ, δ ) and D ′ = ( I ′ + , I ′− , N ′ , Γ ′ , σ ′ , δ ′ )be subgroup data. We say that D ≤ D ′ iff UANTUM SUBGROUPS 23 • I ′ + ⊆ I + and I ′− ⊆ I − .In particular, this condition implies that I ′ ⊆ I , T I ′ ⊆ T I and T I c ⊆ T I ′ c . Since Σ = T I × Ω and Σ ′ = T I ′ × Ω ′ , we have thatΩ ′ ⊆ Ω ⊆ T I c ⊆ T I ′ c . As T I ′ c = T I c × T I ′ c − I c , the restriction map d T I ′ c ։ d T I c admits a canonical section η and η ( N ) ⊆ N ′ . • There exists a morphism of algebraic groups τ : Γ ′ → Γ such that στ = σ ′ . • δ ′ η = t τ δ .Furthermore, we say that D ≃ D ′ iff D ≤ D ′ and D ′ ≤ D . This means that • I + = I ′ + and I − = I ′− . • There exists an isomorphism of algebraic groups τ : Γ ′ → Γ suchthat στ = σ ′ . • N = N ′ and δ ′ = t τ δ . Theorem 2.20.
Let D and D ′ be subgroup data. Then(a) [ A D ] ≤ [ A D ′ ] iff D ≤ D ′ .(b) [ A D ] = [ A D ′ ] iff D ≃ D ′ .Proof. Let q = q D and q ′ = q D ′ . Suppose that [ A D ] ≤ [ A D ′ ], that is, thereexists a surjective Hopf algebra map α : A D → A D ′ such that αq = q ′ .Since by Theorem 2.17, ˆ ι t σ = qι and ˆ ι ′ t σ ′ = q ′ ι , we have that α ˆ ι t σ = αqι = q ′ ι = ˆ ι ′ t σ ′ . Thus, the Hopf algebra map β := α ˆ ι : O (Γ) → O (Γ ′ )is surjective with Im β ⊆ Im t σ and its transpose defines an injective mapof algebraic groups τ : Γ ′ → Γ such that στ = σ ′ .Again by Theorem 2.17, we know that both A D and A D ′ are centralextensions by H ≃ A D /A D O (Γ) + and H ′ ≃ A D ′ /A D ′ O (Γ ′ ) + , respectively.Since ˆ π ′ α ( A D O (Γ) + ) = ˆ π ′ ( A D ′ O (Γ ′ ) + ) = 0, there exists a surjective Hopfalgebra map γ : H → H ′ such that the following diagram commutes1 / / O ( G ) ι / / t σ ′ (cid:29) (cid:29) t σ (cid:15) (cid:15) O ǫ ( G ) π / / q (cid:15) (cid:15) q ′ (cid:30) (cid:30) u ǫ ( g ) ∗ / / r (cid:15) (cid:15) r ′ (cid:0) (cid:0) / / O (Γ) β (cid:15) (cid:15) ˆ ι / / A D ˆ π / / α (cid:15) (cid:15) H / / γ (cid:15) (cid:15) / / O (Γ ′ ) ˆ ι ′ / / A D ′ ˆ π ′ / / H ′ / / . Since t r : H ∗ ֒ → u ǫ ( g ) and t r ′ : ( H ′ ) ∗ ֒ → u ǫ ( g ) are just the inclusions, itfollows that t γ : ( H ′ ) ∗ ֒ → H ∗ is the same inclusion. If H ∗ and ( H ′ ) ∗ aredetermined by the triples (Σ , I + , I − ) and (Σ ′ , I ′ + , I ′− ), it follows that Σ ′ ⊆ Σ, I ′ + ⊆ I + , I ′− ⊆ I − , whence η ( N ) ⊆ N ′ . Thus, u ǫ ( l ′ ) ⊆ u ǫ ( l ) by Lemma 2.4.Now by Theorem 2.17, δ ( z ) = t ( ∂ z ) in A D and δ ′ ( z ′ ) = t ′ ( ∂ z ′ ) in A D ′ , forall z ∈ N and z ′ ∈ N ′ . Thus, for all z ∈ N we have t τ δ ( z ) = αδ ( z ) = αt ( ∂ z ) = αtν ( ψ ∗ ( D z )) = t ′ ν ′ (( ψ ′ ) ∗ η ( D z )) = δ ′ ( η ( z )) , where the fourth equality follows from the construction of the quotients A D , A D ′ and αq = q ′ . All this implies that D ≤ D ′ .Suppose now that D ≤ D ′ . This implies that u ǫ ( l ′ ) ⊆ u ǫ ( l ) and byconstruction, there exists a Hopf algebra map κ : O ǫ ( L ) → O ǫ ( L ′ ) such that O ǫ ( G ) Res / / / / Res ′ $ $ $ $ IIIIIIIII O ǫ ( L ) κ (cid:15) (cid:15) (cid:15) (cid:15) O ǫ ( L ′ )commutes. Since t τ t σ = t σ ′ , there exists a commutative diagram O ( L ) ι L / / t σ (cid:15) (cid:15) O ǫ ( L ) ν (cid:15) (cid:15) t ′ ν ′ κ (cid:22) (cid:22) O (Γ) ¯ ι / / t τ ' ' A l ,σ O (Γ ′ ) ˆ ι ′ / / A D ′ . As A l ,σ is a pushout, there exists a surjective Hopf algebra map ˜ α : A l ,σ → A D ′ such that ˜ αν = t ′ ν ′ κ . Since A D = A l ,σ /J δ , to show the existence of asurjective map α : A D → A D ′ such that αq = q ′ , it is enough to prove that˜ α ( J δ ) = 0. But J δ is the two-sided ideal of A l ,σ generated by δ ( z ) − ∂ z with z ∈ N ; now˜ α ( δ ( z ) − ∂ z ) = t τ δ ( z ) − ˜ α ( νψ ∗ ( D z )) = t τ δ ( z ) − t ′ ν ′ η ( z )= t τ δ ( z ) − δ ′ η ( z ) = 0 , by assumption. Hence, ˜ α ( J δ ) = 0. This finishes the proof of ( a ). Now ( b )follows immediately. (cid:3) Determining quantum subgroups
Let q : O ǫ ( G ) → A be a surjective Hopf algebra map. We prove now thatit is isomorphic to q D : O ǫ ( G ) → A D for some subgroup datum D . Thisconcludes the proof of Theorem 1.The Hopf subalgebra K = q ( O ( G )) is central in A and whence A is an H -extension of K , where H is the Hopf algebra H = A/AK + . Indeed, itfollows directly from [Mo93, Prop. 3.4.3], because A is faithfully flat over K by [S92, Thm. 3.3]. Since K is a quotient of O ( G ), there exists analgebraic group Γ and an injective map of algebraic groups σ : Γ → G suchthat K ≃ O (Γ). Moreover, since q ( O ǫ ( G ) O ( G ) + ) = AK + , we have that O ǫ ( G ) O ( G ) + ⊆ Ker ˆ πq , where ˆ π : A → H is the canonical projection. Since u ǫ ( g ) ∗ ≃ O ǫ ( G ) / [ O ǫ ( G ) O ( G ) + ], there exists a surjective map r : u ǫ ( g ) ∗ → UANTUM SUBGROUPS 25 H and by Proposition 1.12, H ∗ is determined by a triple (Σ , I + , I − ). Inparticular, we have the following commutative diagram(16) 1 / / O ( G ) ι / / t σ (cid:15) (cid:15) O ǫ ( G ) π / / q (cid:15) (cid:15) u ǫ ( g ) ∗ / / r (cid:15) (cid:15) / / O (Γ) ˆ ι / / A ˆ π / / H / / . Let N correspond to Σ as in Remark 2.12. Our aim is to show that thereexists δ such that A ≃ A D for the subgroup datum D = ( I + , I − , N, Γ , σ, δ ).Recall the Lie algebra l from Definition 1.1 and the Hopf algebra u ǫ ( l ) ⊇ H ∗ from 2.1.2. Denote by v : u ǫ ( l ) ∗ → H the surjective Hopf algebra mapinduced by this inclusion. Lemma 3.1.
The diagram (16) factorizes through the exact sequence / / O ( L ) ι L / / O ǫ ( L ) π L / / u ǫ ( l ) ∗ / / , that is, there exist Hopf algebra maps u, w such that the following diagramwith exact rows commutes: / / O ( G ) ι / / t σ (cid:29) (cid:29) res (cid:15) (cid:15) O ǫ ( G ) π / / Res (cid:15) (cid:15) q (cid:30) (cid:30) u ǫ ( g ) ∗ / / p (cid:15) (cid:15) r (cid:127) (cid:127) / / O ( L ) u (cid:15) (cid:15) ι L / / O ǫ ( L ) π L / / w (cid:15) (cid:15) u ǫ ( l ) ∗ / / v (cid:15) (cid:15) / / O (Γ) ˆ ι / / A ˆ π / / H / / . Proof.
To show the existence of the maps u and w it is enough to show thatKer Res ⊆ Ker q , since u is simply wι L . This clearly implies that vπ L = ˆ πw .Let ˇ U ǫ ( b + ) and ˇ U ǫ ( b − ) be the Borel subalgebras of ˇ U ǫ ( g ) (see [DL94] and[J96, Cap. 4]), and let A ǫ be the subalgebra of ˇ U ǫ ( b + ) ⊗ ˇ U ǫ ( b − ) generatedby the elements { ⊗ e j , f j ⊗ , K − λ ⊗ K λ : 1 ≤ j ≤ n, λ ∈ P } , where P is the weight lattice. By [DL94, Sec. 4.3], this algebra has a basisgiven by the set { f K − λ ⊗ K λ e } , where λ ∈ P and e , f are monomials in e α and f β respectively, α, β ∈ Q + . Moreover, A ǫ is a ( Q − , P, Q + )-gradedalgebra whose gradation is given bydeg( f j ⊗
1) = ( − α j , , , deg(1 ⊗ e j ) = (0 , , α j ) , deg( K − λ ⊗ K λ ) = (0 , λ, , for all 1 ≤ j ≤ n , λ ∈ P . By [DL94, 4.3 and 6.5], there exists an injectivealgebra map µ ǫ : O ǫ ( G ) → A ǫ such that µ ǫ ( O ( G )) ⊆ A , where A is thesubalgebra of A ǫ generated by the elements { ⊗ e ℓj , f ℓj ⊗ , K − ℓλ ⊗ K ℓλ : 1 ≤ j ≤ n, λ ∈ P } . Hence, it is enough to show that µ ǫ (Ker Res) ⊆ µ ǫ (Ker q ). Claim: µ ǫ (Ker Res) is the two-sided ideal I generated by the elements { ⊗ e k , f j ⊗ α k / ∈ I − , α j / ∈ I + } . Indeed, let λ ∈ P + and let ψ λ ∈ Γ ǫ ( g ) ◦ such that ψ λ ( F M E ) = δ ,E δ ,F M ( λ ) , ψ − λ ( EM F ) = δ ,E δ ,F M ( − λ ) , for all elements F M E of the PBW basis of Γ ǫ ( g ), where M ∈ Q and the form M ( λ ) is simply the linear extension of the bilinear form < α j , λ > = ǫ d i ( α i ,λ ) for all λ ∈ P , 1 ≤ i ≤ n . By [DL94, Sec. 4.4], there exist matrix coefficients ψ ± α ± λ , and α ∈ Q + such that ψ α − λ ( EM F ) = ψ − λ ( EM F E α ) , ψ − α − λ ( EM F ) = ψ − λ ( F α EM F ) , for all elements EM F of the PBW basis of Γ ǫ ( g ). Moreover, one has that µ ǫ ( ψ − ̟ i ) = K − ̟ i ⊗ K ̟ i , µ ǫ ( ψ α k − ̟ i ) = K − ̟ i ⊗ K ̟ i e k ,µ ǫ ( ψ − α j − ̟ i ) = f j K − ̟ i ⊗ K ̟ i , for all 1 ≤ i, j ≤ n . Through a direct computation one can see that ψ α k − ̟ i , ψ − α j − ̟ i ∈ Ker Res and µ ǫ ( ψ ̟ i ψ α k − ̟ i ) = 1 ⊗ e k µ ǫ ( ψ − α j − ̟ i ψ ̟ i ) = f j ⊗ . for all α k / ∈ I − , α j / ∈ I + . Hence, the generators of I are in µ ǫ (Ker Res).Conversely, if h ∈ Ker Res, then h | Γ ǫ ( l ) = 0 and by definition we have that < µ ǫ ( h ) , EM ⊗ N F > = < h, EM N F > = 0 , for all elements EM N F of the PBW basis of Γ ǫ ( l ). Thus, using the exis-tence of perfect pairings (see [DL94, Sec. 3.2]) and evaluating in adequateelements, it follows that each term of the basis { f K − λ ⊗ K λ e } that appearsin µ ǫ ( h ) must lie in I .Since 0 = π L Res( h ) = rπ ( h ) = ˆ πq ( h ), we have that q ( h ) ∈ Ker ˆ π = O (Γ) + A = q ( O ( G ) + O ǫ ( G )). Then there exist a ∈ O ( G ) + O ǫ ( G ) and c ∈ Ker q such that h = a + c ; in particular, for all generators t of I we havethat t = µ ǫ ( a ) + µ ǫ ( c ), where µ ǫ ( a ) is contained in A . Comparing degreesin both sides of the equality we have that µ ǫ ( a ) = 0, which implies that eachgenerator of I must lie in µ ǫ (Ker q ). (cid:3) The following lemma shows the convenience of characterizing the quo-tients A l ,σ of O ǫ ( G ) as pushouts. UANTUM SUBGROUPS 27
Lemma 3.2. σ (Γ) ⊆ L and therefore A is a quotient of A l ,σ given by thepushout. Moreover, the following diagram commutes (17) 1 / / O ( G ) ι / / res (cid:15) (cid:15) O ǫ ( G ) π / / Res (cid:15) (cid:15) u ǫ ( g ) ∗ / / p (cid:15) (cid:15) / / O ( L ) u (cid:15) (cid:15) ι L / / O ǫ ( L ) π L / / ν (cid:15) (cid:15) u ǫ ( l ) ∗ / / / / O (Γ) j / / A l ,σ ¯ π / / t (cid:15) (cid:15) u ǫ ( l ) ∗ / / v (cid:15) (cid:15) / / O (Γ) ˆ ι / / A ˆ π / / H / / . Proof.
Recall the maps u, w defined in the lemma above; we have that wι L = ˆ ιu , that is, the following diagram commutes O ( L ) ι L / / u (cid:15) (cid:15) O ǫ ( L ) ν (cid:15) (cid:15) w (cid:21) (cid:21) O (Γ) j / / ˆ ι , , A l ,σ A. Since A l ,σ is a pushout, there exists a unique Hopf algebra map t : A l ,σ → A such that ts = w and tj = ˆ ι . This implies that Ker ¯ π = j ( O (Γ)) + A l ,σ ⊆ Ker ˆ πt and therefore the diagram (17) is commutative. (cid:3) Let (Σ , I + , I − ) be the triple that determines H . Recall that by Remark2.12, giving a group Σ such that T I ⊆ Σ ⊆ T is the same as giving asubgroup N ⊆ d T I c . In fact, by Lemma 2.16, we know that the Hopf algebra A l ,σ contains a set of central group-like elements Z = { ∂ z | z ∈ d T I c } suchthat ¯ π ( ∂ z ) = D z for all z ∈ d T I c and H = u ǫ ( l ) ∗ / ( D z − | z ∈ N ). To seethat A = A D for a subgroup datum D = ( I + , I − , N, Γ , σ, δ ) it remains tofind a group map δ : N → b Γ such that A ≃ A l ,σ /J δ . This is given by thelast lemma of the paper. Lemma 3.3.
There exists a group homomorphism δ : N → b Γ such that J δ = ( ∂ z − δ ( z ) | z ∈ N ) is a Hopf ideal of A l ,σ and A ≃ A D = A l ,σ /J δ .Proof. Let ∂ z ∈ Z . Then ˆ πt ( ∂ z ) = v ¯ π ( ∂ z ) = 1 for all z ∈ N , by Lemma 2.14( b ). Since t ( ∂ z ) is a group-like element, this implies that t ( ∂ z ) ∈ A co ˆ π = O (Γ). As G ( O (Γ)) = b Γ, we have a group homomorphism δ given by δ : N → b Γ , δ ( z ) = t ( ∂ z ) ∀ z ∈ N. The two-sided ideal of A l ,σ given by J δ = ( ∂ z − δ ( z ) | z ∈ N ) is clearly aHopf ideal and t ( J δ ) = 0. Consequently we have a surjective Hopf algebramap θ : A D ։ A , which makes the following diagram commutative(18) 1 / / O (Γ) ˜ ι / / A D ˜ π / / θ (cid:15) (cid:15) (cid:15) (cid:15) H / / / / O (Γ) ˆ ι / / A ˆ π / / H / / . Then θ is an isomorphism by Corollary 1.15. (cid:3) Acknowledgments.
We thank Akira Masouka for kindly communicatingus Lemma 1.14.
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