aa r X i v : . [ m a t h . QA ] S e p TWISTING OF AFFINE ALGEBRAIC GROUPS, II
SHLOMO GELAKI
Abstract.
We use [G] to study the algebra structure of twistedcotriangular Hopf algebras J O ( G ) J , where J is a Hopf 2-cocyclefor a connected nilpotent algebraic group G over C . In particu-lar, we show that J O ( G ) J is an affine Noetherian domain withGelfand-Kirillov dimension dim( G ), and that if G is unipotent and J is supported on G , then J O ( G ) J ∼ = U ( g ) as algebras, where g = Lie( G ). We also determine the finite dimensional irreduciblerepresentations of J O ( G ) J , by analyzing twisted function algebrason ( H, H )-double cosets of the support H ⊂ G of J . Finally, wework out several examples to illustrate our results. Contents
1. Introduction 22. Preliminaries 42.1. Hopf 2-cocycles 42.2. Cotriangular Hopf algebras 52.3. Quasi-Frobenius Lie algebras 52.4. Ore extensions 62.5. Unipotent algebraic groups 63. The algebra structure of J O ( G ) J for unipotent G J g O ( G ) J J O ( Z g ) J H ∩ C = { } C ⊂ H J O ( G ) J for unipotent G Date : September 17, 2020.
Key words and phrases. connected nilpotent and unipotent algebraic groups;Hopf 2-cocycles; Cotriangular Hopf algebras; Ore extensions; Weyl algebras.
7. The Hopf algebra J O ( G ) J for connected nilpotent G Introduction
Let G be an affine algebraic group over C , and let O ( G ) be thecoordinate algebra of G . Then O ( G ) is a finitely generated commu-tative Hopf algebra over C . Recall that Drinfeld’s twisting procedureproduces (new) cotriangular Hopf algebra structures on the underlyingcoalgebra of O ( G ). Namely, if J ∈ ( O ( G ) ⊗ ) ∗ is a Hopf 2-cocycle for G , then there is a cotriangular Hopf algebra J O ( G ) J which is obtainedfrom O ( G ) after twisting its ordinary multiplication by means of J andreplacing its R -form 1 ⊗ J − J .In categorical terms, Hopf 2-cocycles for G correspond to tensorstructures on the forgetful functor Rep( G ) → Vec of the tensor cat-egory Rep( G ) of finite dimensional rational representations of G (see,e.g, [EGNO]).If J O ( G ) J = O ( G ) as Hopf algebras then J is called invariant. Invari-ant Hopf 2-cocycles for G form a group, which was described completelyfor connected G [EG4, Theorem 7.8]. However, if J is not invariantthen the situation becomes much more interesting, since the cotrian-gular Hopf algebra J O ( G ) J will be noncommutative. It is thus naturalto study the algebra structure and representation theory of J O ( G ) J incases where the classification of Hopf 2-cocycles for G is known. Forexample, for finite groups G , this was done in [EG3, Theorem 3.2] and[AEGN, Theorem 3.18].The classification of Hopf 2-cocycles for connected nilpotent alge-braic groups G over C is also known [G]. For example, Hopf 2-cocyclesin the unipotent case are classified by pairs ( H, ω ), where H is aclosed subgroup of G , called the support of J , and ω ∈ ∧ Lie( H ) ∗ is anon-degenerate 2-cocycle (equivalently, pairs ( h , r ), where h is a quasi-Frobenius Lie subalgebra of Lie( G ) and r ∈ ∧ h is a non-degeneratesolution to the CYBE (see 2.3)). This was done in [EG2, Theorem 3.2],using Etingof–Kazhdan quantization theory [EK1, EK2, EK3]. Later,we extended Movshev’s theory on twisting of finite groups [Mov] tothe algebraic group case [G, Section 3], and generalized the aforemen-tioned classification to connected nilpotent algebraic groups, without WISTING OF AFFINE ALGEBRAIC GROUPS, II 3 using Etingof–Kazhdan quantization theory [G, Corollary 5.2, Theo-rem 5.3]. Thus our goal in this paper is to study the algebra structure andrepresentation theory of the cotriangular Hopf algebras J O ( G ) J forconnected nilpotent G .The organization of the paper is as follows. In Section 2 we recallsome basic notions and results used in the sequel.In Section 3 we consider the cotriangular Hopf algebras J O ( G ) J for unipotent G . We first show that J O ( G ) J is an iterated Ore extension of C , thus is an affine Noetherian domain with Gelfand-Kirillov dimensiondim( G ) (see Corollary 3.2). Secondly, in Theorem 3.4 we prove thatif J is minimal (i.e., J is supported on G ) then J O ( G ) J ∼ = U ( g ) asalgebras, where g := Lie( G ), while in general, J O ( G ) J is a crossedproduct algebra of J O ( H ) J ∼ = U ( h ) and the algebra J O ( G/H ), where H ⊂ G is the support of J and h := Lie( H ) (see Theorem 3.8).In Section 4 we analyze twisted function algebras on ( H, H )-doublecosets in unipotent G , and use [G], to study the quotient algebra J O ( Z ) J of J O ( G ) J for a double coset Z in H \ G/H . In Theorems4.5 and 4.6 we show that J O ( Z ) J does not contain a Weyl subalgebraif and only if J O ( Z ) J ∼ = U ( h ) as algebras, if and only if, H and J are g -invariants for g ∈ Z .In Section 5 we determine the finite dimensional irreducible repre-sentations of J O ( G ) J (see Theorem 5.2). Namely, in Theorem 5.1 weshow that every irreducible representation of J O ( G ) J factors through aunique quotient algebra J O ( Z ) J , and then deduce from Theorems 4.5and 4.6 that J O ( Z ) J has a finite dimensional irreducible representationif and only if J O ( Z ) J ∼ = U ( h ) as algebras, if and only if, H and J are g -invariants for g ∈ Z .In Section 6 we give several examples that illustrate the results fromSections 3–5 (see Examples 6.1–6.5).Finally, in Section 7 we use the results from Sections 3–5 to describethe algebra structure and representations of the cotriangular Hopf al-gebras J O ( G ) J for connected nilpotent G (see Theorem 7.2). Acknowledgments.
I am grateful to Pavel Etingof for stimulatingand helpful discussions. This material is based upon work supported bythe National Science Foundation under Grant No. DMS-1440140, whilethe author was in residence at the Mathematical Sciences ResearchInstitute in Berkeley, California, during the Spring 2020 semester. We stress however, that both the classification of Hopf 2-cocycles and Movshev’stheory for arbitrary affine algebraic groups over C are still missing (see, e.g., [G]and references therein). SHLOMO GELAKI Preliminaries
Hopf -cocycles. Let A be a Hopf algebra over C . A linear form J : A ⊗ A → C is called a Hopf 2-cocycle for A if it has an inverse J − under the convolution product ∗ in Hom C ( A ⊗ A, C ), and satisfies X J ( a b , c ) J ( a , b ) = X J ( a, b c ) J ( b , c ) ,J ( a,
1) = ε ( a ) = J (1 , a )for all a, b, c ∈ A .Given two Hopf 2-cocycles K, J for A , one constructs a new algebra K A J as follows. As vector spaces, K A J = A , and the new multiplication K m J is given by K m J ( a ⊗ b ) = X K − ( a , b ) a b J ( a , b ) , a, b ∈ A. In particular, J A J is a Hopf algebra, where J A J = A as coalgebrasand the new multiplication J m J is given by(2.1) J m J ( a ⊗ b ) = X J − ( a , b ) a b J ( a , b ) , a, b ∈ A. Equivalently, J defines a tensor structure on the forgetful functorCorep( A ) → Vec.We also have two new unital associative algebras A J := A J and K A := K A , with multiplication rules given respectively by(2.2) m J ( a ⊗ b ) = X a b J ( a , b ) , a, b ∈ A, and(2.3) K m ( a ⊗ b ) = X K − ( a , b ) a b , a, b ∈ A. (For more details, see, e.g, [EGNO].) Remark 2.1.
The algebras A J , K A and K A J are called ( A, J A J )-biGalois, ( K A K , A )-biGalois and ( K A K , J A J )-biGalois algebras, respec-tively. Lemma 2.2.
The comultiplication map ∆ of A determines an injectivealgebra homomorphism ∆ : K A J −→ K A ⊗ A J .Proof. For every a, b ∈ A , we have∆( a )∆( b ) = X K m ( a ⊗ b ) ⊗ m J ( a ⊗ b )= X K − ( a , b ) a b ⊗ a b J ( a , b )= ∆ (cid:16)X K − ( a , b ) a b J ( a b ) (cid:17) = ∆( K m J ( a ⊗ b )) , J A J is denoted also by A J , e.g., in [G]. WISTING OF AFFINE ALGEBRAIC GROUPS, II 5 as claimed. (cid:3)
Cotriangular Hopf algebras.
Recall that (
A, R ) is cotriangularif R : A ⊗ A → C is an invertible linear map under ∗ , such that R − = R , and for every a, b, c ∈ A , we have R ( a, bc ) = X R ( a , b ) R ( a , c ) , R ( ab, c ) = X R ( b, c ) R ( a, c ) , and X R ( a , b ) b a = X a b R ( a , b ) . Recall that if R is non-degenerate, ( A, R ) is called minimal, and inthis case R defines two injective Hopf algebra maps A − −−→ A ∗ fin from A into its finite dual Hopf algebra A ∗ fin . Recall also that any cotriangularHopf algebra ( A, R ) has a unique minimal cotriangular Hopf algebraquotient [G, Proposition 2.1].Given a Hopf 2-cocycle J for A , ( J A J , R J ) is also cotriangular, where R J := J − ∗ R ∗ J . (For more details, see, e.g, [EGNO].) Lemma 2.3.
Assume A is commutative, and let J be a Hopf -cocyclefor A . If p ∈ A is primitive then for every a ∈ A , we have R J ( p, a ) = ( J − J )( p, a ) = ( J − − J − )( p, a ) . Proof.
Since p (1) = 0, we have0 = J ∗ J − ( p, a ) = X J ( p , a ) J − ( p , a )= X J ( p, a ) J − (1 , a ) + X J (1 , a ) J − ( p, a )= ( J + J − )( p, a ) . Thus, we have R J ( p, a ) = X J − ( p , a ) J ( p , a )= X J − ( p, a ) J (1 , a ) + X J − (1 , a ) J ( p, a )= ( J + J − )( p, a ) = ( J − J )( p, a ) , as claimed. (cid:3) Quasi-Frobenius Lie algebras.
Recall that a quasi-FrobeniusLie algebra is a Lie algebra h equipped with a non-degenerate skew-symmetric bilinear form ω : h × h → C satisfying ω ([ x, y ] , z ) + ω ([ z, x ] , y ) + ω ([ y, z ] , x ) = 0 , x, y, z ∈ h (i.e., ω is a symplectic 2-cocycle on h ). SHLOMO GELAKI
Let g be a Lie algebra. Recall that an element r ∈ ∧ g is a solutionof the classical Yang-Baxter equation (CYBE) if[ r , r ] + [ r , r ] + [ r , r ] = 0 . By Drinfeld [D], solutions r of the CYBE in ∧ g are classified by pairs( h , ω ), via r = ω − ∈ ∧ h , where h ⊂ g is a quasi-Frobenius Liesubalgebra with symplectic 2-cocycle ω .2.4. Ore extensions.
Let A be an algebra, and let δ : A → A be analgebra derivation of A . Recall that the Ore extension A [ y ; δ ] of A is thealgebra generated over A by y , subject to the relations ya − ay = δ ( a )for every a ∈ A . (See, e.g., [MR].)2.5. Unipotent algebraic groups.
Let G be a unipotent algebraicgroup of dimension m over C . Recall that A := O ( G ) is a finitelygenerated commutative irreducible pointed Hopf algebra, which is iso-morphic to a polynomial algebra in m variables as an algebra.Assume we have a central extension1 → C ι −→ G π −→ G → , where C ∼ = G a (= additive group). Then we can view O ( G ) as a Hopfsubalgebra of O ( G ) via π ∗ . Let O ( C ) = C [ z ], z is primitive. Choose W in O ( G ) that maps to z under the surjective Hopf algebra map ι ∗ : O ( G ) ։ O ( C ), with minimal possible degree with respect to thecoradical filtration. Set q ( W ) := ∆( W ) − W ⊗ − ⊗ W. Lemma 2.4. q ( W ) is a coalgebra -cocycle in O ( G ) + ⊗ O ( G ) + .Proof. Since the components of q ( W ) have smaller degree than W ,and are mapped to elements of degree ≤ C [ z ], it follows that q ( W )belongs to O ( G ) + ⊗ O ( G ) + . Finally, q ( W ) is a coalgebra 2-cocyclesince (∆ ⊗ id)∆( W ) = (id ⊗ ∆)∆( W ). (cid:3) Lemma 2.5.
The polynomial algebra O ( G )[ W ] has a unique Hopf al-gebra structure such that O ( G ) is a Hopf subalgebra of O ( G )[ W ] , and ∆( W ) = W ⊗ ⊗ W + q ( W ) . Moreover, we have O ( G ) ∼ = O ( G )[ W ] as Hopf algebras.Proof. It is clear that O ( G ) ∼ = O ( G )[ W ] as algebras, and that the Hopfalgebra structure is well defined by Lemma 2.4. Finally, it is clear that O ( G ) ∼ = O ( G )[ W ] as Hopf algebras. (cid:3) WISTING OF AFFINE ALGEBRAIC GROUPS, II 7
Recall that G is obtained from m successive 1-dimensional centralextensions with kernel G a . Thus by Lemma 2.5, A admits a filtration(2.4) C = A ⊂ A ⊂ · · · ⊂ A i ⊂ · · · ⊂ A m = A by Hopf subalgebras A i , such that for every 1 ≤ i ≤ m , A i = C [ y , . . . , y i ]is a polynomial algebra and q ( y i ) is a coalgebra 2-cocycle in A + i − ⊗ A + i − ,with q ( y ) = q ( y ) = 0. We will sometime write q ( y i ) = P Y ′ i ⊗ Y ′′ i ,and (id ⊗ ∆)( q ( y i )) = P Y ′ i ⊗ Y ′′ i ⊗ Y ′′ i .Finally, let H ⊂ G be a closed subgroup of codimension 1. It is wellknown that H is normal in G , so G/H ∼ = G a as algebraic groups. Lemma 2.6.
The exact sequence → H ֒ → G → G/H → splits.Proof. It is sufficient to show that the exact sequence of Lie algebras0 → h ֒ → g → g / h → h is an ideal of g ofcodimension 1, so by choosing a splitting of vector spaces g = h ⊕ C x , x ∈ g / h , we see that x acts on h by derivations. This implies thestatement. (cid:3) The algebra structure of J O ( G ) J for unipotent G In sections 3–5, G will denote a unipotent algebraic group over C ofdimension m , and J will be a Hopf 2-cocycle for G (i.e., for O ( G )).3.1. Ring theoretic properties.
Retain the notation from 2.5, andlet · denote the multiplication in J A J . Set Q := J − J . Lemma 3.1.
The following hold: (1) (2.4) determines a Hopf algebra filtration on J A J : C = A ⊂ A ⊂ · · · ⊂ J ( A i ) J ⊂ · · · ⊂ J ( A m ) J = J A J . (2) For every i , the Hopf algebra J ( A i ) J is generated by y i over J ( A i − ) J . (3) For every i, j , we have J ( y i , y j ) + J − ( y i , y j ) = 0 . (4) For every j < i , we have y i · y j = y i y j + X Y ′ i J ( Y ′′ i , y j ) + X Y ′ j J ( y i , Y ′′ j ) + X Y ′ i Y ′ j J ( Y ′′ i , Y ′′ j )+ X J − ( Y ′ i , Y ′ j ) J ( Y ′′ i , Y ′′ j ) Y ′′ i Y ′′ j . (5) For every j < i , we have [ y i , y j ] := y i · y j − y j · y i = X Y ′ i Q ( Y ′′ i , y j ) + X Y ′ j Q ( y i , Y ′′ j ) + X Y ′ i Y ′ j Q ( Y ′′ i , Y ′′ j )+ X (cid:0) J − ( Y ′ i , Y ′ j ) J ( Y ′′ i , Y ′′ j ) − J − ( Y ′ i , Y ′ j ) J ( Y ′′ i , Y ′′ j ) (cid:1) Y ′′ i Y ′′ j . SHLOMO GELAKI
Hence, [ y i , y j ] belongs to A + i − . (6) y , y are central primitives in J A J . (7) The linear map δ i : J ( A i − ) J → J ( A i − ) J , s [ y i , s ] , is analgebra derivation of J ( A i − ) J for every i . (8) For every i , J ( A i ) J ∼ = J ( A i − ) J [ y i ; δ i ] as Hopf algebras.Proof. (1)–(2) follow from (2.1) and Lemma 2.5 since each A i − is aHopf subalgebra of A i . (3)–(4) follow from (2.1) and ε ( y i · y j ) = 0.(5)–(6) follow from (4), (7) from (5), and (8) from (2) and Lemma2.5. (cid:3) Corollary 3.2.
The Hopf algebra J O ( G ) J is an affine Noetherian do-main with Gelfand-Kirrilov dimension dim( G ) .Proof. It follows from Lemma 3.1(8), by a simple induction, that J O ( G ) J is a finitely generated (i.e., affine) Noetherian domain. The claim aboutGelfand-Kirillov dimension follows from [MR, Proposition 8.2.11] andLemma 3.1(8) by simple induction. (cid:3) Remark 3.3.
One shows similarly that for every Hopf 2-cocycle K for G , the algebra K O ( G ) J is an affine Noetherian domain with Gelfand-Kirillov dimension dim( G ).3.2. The minimal case.
Let H ⊂ G be the support of J (see [G,Section 3.1]). Then J is a minimal Hopf 2-cocycle for H . Let h be theLie algebra of H . Theorem 3.4.
The R -form R J induces algebra isomorphism R + : J O ( H ) J ∼ = −→ U ( h ) . Proof.
Since ( J O ( H ) J , R J ) is a minimal cotriangular Hopf algebra, wehave an injective homomorphism of Hopf algebras R + : J O ( H ) J → ( J O ( H ) J ) ∗ fin , R + ( α )( β ) = R J ( β, α )(see 2.2). Since ( J O ( H ) J ) ∗ fin = J ( O ( H ) ∗ fin ) J (where the right hand sideis s twisted coalgebra), we have ( J O ( H ) J ) ∗ fin = O ( H ) ∗ fin = U ( h ) ⋊ C [ H ]as algebras.Let m = O ( H ) + be the maximal ideal of 1. We first show that theimage of R + is contained in the algebra of distributions O ( H ) ∗ = U ( h )supported at 1. Namely, that R + ( α ) vanishes on some power of m in thealgebra O ( H ) for every α ∈ J O ( H ) J . Indeed, if α ∈ J O ( H ) J has degree n with respect to the coradical filtration of O ( H ), then any summandin (a shortest expression of) ∆ n +1 ( α ) has at least one ε as a tensorand.Since R J ( ε, β ) = R J ( β, ε ) = β (1) = 0 for every β ∈ m , it follows that R + ( α ) vanishes on some power of m in the algebra J O ( H ) J . But it is WISTING OF AFFINE ALGEBRAIC GROUPS, II 9 clear that every such power contains some power of m in the algebra O ( H ), as desired. Thus, we have an injective algebra homomorphism R + : J O ( H ) J −→ U ( h ).To show that R + is surjective, it suffices to show that h belongs tothe image of R + . Indeed, let V ⊂ m be the orthogonal complementof m (with respect to R J ). Then R + maps V injectively into h (as h = ( m / m ) ∗ ), and since R J is non-degenerate, dim( V ) = dim( m / m ).Thus R + ( V ) = h , as required. (cid:3) Corollary 3.5.
We have an equivalence
Rep( J O ( H ) J ) ∼ = Rep( U ( h )) of abelian categories. In particular, J O ( H ) J has a unique finite dimen-sional irreducible representation (as h is nilpotent). (cid:3) Remark 3.6.
By [G, Theorems 4.7, 5.1], O ( H ) J and J O ( H ) are Weylalgebras with left and right action of H by algebra automorphisms,respectively. The induced action of h on O ( H ) J by derivations deter-mines a symplectic 2-cocycle ω ∈ ∧ h ∗ . We have U ω ( h ) ∼ = O ( H ) J as H -algebras, where H acts on U ω ( h ) and O ( H ) J by conjugation and lefttranslations, respectively. Similarly, U − ω ( h op ) ∼ = J O ( H ) as H -algebras.Thus, J O ( H ) ⊗ O ( H ) J ∼ = U ( − ω,ω ) ( h op ⊕ h ) as H -algebras.Now since by Lemma 2.2, we have an algebra isomorphism∆ : J O ( H ) J ∼ = −→ ( J O ( H ) ⊗ O ( H ) J ) H , where h ∈ H acts on J O ( H ) ⊗ O ( H ) J via ρ h ⊗ λ h , it follows that J O ( H ) J ∼ = −→ U ( − ω,ω ) ( h op ⊕ h ) H , as algebras. Thus, by Theorem 3.4, we have an algebra isomorphism U ( h ) ∼ = −→ U ( − ω,ω ) ( h op ⊕ h ) H . The general case.
Recall that O ( G/H ) and O ( H \ G ) are leftand right coideal subalgebras of O ( G ), respectively. It follows that J O ( G/H ) J = J O ( G/H ) is a subalgebra of both J O ( G ) J and J O ( G ). Lemma 3.7.
The subalgebra O ( H \ G/H ) ⊂ J O ( G ) J is central.Proof. Follows from ∆( O ( H \ G/H )) ⊂ O ( H \ G ) ⊗ O ( G/H ). (cid:3) Theorem 3.8.
The algebra J O ( G ) J is isomorphic to a crossed productalgebra J O ( G ) J ∼ = J O ( G/H ) σJ O ( H ) J for some invertible -cocycle σ : ( J O ( H ) J ) ⊗ → J O ( G/H ) and weakaction of J O ( H ) J on J O ( G/H ) . Thus, J O ( G ) J = J O ( G/H ) ⊗ J O ( H ) J I.e., invertible in the convolution algebra Hom(( J O ( H ) J ) ⊗ , J O ( G/H )). I.e., β · ( α ˜ α ) = ( β · α )( β · ˜ α ) and β · ( ˜ β · α ) = σ ( β , ˜ β )( β ˜ β · α ) σ − ( β , ˜ β ). as vector spaces, and the product is given by ( α ⊗ β )( ˜ α ⊗ ˜ β ) = α ( β · ˜ α ) σ ( β , ˜ β ) ⊗ β ˜ β , α, ˜ α ∈ O ( G/H ) , β, ˜ β ∈ J O ( H ) J . Proof.
We have a Hopf quotient ι ∗ : J O ( G ) J ։ J O ( H ) J , with Hopfkernel J O ( G/H ) J = J O ( G/H ). Thus we have an J O ( H ) J -extension J O ( G/H ) ⊂ J O ( G ) J of algebras. We claim it is cleft. Indeed, choosea regular section j to the inclusion morphism ι : H ֒ → G (this ispossible since G is unipotent). Then γ := ϕ : J O ( H ) J → J O ( G ) J is aninvertible J O ( H ) J -comodule map , as required. Hence, the statementfollows from [Mon, Theorem 7.2.2], with σ and weak action given by σ ( β, ˜ β ) = γ ( β ) γ ( ˜ β ) γ − ( β ˜ β ) , β, ˜ β ∈ J O ( H ) J , and β · α = γ ( β ) αγ − ( β ) , β ∈ J O ( H ) J , α ∈ O ( G/H ) , where γ − is the inverse of γ . (cid:3) Remark 3.9. If H is normal in G then J O ( G/H ) = O ( G/H ) is com-mutative. However, if H is not normal in G then the algebra J O ( G/H )is typically not commutative (see Example 6.4).3.4.
One sided twisted algebras.
Let L ⊂ H be a closed subgroup.Since O ( H/L ) is a left coideal subalgebra of O ( H ), it follows that J O ( H/L ) is a subalgebra of J O ( H ). Moreover, J O ( H/L ) = ( J O ( H )) L is a subalgebra of the Weyl algebra J O ( H ) ∼ = U ω ( h ) [G, Theorem 4.7]. Question 3.10.
What is the algebra structure of J O ( H/L )?We have the following partial answers to Question 3.10.
Theorem 3.11.
Let N ⊂ H be a closed normal subgroup. Then thefollowing hold: (1) There exists a closed subgroup L ⊂ H containing N such that J O ( H/N ) ∼ = O ( L \ H ) ⊗ W n as algebras, where n = dim( L ) − dim( N ) . (2) If N ) < dim( H ) , J O ( H/N ) contains a Weyl subalgebra.Proof. (1) Since N is normal in H , O ( H/N ) is a Hopf subalgebra of O ( H ). Thus J restricts to a Hopf 2-cocycle of H/N . By [G, Theorem3.1], there exists a closed subgroup L of H containing N such that L/N ⊂ H/N is the support of J . Then by [G, Theorem 4.7], we havean algebra isomorphism J O ( H/N ) ∼ = O (( L/N ) \ ( H/N )) ⊗ W n ∼ = O ( L \ H ) ⊗ W n , I.e., invertible in the convolution algebra Hom( J O ( H ) J , J O ( G ) J ). WISTING OF AFFINE ALGEBRAIC GROUPS, II 11 as claimed.(2) By (1), it suffices to show that the restriction of J to O ( H/N ) isnot trivial (since then n ≥ J O ( H/N ) J isisotropic with respect to R J . Thus, dim( H/N ) ≤ dim( H ) /
2, which isnot the case. (cid:3)
Corollary 3.12.
Let L ⊂ H be a closed subgroup and let N be itsnormal closure. If N ) < dim( H ) then J O ( H/L ) contains a Weylsubalgebra, or equivalently, J O ( H/L ) is noncommutative. (cid:3) -dimensional central extensions. Suppose we have a centralextension 1 → C −→ G π −→ G → , such that O ( C ) = C [ z ] ( z is primitive), and let W in O ( G ) be as in2.5. Then by Lemma 3.1, we have an isomorphism of Hopf algebras J O ( G ) J ∼ = J O ( G ) J [ W ; δ ] , where the derivation δ is given by δ ( V ) = [ W, V ] for every V ∈ J O ( G ) J ,and q ( W ) is in O ( G ) + ⊗ O ( G ) + .Set H := π ( H ), and let L ⊂ H be the support of the restriction of J to O ( G ). By [G, Proposition 4.6], L has codimension ≤ H .3.5.1. C ∩ H = { } . In this case, L = H . Write q ( W ) = X W ′ ⊗ W ′′ ∈ O ( G ) + ⊗ O ( G ) + in the shortest possible way. Set S := J − − J − . Lemma 3.13.
We can assume that W ∈ O ( G/H ) + , and then have [ W, V ] = P S ( W ′ , V ) W ′′ V for every V ∈ J O ( G ) J .Proof. The first claim follows from C ∩ H = { } . Since I ( H ) is a Hopfideal and O ( G/H ) is a left coideal in O ( G ), (id ⊗ ∆)∆( W ) lies in I ( H ) ⊗ O ( G ) ⊗ O ( G/H ) + O ( G ) ⊗ ⊗ I ( H ) + O ( G ) ⊗ I ( H ) ⊗ O ( G/H ) , which implies the second claim. (cid:3) C ⊂ H . In this case, L has codimension 1. Let A := H/L . Then O ( A ) = C [ x ], x is primitive, and the quotient map σ : H ։ A inducesan injective homomorphism of Hopf algebras σ ∗ : C [ x ] −→ J O ( H ) J .Thus, we can view C [ x ] as a Hopf subalgebra of J O ( H ) J via σ ∗ . Also,by Lemma 2.6, we can choose a splitting homomorphism of groups j : A −→ H of σ , and view A as a subgroup of H ⊂ G via j . Then j ∗ : O ( H ) ։ C [ x ] is a surjective homomorphism of Hopf algebras. Lemma 3.14.
The Hopf algebra surjective map ι ∗ H : O ( G ) ։ O ( H ) restricts to an algebra surjective map ι ∗ H : O ( L \ G/L ) ։ O ( A ) .Proof. Clearly, ι ∗ H maps O ( L \ G/L ) onto O ( L \ H/L ). Since L is normalin H , we have O ( L \ H/L ) = O ( A ), which implies the statement. (cid:3) Consider J O ( G ) J as a left comodule algebra over C [ x ] via( j ∗ ◦ ι ∗ H ⊗ id) ◦ ∆ : J O ( G ) J → C [ x ] ⊗ J O ( G ) J . Let B ⊂ J O ( G ) J be the coinvariant subalgebra. Pick X ∈ O ( L \ G/L )such that ι ∗ H ( X ) = x . Lemma 3.15.
The multiplication map B ⊗ C [ X ] −→ J O ( G ) J is anisomorphism of algebras.Proof. Follows since by Lemma 3.7, X is central in J O ( G ) J . (cid:3) The algebra J g O ( G ) J . Fix g ∈ G . Set Ad g := ρ g ◦ λ g , and J g := J ◦ (Ad g ⊗ Ad g ). Lemma 3.16. λ g − : J O ( G ) J → J g O ( G ) J is an algebra isomorphism.Proof. Straightforward. (cid:3) The quotient algebras J O ( Z g ) J Retain the notation of Section 3. Every double coset Z = HgH in H \ G/H is an orbit of the left action of the unipotent algebraic group H × H on G , given by ( a, b ) · g := agb − . Hence Z is an irreducibleclosed subset of G by the theorem of Kostant and Rosenlicht, and ithas dimension 2 dim( H ) − dim( H ∩ gHg − ) ( g ∈ Z ).Let I ( Z ) ⊂ O ( G ) be the defining ideal of the double coset Z . Since Z is irreducible, I ( Z ) is a prime ideal. Clearly, T Z I ( Z ) = 0.Now fix a double coset Z g = HgH . Set H g := H ∩ gHg − , andconsider the embedding θ : H g → H × H, a ( a, g − ag ) , of H g as a closed subgroup of H × H . The subgroup θ ( H g ) acts on H × H from the right via ( x, y ) θ ( a ) = ( xa, g − a − gy ), x, y ∈ H and a ∈ H g . Let ( x, y ) denote the orbit of ( x, y ) under this action. Thenwe have an isomorphism of affine varieties( H × H ) /θ ( H g ) ∼ = −→ Z g , ( x, y ) xgy. The above right action induces a left action of θ ( H g ) on O ( H ) ⊗ ,given by ( θ ( a )( α ⊗ β )( x, y ) = α ( xa ) β ( g − a − gy ), where x, y ∈ H and a ∈ H g . In other words, the action of θ ( a ) is via ρ a ⊗ λ g − ag , where λ, ρ WISTING OF AFFINE ALGEBRAIC GROUPS, II 13 are the left, right regular actions of G on O ( G ). Let ( O ( H ) ⊗O ( H )) θ ( H g ) denote the subalgebra of invariants under this action. Then we havean algebra isomorphism O ( Z g ) ∼ = −→ ( O ( H ) ⊗ O ( H )) θ ( H g ) . Equivalently, the surjective algebra homomorphism m ∗ g := ( ι ∗ ⊗ ι ∗ )(id ⊗ λ g − )∆ : O ( G ) ։ ( O ( H ) ⊗ O ( H )) θ ( H g ) has kernel I ( Z g ). Proposition 4.1.
The map m ∗ g determines a surjective algebra homo-morphism m ∗ g : J O ( G ) J ։ ( J O ( H ) ⊗ O ( H ) J ) θ ( H g ) . In particular, I ( Z g ) is a two sided ideal in J O ( G ) J .Proof. Since ι ∗ : J O ( G ) J ։ J O ( H ) J is an algebra homomorphism, itsuffices to show that (id ⊗ λ g − )∆ is an algebra homomorphism. Tothis end, notice that we have ∆ ◦ λ g − = ( λ g − ⊗ id) ◦ ∆. Thus, using(2.1)-(2.3), we get that for every α, β ∈ O ( G ),(id ⊗ λ g − )∆( αβ )= (id ⊗ λ g − )∆ (cid:16)X J − ( α , β ) α β J ( α , β ) (cid:17) = X J − ( α , β ) α β ⊗ λ g − ( α β ) J ( α , β )= X J − ( α , β ) α β ⊗ λ g − ( α ) λ g − ( β ) J ( α , β )= X ( α ⊗ λ g − ( α ))( β ⊗ λ g − ( β ))= (id ⊗ λ g − )∆( α )(id ⊗ λ g − )∆( β ) , as required. (cid:3) For g ∈ G , set J O ( Z g ) J := J O ( G ) J / I ( Z g ). Corollary 4.2.
For every g ∈ G , the algebra J O ( Z g ) J is an affineNoetherian domain of Gelfand-Kirillov dimension H ) − dim( H g ) .In particular, I ( Z g ) is a completely prime two sided ideal of J O ( G ) J .Proof. Since by [G, Theorem 4.7], J O ( H ) ⊗ O ( H ) J is a Weyl algebra,the claim follows from Corollary 3.2 and Proposition 4.1. (cid:3) Remark 4.3.
Let K be a Hopf 2-cocycle for O ( G ) with support e H .For every g ∈ G , let Z g := e HgH be the ( e H, H )-double coset of g , let N g := e H ∩ gHg − , and let d g := 12 (cid:16) dim( H ) + dim( e H ) (cid:17) − dim( N g ) . (By [G, Theorem 5.1], dim( H ) and dim( e H ) are even, so d g is an in-teger.) Then I ( Z g ) is a completely prime two sided ideal of K O ( G ) J ,and K O ( Z g ) J := ( K O ( G ) J ) / I ( Z g ) is an affine Noetherian domain withGelfand-Kirillov dimension dim( Z g ) = dim( H ) + dim( e H ) − dim( N g ). Remark 4.4.
By Proposition 4.1 and [G, Theorem 4.7], if H g is trivialthen O ( Z g ) J ∼ = W dim( H ) is a Weyl algebra. Theorem 4.5.
Assume H is g -invariant, and let ω g := ω g − ω . Thenwe have an algebra isomorphism J O ( Z g ) J ∼ = J g O ( H ) J ∼ = U ω g ( h ) . In particular, a maximal Weyl subalgebra of J O ( Z g ) J has dimension r ,where r ∈ Z ≥ is the rank of ω g restricted to h .Proof. Since H is g -invariant if and only if Ad g defines a Hopf algebraisomorphism O ( H ) ∼ = −→ O ( H ), J g is a minimal Hopf 2-cocycle for H .Clearly, J g corresponds to the symplectic 2-cocycle ω g of h .Now since λ g − maps I ( gH ) isomorphically onto I ( H ), it followsfrom Lemma 3.16 that λ g − induces an algebra isomorphism λ g − : J O ( Z g ) J ∼ = J O ( G ) J / I ( gH ) ∼ = −→ J g O ( G ) J / I ( H ) . Finally, by Theorem 3.4, we have algebra isomorphisms J g O ( G ) J / I ( H ) ∼ = J g ( O ( G ) / I ( H )) J ∼ = J g O ( H ) J ∼ = U ω g ( h ) , as desired. (cid:3) Next we consider the case where H is not g -invariant, i.e., the case d := dim( H/H g ) > Theorem 4.6. If H is not g -invariant then the algebra O ( Z g ) J con-tains a Weyl subalgebra.Proof. The proof is by induction on the dimension m of G , m ≥ m = 4. Since G is not commutative, dim( H ) = 2. Since H is properly contained in a proper normal subgroup N of G , it followsthat N is the Heisenberg group of dimension 3. Thus, the inductionbase is given in Example 6.2.Assume m ≥
5. Since G is unipotent, we have a central extension1 → C ι −→ G π −→ G → , where C ∼ = G a as in 3.5. Let O ( C ) = C [ z ], W ∈ O ( G ) and L ⊂ H ⊂ G be as in 3.5. Set ¯ g := π ( g ) and ¯ d := d ¯ g .There are two possible cases: Either H ∩ C is trivial or C ⊂ H . WISTING OF AFFINE ALGEBRAIC GROUPS, II 15 H ∩ C = { } . In this case, we are in the situation of 3.5.1.Consider the regular surjective map π : HgH ։ L ¯ gL . We have π − (¯ g ) = gC ∩ HgH , and dim(
HgH ) − dim( L ¯ gL ) = dim( π − (¯ g )). Lemma 4.7.
Exactly one of the following holds: (1) π − (¯ g ) = { g } . Equivalently, dim( HgH ) = dim( L ¯ gL ) . (2) π − (¯ g ) = gC . Equivalently, dim( HgH ) = dim( L ¯ gL ) + 1 .Proof. Follows since gC ∩ HgH ⊂ gC and dim( gC ) = 1. (cid:3) If Lemma 4.7(1) holds, then O ( Z g ) J ∼ = O ( Z ¯ g ) J and d = ¯ d , so theclaim follows by induction.Otherwise, Lemma 4.7(2) holds. Then d = ¯ d + 1. If ¯ d > O ( Z ¯ g ) J contains a Weyl subalgebra by the induction assumption, andsince O ( Z ¯ g ) J is a subalgebra of O ( Z g ) J , so does O ( Z g ) J .Otherwise ¯ d = 0. So, d = 1. Thus L is ¯ g − invariant, H g is normal in H , HgH = CgH , and dim( L ) = dim( H g ) + 1. Lemma 4.8.
The following hold: (1) W is not constant on HgH = CgH . (2) ρ g ( W ) − W − W ( g ) vanishes on C and H g , but not on H .Proof. Since W = z on C , each W ′ vanishes on C . Hence by Lemma3.13, W ( cgh ) = W ( cg ) = W ( c ) + W ( g ) + P W ′ ( c ) W ′′ ( g ) = c + W ( g )for every c ∈ C and h ∈ H , which implies (1) and the first part of (2).Since ρ g ( W )( ghg − ) = W ( gh ) = W ( g ) for every ghg − in H g , and W vanishes on H , the second part of (2) follows too. (cid:3) For l ∈ L , let ˜ l ∈ L be the unique element such that l = ¯ g ˜ l ¯ g − . Let h, ˜ h ∈ H be the unique elements such that l = π ( h ) and ˜ l = π (˜ h ). Let τ ( l ) := g ˜ hg − h − . Then τ ( l ) ∈ C . Lemma 4.9. τ : L → C is a group homomorphism, and we have asplitting exact sequence of algebraic groups → H g π −→ L τ −→ C → .Proof. Follows from Lemma 2.6 since H g has codimension 1 in H . (cid:3) By Lemma 4.9, we have an injective homomorphism of Hopf algebras τ ∗ : O ( C ) −→ O ( L ). Let p := τ ∗ ( z ). Then p is a nonzero primitiveelement in O ( L ) that generates the defining ideal of π ( H g ) in O ( L ). Lemma 4.10.
We may assume ι ∗ L ( ρ g ( W ) − W − W ( g )) = p .Proof. Consider the surjective Hopf algebra map f : O ( G ) ։ O ( CH )induced by the inclusion of groups CH ⊂ G . Since W vanishes on H , and restricts to z on C , it follows that f ( W ) = z , and f ( W ′ )is a primitive element in O ( CH ) that vanishes on C for every W ′ . Thus, ι ∗ L ( ρ g ( W ) − W − W ( g )) = P ι ∗ L ( W ′ ) W ′′ ( g ) is a nonzero primitiveelement in O ( L ) by Lemma 4.8, and since it vanishes on the definingideal of π ( H g ) in O ( L ), it must be proportional to p . (cid:3) Since R J is non-degenerate on O ( L ) J , there exists X ∈ O ( G ) suchthat R J ( p, X ) = 1. Choose such X with minimal possible degree withrespect to the coradical filtration, and write q ( X ) = P i X i ⊗ Y i in theshortest possible way. Then R J ( p, X i ) = 0 for every i . Proposition 4.11.
We have [ W, X ] ≡ on HgH .Proof.
Since each W ′′ is in O ( G/L ), and R J ( p, X i ) = 0 for every i , wehave by Lemma 3.13,[ W, X ](¯ gl ) = X S ( W ′ , X ) W ′′ (¯ gl ) X (¯ gl ) = X S ( W ′ , X ) W ′′ ( g ) X (¯ gl )= X S ( W ′ , X ) W ′′ ( g ) + X i S ( W ′ , X i ) W ′′ ( g ) Y i (¯ gl )= S ( ρ g ( W ) − W − W ( g ) , X ) + X i S ( ρ g ( W ) − W − W ( g ) , X i ) Y i (¯ gl )= S ( p, X ) + X i S ( p, X i ) Y i (¯ gl ) = R J ( p, X ) + X i R J ( p, X i ) Y i (¯ gl )= R J ( p, X )for every l ∈ L , where the equality before last follows from Lemma 2.3.Thus [ W, X ] ≡ HgH , as claimed. (cid:3) C ⊂ H . In this case, we are in the situation of 3.5.2, and W doesnot vanish on H .4.2.1. A is ¯ g -invariant. In this case, we have H ¯ gH = AL ¯ gL = L ¯ gLA ,and ¯ d = d > Proposition 4.12.
The algebra J O ( Z g ) J contains a Weyl subalgebra.Proof. Since J O ( H ¯ gH ) J is a subalgebra in J O ( Z g ) J via π ∗ , it sufficesto show that J O ( H ¯ gH ) J contains a Weyl subalgebra.Now since ¯ d >
0, the algebra J O ( Z ¯ g ) J contains a Weyl subalgebraby the induction assumption. Let α, β in J O ( G ) J such that [ α, β ] ≡ L ¯ gL . By Lemma 3.15, we can write α = P i α i X i and β = P i β i X i ,where α i , β i are in B , and [ α, β ] = P i,j [ α i , β j ] X i + j .Since X is L -biinvariant, we have X ≡ X (¯ g ) on L ¯ gL . If X (¯ g ) = 0,then X i (¯ g ) = 0 for all i ≥
1, hence [ α , β ] = [ α, β ] ≡ L ¯ gL .But [ α , β ] is in B (as α , β are), so [ α , β ] is A -invariant. Thus,[ α , β ] ≡ H ¯ gH = AL ¯ gL = L ¯ gLA , and we are done. WISTING OF AFFINE ALGEBRAIC GROUPS, II 17
Otherwise, X (¯ g ) = 0. We may assume X (¯ g ) = 1. Then we have P i,j [ α i , β j ] = [ α, β ] ≡ L ¯ gL . Set ˜ α := P i α i and ˜ β := P i β i .Then ˜ α, ˜ β are in B , and we have [ ˜ α, ˜ β ] ≡ L ¯ gL , hence on H ¯ gH , asabove. (cid:3) A is not ¯ g -invariant. In this case, A ¯ gA is 2-dimensional and d = ¯ d + 1. Set ϕ := j ∗ ◦ ι ∗ L : O ( G ) ։ O ( A ). Lemma 4.13.
There exists V ∈ O ( G ) + such that the following hold: (1) ϕ ( ρ ¯ g ( V )) and ϕ ( λ ¯ g − ( V )) are algebraically independent, and V (¯ g ) = 0 . In particular, V is not primitive. (2) ϕ ( q ( V )) = ϕ ( q ( V )) = 0 . In particular, ϕ ( V ) is not primitive.Proof. (1) Since A is not ¯ g -invariant, the first claim follows, and re-placing V by V − V (¯ g ) if necessary, we may assume V (¯ g ) = 0. Sinceeither ϕ ( ρ ¯ g ( V )) or ϕ ( λ ¯ g − ( V )) is not primitive, V is not primitive.(2) The first claim follows from ∆( ϕ ( V )) = ∆ op ( ϕ ( V )) (as A iscommutative). If ϕ ( q ( V )) = ϕ ( q ( V )) = 0, then ϕ ( ρ ¯ g ( V )) = ϕ ( V )and ϕ ( λ ¯ g − ( V )) = ϕ ( V ), which is a contradiction. Thus, ϕ ( V ) is notprimitive, as claimed. (cid:3) Pick V ∈ O ( G ) + as in Lemma 4.13, with minimal possible degree ℓ ≥ V ∈ O ( L \ G/L ) + . Write q ( V ) = P i X i ⊗ Y i . Then for every i ,we have X i ∈ O ( L \ G ) + and Y i ∈ O ( G/L ) + . Lemma 4.14.
We have [ W, V ] = P i S ( W, X i ) Y i − P i S ( W, Y i ) X i .Proof. Since Y i and V lie in O ( G/L ), it follows from (2.1) that[
W, V ] = X i S ( W, X i ) Y i − X i S ( W, Y i ) X i + X S ( W ′ , X i ) W ′′ Y i . Moreover, since X i ∈ O ( L \ G ) + for every i , and W ′ ∈ O ( G ), we have S ( W ′ , X i ) = 0 for every i and W ′ , so the claim follows. (cid:3) Lemma 4.15. q ( V ) = X ⊗ Y , where X and Y are primitive elementsin the defining ideal of L , and ι ∗ H ( X ) = ι ∗ H ( Y ) .Proof. Suppose ι ∗ H ( X i ) = 0. Then since X i vanishes on L , it cannotvanish on A . So, ϕ ( X i ) = 0. Moreover, since the degree of X i is < ℓ , X i must be primitive by minimality of ℓ . Similarly, if ι ∗ H ( Y i ) = 0 then Y i must be primitive. Thus ℓ = 2, which implies the statement. (cid:3) Set p := ι ∗ H ( X ) = ι ∗ H ( Y ). Then p is primitive in O ( H ). Lemma 4.16.
We have [ W, V ] = R J ( W, p )( Y − X ) . Proof.
By Lemmas 4.14 and 4.15(2), we have[
W, V ] = S ( W, ι ∗ H ( X )) Y − S ( W, ι ∗ H ( Y )) X = S ( W, ι ∗ H ( X )) Y − S ( W, ι ∗ H ( X )) X = S ( W, p )( Y − X )= R J ( W, p )( Y − X ) , as claimed, where the last equation holds by Lemma 2.3. (cid:3) Set c := R J ( W, p )( Y − X )( g ) ∈ C . Proposition 4.17.
We have [ W, V ] ≡ c = 0 on HgH . Thus, we mayassume [ W, V ] ≡ on HgH .Proof.
We first show that c = 0. To this end, we have to show that X ( g ) = Y ( g ) and R J ( W, p ) = 0. Since ϕ ( ρ ¯ g ( V )) = ϕ ( V ) + ϕ ( X ) Y ( g ), ϕ ( λ ¯ g − ( V )) = ϕ ( V ) + ϕ ( Y ) X ( g ), ϕ ( ρ ¯ g ( V )) = ϕ ( λ ¯ g − ( V )) by Lemma4.13, and ϕ ( X ) = ϕ ( Y ) by Lemma 4.15, we have X ( g ) = Y ( g ). Fur-thermore, since p vanishes on L by Lemma 4.15, p is orthogonal to J O ( H ) J inside J O ( H ) J . Thus, R J ( W, p ) = 0 by the non-degeneracy of R J on J O ( H ) J .Next we show that [ W, V ] ≡ c on H ¯ gH . Since by Lemma 4.15, X is primitive in I ( L ), we have X ( a l ¯ ga l ) = X ( a ) + X (¯ g ) + X ( a )for every a , a ∈ A and l , l ∈ L , and similarly for Y . Thus, since byLemma 4.15, X = Y on A , we have ( Y − X )( a l ¯ ga l ) = ( Y − X )( g )for every a , a ∈ A and l , l ∈ L , which implies that [ W, V ] ≡ c on H ¯ gH , as claimed. (cid:3) The proof of Theorem 4.6 is complete. (cid:3)
Question 4.18.
Fix g ∈ G , and set A := J O ( Z g ) J .(1) Is it true that A ∼ = U ω g ( h g ) ⊗ W as algebras, where W is a Weylsubalgebra with GKdim( W ) = 2 d g ?(2) Is it true that A contains a subalgebra U ∼ = U ω g ( h g ), and aWeyl subalgebra W with GKdim( W ) = 2 d g , such that U ∩ W is trivial?(3) Is it true that a maximal Weyl subalgebra of A has Gelfand-Kirillov dimension 2 d g + r , where r ∈ Z ≥ is the rank of ω g restricted to h g ?(See, e.g., the end of Example 6.5.) Remark 4.19.
By Proposition 4.1, Question 4.18 is a special case ofQuestion 3.10.
WISTING OF AFFINE ALGEBRAIC GROUPS, II 19 Representations of J O ( G ) J for unipotent G Retain the notation of Sections 3 and 4.
Theorem 5.1.
Every irreducible representation V of J O ( G ) J factorsthrough a unique quotient J O ( Z ) J .Proof. By Lemma 3.7, the central subalgebra O ( H \ G/H ) ⊂ J O ( G ) J acts on V by a certain central character χ : O ( H \ G/H ) → C . Let K := Ker( χ ), let I ⊂ O ( G ) be the ideal generated by K , andlet Z ⊂ G be the closed subscheme defined by I . Then Z is anaffine scheme of finite type (i.e., O ( Z ) can be non-reduced and havenilpotents) with an H × H -action, and all orbits of H × H on theunderlying variety of Z are closed by the theorem of Kostant andRosenlicht since H is unipotent. Pick an orbit Y in Z . If Y = Z then Z = HgH is a single H × H -orbit, so V factors through J O ( Z g ) J , andwe are done.Otherwise, let ˜ I be the ideal of functions on Z vanishing on Y .Then ˜ I is invariant under H × H , and is a union of finite dimensional H × H -modules, so it has a fixed vector f = 0 (as H is unipotent), andthis f cannot be constant since it vanishes on Y . Thus, O ( H \ Z /H )is nontrivial.Now consider the nontrivial central subalgebra O ( H \ Z /H ) ⊂ O ( Z ).It has a maximal ideal n , so O ( H \ Z /H ) / n is a field extension of C .But it is countably dimensional, so has to be C . Thus, we have a cen-tral character χ : O ( H \ Z /H ) → C by which O ( H \ Z /H ) acts on V ,as above. Let K := Ker( χ ), let I ⊂ O ( G ) /I be the ideal generatedby K , and let Z ⊂ Z be the closed subscheme defined by I . Then Z is H × H -stable. Thus we can proceed as above by looking at theorbits of H × H in Z . However, by the Hilbert basis theorem, thesequence Z ⊃ Z ⊃ · · · must stabilize. Hence the process will end,and we will reach a single H × H -orbit HgH , as desired.Finally, since I ( Z ) + I ( Z ′ ) = J O ( G ) J for every two distinct doublecosets Z and Z ′ , I ( Z ) and I ( Z ′ ) cannot both annihilate V . (cid:3) Let N G ( H, J ) be the subgroup of the normalizer N G ( H ) of H in G ,consisting of all elements g ∈ N G ( H ) such that J is g -invariant. Theorem 5.2.
There is one to one correspondence between isomor-phism classes of finite dimensional irreducible representations of J O ( G ) J and elements of the group N G ( H, J ) .Proof. Follows from Theorems 4.5, 4.6 and 5.1, and the facts thatWeyl algebras of degree ≥ and nilpotent Lie algebras have only the trivial finite dimensional irre-ducible representation. (cid:3) Remark 5.3.
Retain the notation from Remark 4.3. Then every irre-ducible representation of the algebra K O ( G ) J factors through a uniquequotient algebra K O ( Z g ) J . Furthermore, if d g = 0 then K and J aregauge equivalent, and K O ( G ) J has finite dimensional irreducible rep-resentations if and only if J is g -invariant. Indeed, note that d g = 0if and only if N g = e H = gHg − . But the later implies that K , J aregauge equivalent. Thus we are reduced to Theorem 5.2.6. Examples
Let A = G a with O ( A ) = C [ X, V ], and let a be the Lie algebra of A , with basis { ∂∂X , ∂∂V } . Let r := ∂∂X ∧ ∂∂V . Then the composition J : O ( A ) ⊗ O ( A ) e r/ −−→ O ( A ) ⊗ O ( A ) ε ⊗ ε −−→ C is a minimal Hopf 2-cocycle for A [EG1, Section 4], and it is straight-forward to verify that(6.1) J ( X, V ) = J − ( V, X ) = 1 / , J ( V, X ) = J − ( X, V ) = − / . Clearly, we have J O ( A ) J = O ( A ) ∼ = U ( a ) as Hopf algebras. Example 6.1.
Let G = { xE + vE + yE | x, v, y ∈ C } ⊂ U be the Heisenberg group of dimension 3. Its coordinate Hopf algebrais a polynomial algebra O ( G ) = C [ X, Y, V ], with
X, Y being primitive,and ∆( V ) = V ⊗ ⊗ V + X ⊗ Y .Set a := ∂∂X , b := ∂∂Y , c := ∂∂V . Then g := Lie( G ) has basis a, b, c , with bracket [ a, b ] = c . The element r := a ∧ c is a non-degenerate g -invariant solution to the CYBE in ∧ h ,where h ⊂ g is the (abelian) Lie subalgebra spanned by a, c . Thus, J := e r/ is a minimal Hopf 2-cocycle for H , where H = { xE + vE | x, v ∈ C } ⊂ G is the (normal abelian) subgroup with Lie( H ) = h , and we have that J O ( H ) J = O ( H ) ∼ = U ( h ) as Hopf algebras.We now view J as a (non-minimal) Hopf 2-cocycle for G by pullingit back along the obvious Hopf algebra surjective map O ( G ) ։ O ( H )determined by Y X X , and V V . Since J is an invariant WISTING OF AFFINE ALGEBRAIC GROUPS, II 21
Hopf 2-cocycle for G , J O ( G ) J = O ( G ) ∼ = O ( G/H ) ⊗ J O ( H ) J as al-gebras (so, J O ( G ) J is isomorphic to U ( C ) as an algebra, but not to U ( g )), and J O ( Z g ) J ∼ = J O ( H ) J = O ( H ) ∼ = U ( h ) for every g ∈ G since H is normal and r is g -invariant. Example 6.2. (The induction base in the proof of Theorem 4.6.) Let G = x v w y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v, w, y ∈ C ⊂ U . Then G is a 4-dimensional unipotent algebraic group over C . Its co-ordinate Hopf algebra O ( G ) = C [ X, Y, V, W ] is a polynomial algebra,with
X, Y being primitive, and∆( V ) = V ⊗ ⊗ V + X ⊗ Y, ∆( W ) = W ⊗ ⊗ W + V ⊗ Y + X ⊗ Y / . Let C := { wE | w ∈ C } . Then C ∼ = G a is a closed centralsubgroup of G .Set a := ∂∂X , b := ∂∂Y , c := ∂∂V , d := ∂∂W . Then g := Lie( G ) has basis a, b, c, d , with brackets [ a, b ] = c , [ c, b ] = d .Let H := { xE + vE | x, v ∈ C } ⊂ G. Then H ∼ = G a and O ( H ) = C [ X, V ] is a polynomial Hopf algebra.Since C ∩ H = { } , we have H ∼ = L (see 4.1). Let h := Lie( H )with basis a, c , let r := a ∧ c , and let J := e r/ as above. We have J O ( H ) J = O ( H ) as algebras.Pull J back to O ( G ) along the obvious Hopf algebra surjective map O ( G ) ։ O ( H ) determined by Y, W X X , and V V .By (2.1) and (6.1), it is straightforward to find out that J O ( G ) J isgenerated as an algebra by X, Y, V, W , subject to the relations[
X, Y ] = [
X, V ] = [
Y, V ] = [
Y, W ] = 0 , [ W, X ] = Y, [ W, V ] = Y / . In particular
X, V span a lie algebra isomorphic to h , W, Y span anabelian lie algebra a , and we have algebra isomorphisms J O ( G ) J ∼ = U ( a ) ⋊ U ( h ) ∼ = O ( G/H ) J O ( H ) J , where [ X, W ] = − Y , [ X, Y ] = 0, [
V, Y ] = 0 and [
V, W ] = − Y /
2. (SeeTheorem 3.8.)Take g := g ( x , v , w , ∈ G . Then H g = H , I ( Z g ) = ( Y, W − w ),and J O ( Z g ) J ∼ = C [ X, V ] ∼ = U ( h g ) = U ( h ) as algebras. Take g := g (0 , , , y ), y = 0. Then g − = g (0 , , , − y ). We have H g = { xE − y xE | x ∈ C } ∼ = G a , and HgH = x v w y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v, w ∈ C . It follows that I ( Z g ) = ( Y − y ), and [ W, V ] ≡ y / = 0 on HgH .Thus, we have J O ( Z g ) J ∼ = C [ X ] ⊗ C [ V ][ W ; d/dV ] ∼ = U ( h g ) ⊗ W as algebras.Finally, note that we have p = y X + V , ι ∗ L ( ρ g ( W ) − W − W ( g )) = y p , p vanishes on H g , and R J ( p, ι ∗ ( X )) = 1. Also, for l = l ( x, v ) ∈ L , wehave τ ( l ) = 1 − y ( y x/ v ) E ∈ C (see 4.1). Example 6.3.
Let G and g be as in Example 6.2. Set r := a ∧ c + d ∧ b. Then r is a non-degenerate solution of the CYBE in ∧ g , correspondingto the symplectic structure ω on g determined by ω ( a, c ) = ω ( d, b ) = 1.Let J := 1 + r/ · · · be the corresponding minimal Hopf 2-cocyclefor G . It is straightforward to verify that J ( X, V ) = J − ( V, X ) = J ( W, Y ) = J − ( Y, W ) = 1 / ,J ( V, X ) = J − ( X, V ) = J ( Y, W ) = J − ( W, Y ) = − / , and J, J − vanish on other pairs of generators.By (2.1), it is straightforward to find out that the minimal cotrian-gular Hopf algebra J O ( G ) J is generated as an algebra by X, Y, V, W ,such that [
W, X ] = Y, [ W, V ] = Y / X, and other pairs of generators commute. Set X ′ := Y / X . Then X ′ , Y, V, W span a Lie algebra of dimension 4 such that [ W, V ] = X ′ and [ W, X ′ ] = Y , hence isomorphic to g . Thus, J O ( G ) J ∼ = U ( g ) asalgebras (see Theorem 3.4). Example 6.4.
Let G := U be the 6-dimensional unipotent algebraicgroup of 4 by 4 upper triangular matrices over C . Its coordinate Hopf It is known that J has this form (see, e.g., [EG1]). WISTING OF AFFINE ALGEBRAIC GROUPS, II 23 algebra O ( G ) = C [ F , F , F , F , F , F ] is a polynomial algebra,with F , F , F being primitive,∆( F ) = F ⊗ ⊗ F + F ⊗ F , ∆( F ) = F ⊗ ⊗ F + F ⊗ F and ∆( F ) = F ⊗ ⊗ F + F ⊗ F + F ⊗ F . Let H := { xE + uE | x, u ∈ C } ⊂ G . Then H ∼ = G a is a closedsubgroup of G , and O ( H ) = C [ X, U ] is a polynomial Hopf algebra. Let a := ∂∂X , c := ∂∂U , r := a ∧ c , and J := e r/ .Pull J back to O ( G ) along the Hopf algebra surjective homomor-phism O ( G ) ։ O ( H ) determined by F , F , F , F F X ,and F U . Then it is straightforward to verify that J O ( G/H ) isgenerated as an algebra by F , F , Y, V , where Y := F − F F and V := F − F F , such that[ Y, F ] = F , [ F , V ] = F F , [ Y, V ] = F Y, and other pairs of generators commute. In particular, J O ( G/H ) is notcommutative (see Remark 3.9).
Example 6.5.
Let G := U be as in Example 6.4. Let H = x v w y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v, w, y ∈ C , and J be as in Example 6.3 (where H is denoted there by G ).Pull J back to O ( G ) along the surjective Hopf algebra homomor-phism O ( G ) ։ O ( H ) determined by F X, F V, F W, F , F Y, F Y / . Then using (6.1), it is straightforward to verify that J O ( G ) J is gener-ated as an algebra by { F ij } , such that[ F , F ] = F , [ F , F ] = F − F − F F , [ F , F ] = F − F , and other pairs of generators commute.Let C := { wE | w ∈ C } . Then C ⊂ H is central in G (see 4.2).We have L = x v
00 1 0 00 0 1 00 0 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v ∈ C ⊂ H = x v
00 1 y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v, y ∈ C and A = y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ∈ C ∼ = G a . Now take g := 1 + E . Then we have H g = { xE + vE + wE | x, v, w ∈ C } ∼ = G a , dim( HgH ) = 5 , d = 1 ,H ¯ g = { xE + vE | x, v ∈ C } ∼ = G a , dim( H ¯ gH ) = 4 ,L = L ¯ g ∼ = G a , dim( L ¯ gL ) = 2 , ¯ d = 0 , ¯ gA = y y y + 10 0 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ∈ C , A ¯ g = y y + y y + 10 0 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ∈ C and A ¯ gA = y z + y y + 10 0 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y, z ∈ C . In particular, A is not ¯ g -invariant (see 4.2.2).Now it is straightforward to verify that HgH = x v w y z y + 10 0 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x, v, w, y, z ∈ C . Thus, I ( HgH ) = h F − F − i , and we see that F , F generatea Weyl subalgebra in J O ( Z g ) J . (In the notation of 4.2.2, we have W = F , V = F , X = F , and Y = F .)Finally, let A, B, C, T, Q be the images of F , F , F , F , F in J O ( Z g ) J , respectively. Then J O ( Z g ) J is generated as an algebra by A, B, C, T, Q , such that [
A, Q ] = − C , [ B, Q ] = T − A − C ( C − T, Q ] = 1, and other pairs of generators commute. Thus, replacing A with − A and setting P := T − C ( C − J O ( Z g ) J ∼ = C [ A, B, C, P ][ Q ; δ ] , δ := C ∂∂A + ( P + A ) ∂∂B + ∂∂P . Set A ′ := A − CP and B ′ := B − ( P + A ) P + ( C + 1) P /
2, and let W denote the Weyl subalgebra generated by P, Q . Then we have an
WISTING OF AFFINE ALGEBRAIC GROUPS, II 25 algebra isomorphism J O ( Z g ) J ∼ = C [ A ′ , B ′ , C ] ⊗ W ∼ = U ω g ( h g ) ⊗ W . (See Question 4.18.)7. The Hopf algebra J O ( G ) J for connected nilpotent G In this section, we let G = T × U be a connected nilpotent algebraicgroup over C , where T is the maximal torus of G and U is the unipotentradical of G . Let A := O ( G ). Let O ( U ) = C [ y , . . . , y m ] be as in Sec-tion 3, and let A := O ( T ) = C [ X ( T )] = C [ x ± , . . . , x ± k ]. Recall that A = O ( T ) ⊗ O ( U ) as Hopf algebras. Finally, set A i := A [ y , . . . , y i ],1 ≤ i ≤ m (so, A m = A ). Lemma 7.1.
Let J be a Hopf -cocycle for A , and let · denote themultiplication in J A J . The following hold: (1) We have a Hopf filtration on J A J : A ⊂ J ( A ) J ⊂ · · · ⊂ J ( A i ) J ⊂ · · · ⊂ J ( A m ) J = J A J . (2) For every i , the Hopf algebra J ( A i ) J is generated by y i over J ( A i − ) J . (3) For every j < i , we have [ y i , y j ] := y i · y j − y j · y i = X Y ′ i Q ( Y ′′ i , y j ) + X Y ′ j Q ( y i , Y ′′ j ) + X Y ′ i Y ′ j Q ( Y ′′ i , Y ′′ j )+ X (cid:0) J − ( Y ′ i , Y ′ j ) J ( Y ′′ i , Y ′′ j ) − J − ( Y ′ i , Y ′ j ) J ( Y ′′ i , Y ′′ j ) (cid:1) Y ′′ i Y ′′ j . Hence, [ y i , y j ] belongs to A + i − . (4) y , y are central primitives in J A J . (5) For every i, j , we have [ y i , x j ] = x j X Q ( Y ′′ i , x j ) Y ′ i + x j X (cid:0) J − ( Y ′ i , x j ) J ( Y ′′ i , x j ) − J − ( Y ′ i , x j ) J ( Y ′′ i , x j ) (cid:1) Y ′′ i . (6) The linear map δ i : J ( A i − ) J → J ( A i − ) J , s [ y i , s ] , is analgebra derivation of J ( A i − ) J for every i . (7) For every i , J ( A i ) J ∼ = J ( A i − ) J [ y i ; δ i ] as Hopf algebras.Proof. (1)–(4) are similar to Lemma 3.1. As for (5), we have[ y i , x j ] = x j (cid:0) J − ( y j , x j ) − J − ( y j , x j ) + Q ( y j , x j ) (cid:1) + x j (cid:16)X Q ( Y ′′ i , x j ) Y ′ i + X J − ( Y ′ i , x j ) Y ′′ i J ( Y ′′ i , x j ) (cid:17) − x j (cid:16)X J − ( Y ′ i , x j ) Y ′′ i J , ( Y ′′ i , x j ) (cid:17) , and since ǫ ([ y i , x j ]) = 0, J − ( y j , x j ) − J − ( y j , x j ) + Q ( y j , x j ) = 0.Finally, (6) follows from (3) and (5), and (7) from (2) and (6). (cid:3) For every i, j , define p ij ∈ O ( U ) + as follows: p ij := X Q ( Y ′′ i , x j ) Y ′ i + X (cid:0) J − ( Y ′ i , x j ) J ( Y ′′ i , x j ) − J − ( Y ′ i , x j ) J ( Y ′′ i , x j ) (cid:1) Y ′′ i . Theorem 7.2.
The following hold: (1) O ( T ) and J O ( U ) J are Hopf subalgebras of J O ( G ) J . (2) The group X ( T ) acts on J O ( U ) J by automorphisms via x − j y i x j = y i + p ij . (3) J O ( G ) J ∼ = J O ( U ) J ⋊ C [ X ( T )] is a smash product algebra. (4) We have
Rep( J O ( G ) J ) = Rep( J O ( U ) J ) X ( T ) . (5) The Hopf algebra J O ( G ) J is an affine Noetherian domain withGelfand-Kirillov dimension dim( G ) .Proof. (1) and (2) follow from Lemma 7.1, (3) follows from (2), and(4)–(5) follow from (3) and Corollary 3.2. (cid:3) Remark 7.3.
Theorems 5.2, 7.2(4) imply a classification of finite di-mensional irreducible representations of J O ( G ) J . Example 7.4.
Let U be the Heisenberg group as in Example 6.1 (ex-cept, there it is denoted by G ). Let G := G m × U . Then G is a con-nected (non-unipotent) nilpotent algebraic group over C , and we have O ( G ) = C [ F ± , X, Y, V ], where F is a grouplike element and X, Y, V are as in Example 6.1. The Lie algebra g of G has basis f, a, b, c , where f := F ∂∂F , and a, b, c are as in Example 6.1. By [G, Theorem 5.3 &Proposition 5.4], the classical r -matrix r := f ∧ ( a + b ) for g correspondsto a Hopf 2-cocycle J for G . It is straightforward to verify that J O ( G ) J is generated as an algebra by F, X, Y, V , such that [
V, F ] = F ( Y − X ),or equivalently, F − V F = V + Y − X , and other pairs of generatorscommute. Thus, we have J O ( G ) J ∼ = C [ X, Y, V ] ⋊ C [ F ± ] as algebras,with nontrivial action. References [AEGN] E. Aljadeff, P. Etingof, S. Gelaki and D. Nikshych. On twisting of finite-dimensional Hopf algebras.
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