aa r X i v : . [ m a t h . QA ] J u l AFFINE COMMUTATIVE-BY-FINITE HOPF ALGEBRAS
K. A. BROWN AND M. COUTO
Abstract.
The objects of study in this paper are Hopf algebras H which are finitelygenerated algebras over an algebraically closed field and are extensions of a commuta-tive Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homolog-ical properties are recalled and classes of examples are listed. Bounds are obtained onthe dimensions of simple H -modules, and the structure of H is shown to be severelyconstrained when the finite dimensional extension is semisimple and cosemisimple. Introduction H over the algebraically closed field k is affine commutative-by-finite if it is a finitely generated module over a normal commutative finitely generatedHopf subalgebra A . In this context, to say that A is normal means that it is closedunder the adjoint actions of H , see § § H should be viewed as an extension of the affine commutativeHopf subalgebra A by the finite dimensional Hopf algebra H := H/A + H , where A + denotes the augmentation ideal of A .1.2. This paper is first of a series in which we treat the class of affine commutative-by-finite Hopf k -algebras as a laboratory for testing hypotheses about all Hopf algebras offinite Gelfand-Kirillov dimension.With that more general aim in mind, we first (in §
2) review and organise the knownproperties of commutative-by-finite Hopf algebras, including finiteness conditions, ho-mological properties and representation theory. Most of the results of this section arenot new, but are gathered from a number of sources, for example [6], [8], [41], [63],[75], [76]. Then, in §
3, we list and describe many important families of these algebras.The sources here include [15], [20], [29], [31], [39]. We also recall an example due toGelaki and Letzter [28] of a prime Hopf algebra, finite over its affine centre, which is not commutative-by-finite.The material in §§ § H on Maxspec( A ), focussing on the concept of an H - orbit of maximal ideals. In particular, the H - core m ( H ) of a maximal ideal m of A ,which features in Theorem 1.2 below, is defined here as the biggest H -invariant ideal of A contained in m . The (finite) set of maximal ideals of A which contain m ( H ) is - bydefinition - the H - orbit of m . § H on the nilradical and Mathematics Subject Classification.
Primary 16T05; Secondary 16T20, 16Rxx, 16Gxx.
Key words and phrases.
Hopf algebra, polynomial identity.The research of Ken Brown was partially supported by a grant from the Leverhulme Foundation,Emeritus Fellowship EM-2017-081. The PhD research of Miguel Couto was supported by a grant of thePortuguese Foundation for Science and Technology, SFRH/BD/102119/2014. minimal primes of A , and applies this to study the surprisingly strict relation between(semi)primeness of H and of A .Most of the new results are in §§ § H can be chosen to be semisimple and cosemisimple. Resultsof Etingof, Walton and Skryabin [25, 67] are crucial to the key message about this classof algebras, which is that the underlying noncommutativity is generated by the actionof a finite group. Theorem 7.2, the main result of §
7, is rather complex to state; thefollowing is the special case where H is assumed to be prime, avoiding many of thetechnicalities. Theorem 1.1 (Corollary 7.5) . Let H be a prime commutative-by-finite Hopf algebra withaffine commutative normal Hopf subalgebra A , such that H = H/A + H is semisimpleand cosemisimple. Then A is a domain, and after replacing A by a larger smoothcommutative affine domain D which is a normal left coideal subalgebra of H ,(1) H/D + H ∼ = k Γ for a finite group Γ whose order is a unit in k ;(2) the adjoint action of H on A factors through k Γ , and Γ acts faithfully on A viathe adjoint action;(3) There exists another group algebra factor k Λ of H , such that Λ acts faithfullyon D via the left adjoint action, and Γ is a factor of Λ .(4) Suppose in addition that H is pointed. Then H is a crossed product of D by k Γ ,that is, H ∼ = D σ k Γ for some cocycle σ . The results of § H with commutative normal Hopf subalgebra A , and a simple H -module V ,there exists an H -invariant Frobenius algebra factor of A , denoted by A/ m ( H ) , whichacts faithfully on V . The following consequence is a simplified special case of Theo-rems 6.1 and 7.7. Recall that if R is a prime affine noetherian algebra which is a finitemodule over its centre, then the PI-degree of R , denoted by PI . deg( R ), is the maximumdimension over k of the simple R -modules, [9, Theorem I.13.5, Lemma III.1.2]. Theorem 1.2.
Let H , A and V be as above, and assume that A is semiprime (as isthe case in characteristic 0, for example) and that H is prime.(1) There is an H -invariant ideal m ( H ) of A , an invariant of V , such that the Frobe-nius algebra A/ m ( H ) embeds in V as an A -module. Hence dim k ( A/ m ( H ) ) ≤ dim k ( V ) ≤ dim k ( H ) . (2) Suppose that H is semisimple and cosemisimple. Then, in the notation of The-orem 1.1, PI . deg( H ) = | Γ | . (3) Suppose that H is semisimple and cosemisimple and that H is pointed. Then(using again the notation of Theorem 1.1) there is a maximal ideal m of D anda positive integer ℓ , both depending on V , such that ℓ | Γ : C Γ ( m ) | = dim k ( V ) ≤ | Γ | , where C Γ ( m ) denotes the centraliser of m in Γ . I HOPF ALGEBRAS 3
Notation.
Throughout this paper k will denote an algebraically closed field, allvector spaces are over k unless stated otherwise, and all unadorned tensor products areover k . The Gelfand-Kirillov dimension of an algebra R will be denoted by GK dim( R ).For details, refer to [34]. Recall that, for an affine noetherian algebra R satisfying apolynomial identity, this dimension is always a non-negative integer, coinciding with theclassical Krull dimension defined in terms of the maximum length of a chain of primeideals, [34, Corollary 10.16]. The global dimension of R is denoted gl . dim( R ), andthe projective dimension of an R -module M by pr . dim( M ). For a Hopf algebra H weuse the usual notation of ∆ , ǫ for the coalgebra structure, with ∆( h ) = P h ⊗ h for h ∈ H , and we use S to denote its antipode. The augmentation ideal ker ǫ of H will bedenoted by H + . Given a Hopf surjection π : H → T , H is canonically a right (and left) T -comodule algebra with coaction ρ = ( id ⊗ π )∆. The subspace of right coinvariants { h ∈ H : ρ ( h ) = h ⊗ } will be denoted by either H co π or H co T . Unexplained Hopfalgebra terminology can be found in [48] or [57], for example.2. Basic properties
Definition and initial properties.
Recall [48, § K of aHopf algebra H is normal if it is invariant under the left and right adjoint actions of H ;that is, for all k ∈ K and h ∈ H , ad l ( h )( k ) = X h kS ( h ) ∈ K and ad r ( h )( k ) = X S ( h ) kh ∈ K. Definition 2.1.
A Hopf k -algebra H is commutative-by-finite if it is a finite (left orright) module over a commutative normal Hopf subalgebra A . Remark 2.2.
For an affine commutative-by-finite Hopf algebra H , it is enough torequire that A is normal on one side only. This follows from Lemma 4.11(1).A commutative-by-finite Hopf algebra H is an extension of a commutative Hopf al-gebra by a finite dimensional Hopf algebra. Making this obvious but fundamental ob-servation precise, we have the following basic facts, with corresponding notation whichwe shall retain henceforth. Theorem 2.3.
Let H be a commutative-by-finite Hopf algebra, finite over the normalcommutative Hopf subalgebra A with augmentation ideal A + .(1) The following are equivalent:(a) H is noetherian.(b) H is affine.(c) A is affine.(d) A is noetherian.(2) A + H = HA + is a Hopf ideal of H , and H := H/A + H is a finite dimensional quotient Hopf algebra of H .(3) The left (resp. right) adjoint action of H on A factors through H , so that A isa left (resp. right) H -module algebra.(4) H satisfies a polynomial identity.Assume in the rest of the theorem that H satisfies the equivalent conditions of (1).Denote by π : H −→ H the Hopf algebra surjection from H to H given by (2). K. A. BROWN AND M. COUTO (5) The antipode S of H is bijective.(6) GK dim( H ) = GK dim( A ) = K dim( H ) = K dim( A ) < ∞ . (7) Assume one of the following hypotheses:(i) char k = 0 ;(ii) A is semiprime;(iii) H is pointed;(iv) A is central in H .Then:(a) A ⊆ H is a faithfully flat H -Galois extension;(b) A equals the right and the left H -coinvariants of the H -comodule H ; thatis, H co π = co π H = A ; (c) H is a finitely generated projective generator as left and right A -module;(d) A is a left (resp. right) A -module direct summand of H .Proof. (1) (c) ⇔ (d): Molnar’s theorem [47] ensures that (d) implies (c), and the converseis Hilbert’s Basis Theorem.(d) ⇔ (a): That (d) implies (a) is clear. The converse follows from [27].(b) ⇔ (c): That (c) ⇒ (b) is trivial. The converse follows from a generalized version ofthe Artin-Tate lemma attributed to Small, which states that, if A ⊆ H is any extensionof k -algebras where H is affine and is a finitely-generated left module over a commutativesubalgebra A , then A is affine. A proof can be found at [59, Lemma 1.3].(2) Normality of A ensures that A + H is a Hopf ideal by [48, Lemma 3.4.2(1)].(3) This is clear.(4) Every k -algebra which is a finite module over a commutative subalgebra satisfies aPI by [46, Corollary 13.1.13(iii)].(5) Every affine noetherian PI Hopf algebra has bijective antipode by [64, Corollary 2].(6) The fact that the Gelfand-Kirillov and Krull dimensions of H and A are finite andcoincide follows from [34, Proposition 5.5 and Corollary 10.16].(7) (a) Hypothesis (i) is a particular case of (ii) by [48, Corollary 9.2.11]. Assume (ii).Then A has finite global dimension by [74, 11.6, 11.7]. Then, by [76, Theorem 0.3],together with (6), H is a projective left and right A -module. A flat extension of Hopfalgebras with bijective antipodes is faithfully flat, [45, Corollary 2.9]. Together with (5),this proves right and left faithful flatness. If H is pointed (iii) or A is central in H (iv),then faithful flatness follows from [72, Theorem 3.2] and [61, Theorem 3.3] respectively.The H -Galois property follows from faithful flatness, as is shown in the proof of [48,Proposition 3.4.3].(b) is immediate from (a) and [48, Proposition 3.4.3].(c) follows from (a) by [45, Corollary 2.9].(d) The left A -module H/A is in the category A M H , in the notation of [45]. Hence H/A is left A -projective by (c) and [45, Corollary 2.9], so the exact sequence0 −→ A −→ H −→ H/A −→ A -modules splits, as required. The argument on the right is identical. (cid:3) I HOPF ALGEBRAS 5
Remarks 2.4.
Keep the notation and hypotheses of Theorem 2.3.(1) Parts (1), (4), (5) and (6) of the theorem are valid (with the same proofs) withoutthe hypothesis that A is normal in H .(2) (Radford [56]) In general, H is not a free A -module. For example, let H = O ( SL ( k )). This commutative Hopf algebra is a finite module over the Hopf subal-gebra A generated by the monomials of even degree, but it is not a free A -module.(3) Notwithstanding (2), H is A -free when H is pointed [54] or when A contains thecoradical of H [53, Corollary 2.3].(4) A further important setting where H is A -free is when H decomposes as a crossedproduct H = A σ H . By a celebrated result of Doi and Takeuchi [23], this happens ifand only if there is a cleaving map γ : H −→ H - that is, γ is a convolution invertibleright H -comodule map. Moreover, when the extension A ⊂ H is H -Galois, such acleaving map exists if and only if A ⊂ H has the normal basis property . For details, seefor example [48, Propositions 7.2.3, 7.2.7, Theorem 8.2.4].Such a crossed product decomposition of H is guaranteed when H is pointed or whenits coradical is contained in AG ( H ) by [60, Corollary 4.3]. In particular, in these casesall conclusions of Theorem 2.3(7) hold.2.2. Finiteness over the centre.
The following remains at present a very naturalopen question.
Question 2.5. [6, Question E], [7, Question C(i)] Is every affine or noetherian Hopfalgebra satisfying a polynomial identity a finite module over its centre?However, for the special case of affine commutative-by-finite Hopf algebras, the answerto Question 2.5 is “yes”. This follows easily from an important result of Skryabin [63,Proposition 2.7] on integrality over invariants, as we now show.
Definition 2.6.
Let T be a Hopf algebra and R a left T -module algebra. The subalgebraof T - invariants of R is R T = { r ∈ R : t · r = ǫ ( t ) r, ∀ t ∈ T } . We give some details here on the proof of this result by Skryabin, since this statementis not explicitly enunciated in [63].
Theorem 2.7. [63, Proposition 2.7]
Let T be a finite-dimensional Hopf algebra and A an affine commutative left T -module algebra. Then, A is a finitely generated moduleover A T .Proof. On one hand, if char k >
0, then clearly A is Z -torsion, so [63, Proposition 2.7(b)]applies to show that A is integral over A T . If on the other hand char k = 0, then A issemiprime by [48, Corollary 9.2.11], so that, in the terminology of [63], A is T -reduced,and again [63, Theorem 2.5, Proposition 2.7(a)] give A integral over A T . Since A isaffine, it is a finitely generated A T -module. (cid:3) Corollary 2.8.
Let H be an affine commutative-by-finite Hopf k -algebra, finite over thenormal commutative Hopf subalgebra A . Then, H is a finitely-generated module over itscentre Z ( H ) , which is affine.Proof. Note first that Z ( H ) ∩ A = A H . For it is clear that Z ( H ) ∩ A ⊆ A H . Conversely,if a ∈ A H and h ∈ H then ha = X h aǫ ( h ) = X h aS ( h ) h = X ǫ ( h ) ah = ah. K. A. BROWN AND M. COUTO
By Theorem 2.7, A is a finite-module over A H . Since H is a finite A -module, it is finite Z ( H )-module. Lastly, Z ( H ) is affine by the Artin-Tate lemma, [46, Lemma 13.9.10]. (cid:3) Homological properties and consequences.
The class of affine commutative-by-finite Hopf k -algebras includes as subclasses the affine commutative Hopf k -algebrasand the finite dimensional Hopf k -algebras. Both these classes exhibit important ho-mological properties - affine commutative Hopf algebras are Gorenstein, [6, 2.3, Step1], and in characteristic 0 they have finite global dimension - that is, they are regular [74, 11.4, 11.6, 11.7]; and finite dimensional Hopf algebras are Frobenius [48, 2.1.3]. Wereview here how these features partially extend to the commutative-by-finite setting.The definitions, for a Hopf k -algebra H , of the Auslander Gorenstein, Auslanderregular, AS-Gorenstein, AS-regular and GK-Cohen Macaulay properties can be found,for example, in [76, 3.1, 3.6]. The first and third [resp. second and fourth] of theseproperties are strengthened versions of the requirement to have finite injective [resp.finite global] dimension.It’s immediate from the definitions that a Hopf algebra which is both AS-Gorensteinand GK-Cohen Macaulay must have its injective dimension equal to its GK-dimension.By results of Wu and Zhang, [76, Proposition 3.7] and [14, Lemma 6.1, § injective homogeneity , see [12], [13]. Theorem 2.9.
Let H be an affine commutative-by-finite Hopf algebra, finite over thecommutative normal Hopf subalgebra A . Let GK dim( H ) = d .(1) H is AS-Gorenstein and Auslander Gorenstein, of injective dimension d .(2) H is GK-Cohen Macaulay.(3) H is left and right GK-pure; that is, every non-zero left or right ideal of H hasGK-dimension d .(4) H is injectively homogeneous. As a consequence, H is a Cohen-Macaulay Z ( H ) -module.(5) H , Z ( H ) and A H each have artinian classical rings of fractions, Q ( H ) , Q ( Z ( H )) and Q ( A H ) .(6) The regular elements Z of Z ( H ) and A of A H are also non-zero divisors in H ,and (2.1) Q ( H ) = H [ Z ] − = H [ A ] − . (7) Q ( H ) is quasi-Frobenius.Proof. Parts (1) and (2) are discussed above, and (3) is an immediate consequence of(2), since a non-zero left or right ideal of H has homological grade 0, by definition ofthe latter.By Corollary 2.8 H is finite over Z ( H ). That (4) is a consequence of (1) and (2) foralgebras which are finite over their centres is shown in [13, Theorems 5.3 and 4.8].For (5), it is a standard consequence of Small’s theorem [46, 4.1.4] that GK-purenoetherian algebras have artinian rings of fractions [46, 6.8.16]. The claim that Q ( H )exists and is artinian is an immediate consequence of this and (3). The arguments for Z ( H ) and for A H are identical; we deal here with Z ( H ). As with the proof for H , it I HOPF ALGEBRAS 7 is enough to prove that Z ( H ) is GK-pure. By Corollary 2.8 H is a finitely generated Z ( H )-module. Let 0 = I ⊳ Z ( H ), so GK dim H ( HI ) = d by (3). On the other hand(2.2) GK dim Z ( H ) ( HI ) = GK dim H ( HI ) , by [34, Corollary 5.4], and(2.3) GK dim Z ( H ) ( HI ) = GK dim Z ( H ) ( I ) , since I ⊆ HI and HI is a homomorphic image of a finite direct sum of copies of I as Z ( H )-module, so that [34, Proposition 5.1(a),(b)] applies. Therefore GK dim Z ( H ) ( I ) = d as required.(6) Let z ∈ Z . Then GK dim( Z ( H ) /Z ( H ) z ) < d by [34, Proposition 3.15]. Therefore,if zh = 0 for some h ∈ H , the Z ( H )-module Z ( H ) h has GK-dimension strictly less than d , by [34, Proposition 5.1(c)]. As in the proof of (5), GK dim H ( Hh ) < d , so h = 0 by(3). The same argument works for elements of A . The last two partial quotient rings in(2.1) therefore exist; since they are clearly artinian, they must equal Q ( H ).(7) That Q ( H ) is quasi-Frobenius is proved in [76, Theorem 0.2(2)]. (cid:3) Smooth commutative-by-finite Hopf algebras.
An affine commutative Hopfalgebra has finite global dimension if and only if it has no non-zero nilpotent elements[74, 11.6, 11.7]; while a finite dimensional Hopf algebra has finite global dimension ifand only if it is semisimple, if and only if ǫ ( R ) = 0, for a left or right integral R , [48,2.2.1]. It’s thus natural to look for an easily checked necessary and sufficient criterion foran affine commutative-by-finite Hopf algebra to have finite global dimension. Examplessuggest there may be no such simple condition, but sufficient conditions for smoothnessare not hard to obtain, as follows. The hypothesis on H as A -module in (3) is presumablyredundant - for situations where it is known to be satisfied, see Theorem 2.3(7). Proposition 2.10.
Let H be an affine commutative-by-finite Hopf k -algebra, with nor-mal commutative Hopf subalgebra A such that H is a finite A -module.(1) If A has no nonzero nilpotent elements and H is semisimple, then H has finiteglobal dimension.(2) If k has characteristic 0 and H is semisimple, then H has finite global dimension.(3) If H has finite global dimension and H is A -flat then A has finite global dimen-sion, so A has no nonzero nilpotent elements.Proof. (1) Since A is semiprime and affine, by Theorem 2.3(1), it has finite global dimen-sion by [74, 11.6, 11.7]. By Theorem 2.3(7) H A is faithfully flat, so the finite resolution byprojective A -modules of the trivial A -module k induces a finite projective resolution by H -modules of H ⊗ A k ∼ = H . The trivial H -module is a direct summand of H by semisim-plicity of H , hence pr . dim H ( k ) ≤ pr . dim A ( k ) < ∞ . But gl . dim( H ) = pr . dim H ( k ) by[40, Section 2.4], so the result follows.(2) This follows from (1), since A is always reduced in characteristic 0 [74, 11.4].(3) If gl . dim( H ) < ∞ , then the finite projective resolution of the trivial H -module isalso a projective A -resolution, as A H is projective by Theorem 2.3(7). So gl . dim( A ) < ∞ by [40, Section 2.4]. (cid:3) Remarks 2.11. (1) As recalled before the proposition, a semiprime affine commutativeHopf algebra H has finite global dimension. But this fails abysmally to generalise tothe commutative-by-finite setting. For example, the algebras B = B ( n, p , . . . , p s , q )constructed by Goodearl and Zhang in [29] and discussed in § K. A. BROWN AND M. COUTO commutative-by-finite Hopf algebras and are domains of GK-dimension 2, but haveinfinite global dimension.(2) The converses of Propositions 2.10(1) and (2) are false even when A is centralor H is cocommutative. For the case when A is central one can take the quantisedenveloping algebra U ǫ ( g ) of any simple Lie algebra g at a root of unity ǫ , see § G := h x, y : x − y x = y − , y − x y = x − i discussed in [51, Lemma 13.3.3]. As explained by Passman, G has a normal subgroup N which is free abelian of rank 3, with G/N a Klein 4-group. Moreover, ( ∗ ) G does not have any normal abelian subgroup W of finite index with | G : W | prime to 2. Take a field k of characteristic 2 and let H = kG . Then H is an affine commutative-by-finite domainby [51, Theorem 13.4.1], and gl . dim( H ) = 3 by Serre’s theorem on finite extensions, [51,Theorem 10.3.13]. The normal Hopf subalgebras of H are just the group algebras kT with T a normal subgroup of G . Thus there is no way to present H as a finite extensionof a commutative normal Hopf subalgebra A , with H semisimple, in view of property( ∗ ) and Maschke’s theorem.We now gather results from the literature to show that smooth affine commutative-by-finite Hopf algebras share many of the attractive properties of commutative noetherianrings of finite global dimension. For the definition of a homologically homogeneous (hom.hom.) ring, see [12] or [13]. Theorem 2.12.
Let H be an affine commutative-by-finite Hopf k -algebra of finite globaldimension d .(1) H is Auslander regular, AS-regular and GK-Cohen Macaulay of GK dim( H ) = d .(2) H is homologically homogeneous.(3) H is a finite direct sum of prime rings, H = t M ℓ =1 H ℓ with H ℓ a prime hom. hom. algebra of dimension d for all ℓ .(4) Z ( H ) = L tℓ =1 Z ( H ℓ ) , where Z ( H ℓ ) is an affine integrally closed domain of GK-dimension d for all ℓ .Proof. (1) This follows from Theorem 2.9(1),(2) and the definitions of these concepts.(2), (3) Since H is a noetherian PI ring, this follows from (1) and [70, Theorem 5.4and discussion on p. 1013], or alternatively from (1), Corollary 2.8, [13, Theorem 4.8and Corollary 5.4] and [11, Theorem 5.3].(4) From (3), Z ( H ) = L tℓ =1 Z ( H ℓ ), with each Z ( H ℓ ) an affine domain, since H ℓ isprime and Z ( H ℓ ) is a factor of the affine algebra Z ( H ). Since H is a finite Z ( H )-moduleby (2) or Corollary 2.8, each H ℓ is a finite module over the image Z ( H ℓ ) of Z ( H ) in H ℓ .GK dim( Z ( H ℓ )) = d by (3) and [34, Proposition 5.5], since H ℓ is a finite Z ( H ℓ )-module.Finally, the summands H ℓ of H are all hom. hom. by (3), so Z ( H ℓ ) is integrally closedby [11, 6.1]. (cid:3) Involutory commutative-by-finite Hopf algebras.
Recall that a Hopf algebra H is involutory if the antipode S of H satisfies S = id H . Commutative or cocommu-tative Hopf algebras are involutory [48, 1.5.12]; and when k has characteristic 0 and H is finite dimensional, H is involutory if and only if it is semisimple if and only if it is I HOPF ALGEBRAS 9 cosemisimple, [35, 2.6] and [36, Theorems 3 and 4]. The following results are effectivelyspecial cases of results from [75].
Proposition 2.13.
Let H be an involutory affine commutative-by-finite Hopf k -algebra.(1) If char k = 0 then H is semisimple and gl . dim( H ) < ∞ , so Theorem 2.12 appliesto H .(2) Suppose that char k = p > and that either(i) H is semiprime and p ∤ PI − degree( H/P ) for some minimal prime ideal P of H ; or(ii) A is semiprime and p ∤ dim k ( H ) .Then gl . dim( H ) < ∞ , so the conclusions of Theorem 2.12 apply to H .Proof. (1) When char k = 0, H is semisimple by the result of Larson and Radfordrecalled before the proposition, so the result follows from Proposition 2.10(2).(2) If (i) holds then this is a special case of [75, Proposition 3.5]. Suppose that (ii)holds. Then pr . dim H ( H ) < ∞ by the argument used in the proof of Proposition 2.10(1),so the conclusion follows from [75, Lemma 1.6(3)]. (cid:3) Examples of affine commutative-by-finite Hopf algebras
Beyond the commutative and the finite dimensional Hopf algebras, here are someother families of examples, and - in § Enveloping algebras of Lie algebras in positive characteristic.
Assume inthis paragraph that k has positive characteristic p . The universal enveloping algebra U ( g ) of a finite-dimensional Lie algebra g is a noetherian Hopf algebra and in positivecharacteristic it is finitely-generated over its centre [31]. When g = L mi =1 kx i is re-stricted, with restriction map x x [ p ] , U ( g ) is a free module of finite rank over thecentral Hopf subalgebra A = k h x pi − x [ p ] i : 1 ≤ i ≤ m i , which is a polynomial algebraon these primitive generators ([30, Section 2.3] or [9, Theorem I.13.2(8)]). The Hopfquotient H is the restricted enveloping algebra of g , usually denoted by u [ p ] ( g ). Moregenerally, for any finite-dimensional k -Lie algebra g = L mi =1 kx i , U ( g ) is a free mod-ule of finite rank over a central Hopf subalgebra A = k h y , . . . , y m i , where each y i isa p -polynomial in x i , [31, Proposition 2]. In fact, A is a polynomial algebra on theseprimitive generators. These algebras U ( g ) are involutory, being cocommutative, and aresmooth domains - by [16, Theorem XIII.8.2],gl . dim( U ( g )) = dim g = GK dim( U ( g )) . Quantized enveloping algebras and quantized coordinate rings at a rootof unity.
Quantized enveloping algebras U q ( g ) of semisimple finite-dimensional Lie al-gebras g are noetherian Hopf algebras, [20, Section 9.1], [17, Section 9.1A]. When q = ǫ is a primitive ℓ th root of unity, U ǫ ( g ) is a free module of rank ℓ dim g over a central Hopfsubalgebra Z , [20, Corollary and Theorem 19.1], [17, Propositions 9.2.7 and 9.2.11].Its finite-dimensional Hopf quotient H = U ǫ ( g ) /Z +0 U ǫ ( g ) is the restricted quantizedenveloping algebra denoted by u ǫ ( g ).Quantized coordinate rings O q ( G ) of connected, simply connected, semisimple Liegroups G are noetherian Hopf algebras, [19, Sections 4.1 and 6.1]. If q = ǫ is a primitive ℓ th root of unity, O ǫ ( G ) contains a central Hopf subalgebra isomorphic to O ( G ), [19,Proposition 6.4], [9, Theorem III.7.2], and O ǫ ( G ) is a free O ( G )-module of rank ℓ dim( G ) [10]. Better, the extension O ( G ) ⊆ O ǫ ( G ) is cleft in the sense of Remark 2.4(4) (see [2, Remark 2.18(b)]). The finite-dimensional Hopf quotient H = O ǫ ( G ) / O ( G ) + O ǫ ( G )is the restricted quantized coordinate ring, sometimes denoted by o ǫ ( G ).The algebras in these families are thus commutative-by-finite, and are smooth ofglobal dimensions dim( g ) and dim( G ) respectively, [16, Theorem XIII.8.2], [8, Theorem2.8].3.3. Group algebras of finitely-generated abelian-by-finite groups.
Let G be afinitely generated group with an abelian normal subgroup N of finite index. Then N isfinitely generated, so G is polycyclic-by-finite, and hence kG is an affine cocommutative(and so involutory) Hopf algebra, noetherian by [51, Corollary 10.2.8]. Since kG is afinite kN -module, it is commutative-by-finite. Further, kG is cleft, (see Remark 2.4(4)),decomposing as crossed products kG ∼ = kN σ k ( G/N ) [48, Example 7.1.6]. Letting d denote the torsion-free rank of N , inj . dim( kG ) = d. By [51, Theorem 10.3.13], kG has finite global dimension (necessarily d ) if and only ifchar k = 0, or char k = p > G has no elements of order p .3.4. Prime regular affine Hopf algebras of Gelfand-Kirillov dimension 1.
TheseHopf k -algebras were completely classified when k is an algebraically closed field ofcharacteristic 0 by Wu, Liu and Ding in [22], building on [15] and [41]. By a fundamentalresult of Small, Stafford and Warfield [69], a semiprime affine algebra of GK-dimensionone is a finite module over its centre. But in fact more is true for these Hopf algebras -they are all commutative-by-finite, as can be checked on a case-by-case basis.With k algebraically closed of characteristic 0, there are 2 finite families and 3 infinitefamilies, as follows.(I) The commutative algebras k [ x ] and k [ x ± ].(II) A single cocommutative noncommutative example, the group algebra H = kD of the infinite dihedral group D = h a, b : a = 1 , aba = b − i . Here, H is finiteover the normal commutative Hopf subalgebra A = k h b i .(III) The infinite dimensional Taft algebras T ( n, t, q ) = k h g, x : g n = 1 , xg = qgx i ,where q is a primitive n th root of 1 in k , with g group-like and x (1 , g t )-primitive.The commutative normal Hopf subalgebra A is k [ x n ′ ], where n ′ = n/ gcd( n, t ).(IV) The generalised Liu algebras B ( n, w, q ), where n and w are positive integers and q is a primitive n th root of 1. We define B ( n, w, q ) := k h x ± , g ± , y : x central , yg = qgy, g n = x w = 1 − y n i , with x and g group-like and y (1 , g )-primitive. Clearly, A := k [ x ± ] is a centralHopf subalgebra over which B ( n, w, q ) is free of rank n .(V) Let m and d be positive integers with (1 + m ) d even, and let q be a primitive2 m th root of 1 in k . The Hopf algebras D ( m, d, q ) are defined in [22, Section 4.1]. D ( m, d, q ) is finitely generated over the normal commutative Hopf subalgebra A = k [ x ± ], [22, (4.7)].The above algebras are all free over their respective normal commutative Hopf subal-gebras. Families (I)-(IV) are pointed and decompose as crossed products H ∼ = A σ H ;but D ( m, d, q ) is not pointed, [22, Proposition 4.9].Given that these algebras are all regular, Theorem 2.12 applies to them - in particularthey are all hereditary, by (1) of that result. The following questions are now obvious: I HOPF ALGEBRAS 11
Question 3.1. (i) What is the classification corresponding to the above when k isalgebraically closed of positive characteristic p ?(ii) Can the classification be completed in characteristic 0 if the hypothesis of regu-larity is omitted?Regarding (ii), considerable progress is made in [39], including the construction ofmany examples. However, all these new examples are commutative-by-finite, as is ex-plicitly noted in [39]. Indeed, Liu conjectures [39, Conjecture 7.19] that, in characteristic0, every prime affine Hopf k -algebra of GK-dimension 1 is commutative-by-finite.3.5. Noetherian PI Hopf domains of Gelfand-Kirillov dimension two.
Continueto assume that k is algebraically closed of characteristic 0. Let H be a noetherian Hopf k -algebra domain with GK dim( H ) = 2. These were classified in [29] under the additionalassumption that Ext H ( H k, H k ) = 0 . ( ♯ )By [29, Proposition 3.8(c)], ( ♯ ) is equivalent to H having an infinite dimensional com-mutative Hopf factor; that is, to the quantum group H containing a one-dimensionalclassical subgroup. There are 5 classes of such Hopf algebras, as follows.(I) The group algebras over k of Z × Z and Z ⋊ Z .(II) The enveloping algebras of the two 2-dimensional k -Lie algebras.(III) Hopf algebras A ( n, q ), for n ∈ Z and q ∈ k × , as algebras the localised quantumplane k h x ± , y : xy = qyx i , with x group-like and y (1 , x n )-primitive.(IV) Hopf algebras B ( n, p , . . . , p s , q ), where s ≥ n, p , . . . , p s are positive integerswith p | n and { p i : i ≥ } strictly increasing and pairwise relatively prime, and q is a primitive ℓ th root of 1 where ℓ = ( n/p ) p . . . p s . Then B ( n, p , . . . , p s , q )is the subalgebra of the localised quantum plane from (III) generated by x ± with { y m i : 1 ≤ i ≤ s } , where m i := Π j = i p j . This supports a Hopf structurewith x group-like and y m i being (1 , x m i n )-primitive.(V) Hopf algebras C ( n ) for each integer n , n ≥
2, where C ( n ) := k [ y ± ][ x ; ( y n − y ) ddy ],with y group-like and x being ( y n − , • the group algebras in (I). The group algebra of Z ⋊ Z is module-finite over thenormal commutative Hopf subalgebra Z × Z . • the enveloping algebra of the 2-dimensional abelian Lie algebra in (II). • the algebras A ( n, q ) from (III) where q is a root of unity. These algebras aremodule-finite over the normal commutative Hopf subalgebra A = k h ( x l ) ± , y l ′ i ,where q is a primitive l th root of 1 and l ′ = l/ gcd( n, l ). • the algebras in (IV). It is not hard to see that A := k h ( y m i ) p i , x ± ℓ i is a normalcommutative Hopf subalgebra over which B ( n, p , . . . , p s , q ) is a finite module.As was shown in [29, Propositions 1.6 and 0.2a], all the families have global dimension2, except for (IV), whose members have infinite global dimension, being free over thecoordinate ring k h y m i : 1 ≤ i ≤ s i of a singular curve.In [73], it was shown that not all noetherian Hopf k -algebra domains of GK-dimension2 satisfy ( ♯ ). More precisely, a key discovery of [73] was the existence of:(VI) an infinite family of noetherian Hopf algebra domains of GK-dimension 2, withExt H ( H k, H k ) = 0 for all members of the family. The algebras in (VI) are commutative-by-finite - by [73, Theorem 2.7], each of them isa finite-rank free module over a central Hopf subalgebra which is the coordinate ring ofa 2-dimensional solvable group. Almost all of them have infinite global dimension, someof them being finite-rank free modules over an algebra in (IV).It is conjectured in [73, Introduction] that, when k has characteristic 0, the families(I)-(VI) constitute all the affine Hopf k -algebra domains of GK-dimension 2, at least inthe pointed case. Namely, the authors ask: Question 3.2. (Wang, Zhang, Zhuang, [73]) Let k be algebraically closed of charac-teristic 0, and let H be an affine Hopf k -algebra domain of GK-dimension two. If H ispointed, does it belong to the families (I)-(VI)?At present every known affine Hopf k -algebra domain of GK-dimension two is gener-ated by group-like and skew primitive elements. The answer to Question 3.2 is affirma-tive for Hopf algebras so generated, [73, Corollary 0.2].3.6. Affine PI but not commutative-by-finite example.
Gelaki and Letzter gavean example [28] of a prime noetherian Hopf k -algebra U of Gelfand-Kirillov dimension2 which is a finite module over its centre, but which is not commutative-by-finite. Theirexample is the bosonisation of the enveloping algebra of the Lie superalgebra pl(1 , U contains a nonzero element u forming part of a PBW basis of U , with u = 0. Thus U is not a domain, so does notfeature in the list in § U is a free k h u i -module, so thatpr . dim U ( k ) ≥ pr . dim k h u i ( k ) = ∞ , forcing gl . dim( U ) = ∞ . Thus, the following question remains open at present: Question 3.3.
Is every affine noetherian regular PI Hopf algebra commutative-by-finite? 4. H -Stability and orbital semisimplicity We study here the action of H on the ideals of A . It is convenient in § § § Hopf orbits in
Maxspec( A ) .Definition 4.1. Let T be a Hopf algebra, R a left T -module algebra.(1) A subspace V of R is T - stable if t · v ∈ V for all t ∈ T and v ∈ V .(2) The T - core of an ideal I of R is the ideal I ( T ) = { x ∈ I : t · x ∈ I, ∀ t ∈ T } . Note that the T -core I ( T ) of I will in general be larger than the subspace R T ∩ I of T -invariants of I - hence the use of brackets in our notation. The following lemma iseasily checked. Lemma 4.2.
With the above notation, I ( T ) is the largest T -stable subspace of R con-tained in I . Let A be a commutative left T -module algebra. The notion of the T -core of an idealof A leads to an equivalence relation on the prime spectrum of A , as discussed in a moregeneral setting by Skryabin in [66]. However we limit attention here to the case of themaximal ideals of a commutative T -module algebra A , although everything could bedone under weaker hypotheses, as in [66]. I HOPF ALGEBRAS 13
Definition 4.3.
Let T be a Hopf algebra and A a commutative left T -module algebra.(1) Define an equivalence relation ∼ ( T ) on Maxspec( A ) as follows: for m , m ′ ∈ Maxspec( A ), m ∼ ( T ) m ′ if and only if m ( T ) = m ′ ( T ) . (2) For each m ∈ Maxspec( A ), its T - orbit is the set of maximal ideals with the same T -core, O m = { m ′ ∈ Maxspec( A ) : m ′ ( T ) = m ( T ) } . The following key result is essentially due to Skryabin [66, Theorems 1.1, 1.3]. Foraffine commutative-by-finite Hopf algebras, we shall in due course (Theorem 6.1) ob-tain an upper bound for the dimensions of the finite dimensional commutative algebras A/ m ( H ) appearing below, and hence for the cardinalities of the orbits O m . Proposition 4.4.
Let T be a finite-dimensional Hopf algebra, A an affine commutativeleft T -module algebra and m ∈ Maxspec( A ) .(1) A/ m ( T ) is a T -simple algebra. That is, the only T -stable ideals of A/ m ( T ) arethe trivial ones.(2) A/ m ( T ) is a Frobenius algebra. In particular, A/ m ( T ) is finite dimensional.(3) O m = { m ′ ∈ Maxspec( A ) : m ( T ) ⊆ m ′ } .(4) O m is finite.Proof. (1) This is a special case of [66, Proposition 3.5].(2) Since A/ m = k , then A T / ( m ∩ A T ) = k . Thus, by Theorem 2.7, A/ ( m ∩ A T ) A is afinite dimensional algebra, and therefore so is its factor algebra A/ m ( T ) . In particular,being artinian, it equals its classical ring of quotients:(4.4) A/ m ( T ) = Q ( A/ m ( T ) ) . Moreover, by (1), A/ m ( T ) is T -semiprime. Thus, by [66, Theorem 1.3], Q ( A/ m ( T ) ) isquasi-Frobenius. The result follows from this and (4.4), noting that a commutativequasi-Frobenius k -algebra is actually Frobenius.(3) If m ′ ∈ O m , m ( T ) = m ′ ( T ) ⊆ m ′ and so O m ⊆ { m ′ ∈ Maxspec( A ) : m ( T ) ⊆ m ′ } . For thereverse inclusion, suppose some m ′ ∈ Maxspec( A ) contains m ( T ) , so that m ( T ) ⊆ m ′ ( T ) .Hence, by (1), m ′ ( T ) = m ( T ) ; that is, m ′ ∈ O m .(4) is an immediate consequence of (2) and (3). (cid:3) Orbital semisimplicity.
When the Hopf algebra T of Definition 4.3 is a groupalgebra kG of a finite group G , the setting is the familiar one of classical invarianttheory. In particular, G acts by k -algebra automorphisms on A and for m ∈ Maxspec( A ), O m = { m g : g ∈ G } and m ( kG ) = T { m g : g ∈ G } , so that A/ m ( kG ) is a finite direct sumof copies of k . We isolate this desirable state of affairs in the following definition. Definition 4.5.
Let T be a finite dimensional Hopf algebra and A an affine commutativeleft T -module algebra. Then, A is T - orbitally semisimple if A/ m ( T ) is semisimple forevery m ∈ Maxspec( A ). The prefix T will be omitted when this is clear from the context.In view of Proposition 4.4(3), A is T -orbitally semisimple if and only if(4.5) m ( T ) = \ m ′ ∈O m m ′ for every m ∈ Maxspec( A ). At first glance the following result of Montgomery and Schneider, [49, Theorem 3.7,Corollary 3.9], (generalising an earlier result of Chin [18, Lemma 2.2] and specialisedto the present setting), might suggest that orbital semisimplicity always holds, at leastwhen T is pointed. However this is not the case, as the example which follows illustrates. Proposition 4.6.
Let T be a finite-dimensional Hopf algebra and A an affine commu-tative T -module algebra. Let T ⊂ T ⊂ . . . ⊂ T m = T be the coradical filtration of T ,and let h T i be the Hopf subalgebra of T generated by T . Then,(1) For an ideal I of A , ( I ( h T i ) ) m +1 ⊆ I ( T ) . (2) For all m , m ′ ∈ Maxspec( A ) , m ( T ) = m ′ ( T ) if and only if m ( h T i ) = m ′ ( h T i ) .(3) Suppose T is pointed, so that T = h T i . For any m , m ′ ∈ Maxspec( A ) , m ( T ) = m ′ ( T ) if and only if m ′ = g · m for some group-like element g of T . Example 4.7.
Fix an integer n , n ≥
2, and let q be a primitive n th root of 1 in k . Let T be the n -dimensional Taft algebra, T := k h g, x : g n = 1 , x n = 0 , xg = qgx i , with g group-like and x (1 , g )-primitive. Let A be the polynomial algebra k [ u, v ].As shown by Allman, [1, § A is a left T -module algebra with the action defined by g · u = u, g · v = qv, x · u = 0 , x · v = u. This action is not orbitally semisimple. Indeed, one can easily check with the aid ofProposition 4.6(3) that, for a ∈ k × and m := h u − a, v i , m ( T ) = h u − a, v n i . In the positive direction, we have the following result.
Theorem 4.8.
Let T be a finite dimensional Hopf algebra and A an affine commutative T -module algebra. Then, A is T -orbitally semisimple in each of the following cases:(1) the action is trivial;(2) the action factors through a group;(3) T is cosemisimple;(4) T is involutory and char k = 0 or char k = p > dim k ( A/ m ( T ) ) for all m ∈ Maxspec( A ) .Proof. (1) and (2) are clear.(3) Let m ∈ Maxspec( A ), so that A/ m ( T ) is a T -module algebra, and is T -simple byProposition 4.4(1). Thus A/ m ( T ) is semisimple by [68, Theorem 0.5(ii)].(4) If k has characteristic 0 this is simply a restatement of (3), since in this case T isinvolutory if and only if it is cosemisimple if and only if it is semisimple, [35, 2.6] and[36, Theorems 3 and 4]. Suppose that char k = p > m ∈ Maxspec( A ). Ifdim k ( A/ m ( T ) ) < p then its Jacobson radical is T -stable, by [37, Theorem]. By the T -simplicity of A/ m ( T ) ensured by Proposition 4.4(1), this forces A/ m ( T ) to be semisimple. (cid:3) Remark 4.9.
There is a considerable overlap between cases (2), (3) and (4) of theabove result. As noted in the proof, being semisimple, cosemisimple and involutoryare equivalent conditions on T when k has characteristic 0. Moreover, when A is acommutative T -module domain and T is cosemisimple, Skryabin, in [67, Theorem 2],showed that the action factors through a group, extending earlier work of Etingof andWalton [25]. I HOPF ALGEBRAS 15 H -stability. The equivalence relation of Definition 4.3 was studied by Montgomeryand Schneider [49], before Skryabin [66], in the special setting of a faithfully flat T -Galois extension R ⊂ S . In fact, the equivalence relation as defined on Spec( R ) in [49,Definition 2.3(2)] is different from the one given above, defining instead an ideal I of R to be T -stable if IS = SI , and then using this to define an equivalence relation. Thenext lemma examines the relation between these two notions of stability. Lemma 4.10.
Let U be a Hopf algebra with bijective antipode and W a normal Hopfsubalgebra of U . Write U := U/W + U . Let I be an ideal of W .(1) If ad l ( u )( x ) ∈ I for all u ∈ U, x ∈ I , then U I ⊆ IU , and IU is an ideal of U .(2) If ad r ( u )( x ) ∈ I for all u ∈ U, x ∈ I , then IU ⊆ U I , and
U I is an ideal of U .If the extension W ⊆ U is faithfully flat U -Galois, the following are equivalent:(i) ad l ( u )( x ) ∈ I , for all u ∈ U, x ∈ I .(ii) ad r ( u )( x ) ∈ I , for all u ∈ U, x ∈ I .(iii) U I = IU .Proof. If I is invariant under left adjoint action, then for all x ∈ I, u ∈ U we have ux = P u xS ( u ) u = P ad l ( u )( x ) u ∈ IU , proving (1). (2) is proved similarly.Suppose in addition that the extension W ⊆ U is faithfully flat U -Galois. This impliesthat, if (i) or (ii) holds, then U I = IU by [49, Remark 1.2(ii)]. For the converse, supposethat IU = U I , and let x ∈ I and u ∈ U . Then ad l ( u )( x ) = X u xS ( u ) ∈ IU ∩ W, since W is normal. But the extension W ⊆ U being faithfully flat gives IU ∩ W = I .This proves ( iii ) = ⇒ ( i ) and the proof is analogous for the right adjoint action. (cid:3) Returning to our primary focus, let H again be an affine commutative-by-finite Hopf k -algebra, with k algebraically closed and A a commutative normal Hopf subalgebraover which H is a finite module. Let H = H/A + H and let π be the Hopf surjectionfrom H to H as in Theorem 2.3. Recall from Theorem 2.3(3) that the adjoint actionsof H on A factor over A + H , so that A is a left and right H -module algebra.Using the terminology introduced at Definition 4.1(1), the left H -core of an ideal I of A with respect to the left adjoint action will be denoted by ( H ) I and we will define I as left H - stable if it is invariant under left adjoint action. Right H -cores are denoted by I ( H ) and right H -stable ideals are defined analogously. When A is semiprime or H ispointed, recall from Theorem 2.3(7) that A ⊆ H is a faithfully flat H -Galois extension,hence left and right stability of ideals are equivalent by Lemma 4.10 and we refer to I simply as H -stable. The non-semiprime case will be dealt with in the next section.In view of Definition 4.5, we say A is left orbitally semisimple if A/ ( H ) m is semisimplefor every m ∈ Maxspec( A ). One similarly obtains a notion of right orbital semisimplicity. Lemma 4.11.
Let H be an affine commutative-by-finite Hopf algebra, with commutativenormal Hopf subalgebra A .(1) Let V be a subspace of A . Then, S (cid:16) ( H ) V (cid:17) = S ( V ) ( H ) . (2) A is left orbitally semisimple if and only if it is right orbitally semisimple.(3) Let I be an ideal of A such that S ( I ) = I . Then, I is left H -stable if and onlyif it is right H -stable. Proof. (1) Let v ∈ V and h ∈ H . Then, ad r ( h )( Sv ) = X S ( h ) S ( v ) h = S (cid:16)X S − ( h ) vh (cid:17) = S (cid:16)X S − ( h ) vS ( S − ( h ) ) (cid:17) = S ( ad l ( S − h )( v )) . The statement now easily follows from this.(2) By (1), S induces an isomorphism between A/ ( H ) m and A/S ( m ) ( H ) . Considering S acts on Maxspec( A ), the statement follows.(3) Suppose S ( I ) = I . If I is left H -stable, then I ( H ) = S ( ( H ) I ) = S ( I ) = I . Conversely,if I is right H -stable, ( H ) I = S − ( I ( H ) ) = S − ( I ) = I . (cid:3) Given part (2) of the previous lemma, the adjectives left and right will be omittedfrom orbital semisimplicity.In view of Example 4.7 it seems likely that not all affine commutative-by-finite Hopfalgebras are orbitally semisimple, but we know of no example at present. All the affinecommutative-by-finite Hopf algebras described in § § § § A arecentral. The group algebras in § § § § § Prime and semiprime commutative-by-finite Hopf algebras
We study here the primeness and semiprimeness of commutative-by-finite Hopf alge-bras, starting with the classical commutative case in § § Lemma 5.1.
Let I be a Hopf ideal of the Hopf algebra H . Then T ∞ n =1 I n is a Hopfideal. The nilradical and primeness: commutative case.
The proof of the nextlemma is included here to facilitate later discussion of noncommutative versions.
Lemma 5.2.
Let A be an affine commutative Hopf algebra.(1) The nilradical N ( A ) of A is a Hopf ideal.(2) The left coideal subalgebra C := A co A/N ( A ) of A has the following properties:(i) C is a local Frobenius subalgebra of A with C + A ⊆ N ( A ) .(ii) A is a free C -module.(3) There is a unique minimal prime ideal P of A for which P ⊆ A + ;(4) P = T n ( A + ) n + N ( A ) is a Hopf ideal.Let T be a Hopf algebra such that A is a left T -module algebra. Suppose that N ( A ) is T -stable. Then:(5) the prime ideal P is T -stable.Proof. (1) By the Nullstellensatz, N ( A ) = T V { Ann( V ) } as V runs through the irre-ducible A -modules. If V and W are irreducible A -modules, then they have dimension 1since k is algebraically closed, so that V ⊗ W is also irreducible. Hence N ( A )( V ⊗ W ) = 0 I HOPF ALGEBRAS 17 and it follows that ∆( N ( A )) ⊆ N ( A ) ⊗ A + A ⊗ N ( A ) . Clearly, S ( N ( A )) = N ( A ) as S is an automorphism, and ǫ ( N ( A )) = 0, since N ( A ) is nilpotent.(2) Let C = A co A/N ( A ) . Clearly this is a left coideal subalgebra of A such that C + A ⊆ N ( A ). In particular, C + is a nilpotent ideal of C , hence C is local. A commutativeHopf algebra is flat over its coideal subalgebras by [45, Theorem 3.4], hence A is C -flat. Moreover, when checking faithfulness, it suffices to prove V ⊗ C A = 0 for allsimple C -modules V , [74, § C is local with unique simple k , and k ⊗ C A ∼ = A/C + A = 0, A is a faithfully flat C -module.Since A is noetherian, faithful flatness of A C guarantees that C is also noetherian,as a strictly ascending chain of ideals of C would induce a strictly ascending chain ofideals of A . In particular, C + is a finitely generated nilpotent maximal ideal, so thatdim k C < ∞ . Since C is finite dimensional local, its only flat modules are free [58,Corollary 4.3, Theorem 4.44], so A is C -free.Finally, to see that C is Frobenius, note that A , being an affine commutative Hopfalgebra, is Gorenstein [6, Proposition 2.3, Step 1]. Viewing an injective resolutionof A | A as a resolution of C -modules by restriction, we see that the free C -module A has finite injective dimension. Hence inj . dim C < ∞ . But commutative Gorensteinfinite dimensional rings are self-injective by [24, Proposition 21.5], thus quasi-Frobenius.Finally, commutative quasi-Frobenius algebras are Frobenius, proving (2).(3),(4) In view of (1), we may factor by N ( A ) and so assume that A is semiprime. Hence A has finite global dimension [74, 11.6, 11.7], and so is a finite direct sum of domainsby [24, Corollary 10.14] and the Chinese Remainder Theorem. In particular, distinctminimal primes of A are comaximal, so there is a unique such prime P contained in A + .As P = Ae is idempotently generated, e = e n ∈ ( A + ) n for all n ≥
1, so that P ⊆ ( A + ) n for all n . Since A/P is a commutative affine domain, T n ( A + ) n ⊆ P byKrull’s Intersection Theorem, [24, Corollary 5.4]. Combining this with the previousinclusion yields(5.6) P = \ n ≥ ( A + ) n . But the right hand side of (5.6) is a Hopf ideal by Lemma 5.1, so (4) is proved.(5) Since N ( A ) is T -stable, we can again factor by it in proving the T -stability of P .Since T n ( A + ) n is an intersection of T -stable ideals, the result follows from (5.6). (cid:3) Remarks 5.3.
Three points concerning Lemma 5.2: (1) C is not in general a Hopf sub-algebra of A . For instance, following [5, Example 2.5.3], let k be a field of characteristic p > n . Let G = ( k, +) ⋊ k × , where k × acts on ( k, +) bymultiplication. Its coordinate ring is O ( G ) = k [ x, y ± ] where x is (1 , y )-primitive and y is group-like. Let A be the Hopf quotient O ( G ) / ( x p n ). Its nilradical is N ( A ) = xA andthe coinvariants are C = A co A/N ( A ) = k h x i .(2) Since A/N ( A ) is an affine commutative semiprime Hopf k -algebra it is the coordinatering O ( G ) of an affine algebraic k -group G . The ideal P of A is the defining ideal of G ◦ ,the connected component of the identity of G , so A/P = O ( G ◦ ) [74, § C + A ⊆ N ( A ) of Lemma 5.2(2)(i) isstrict. The equality C + A = N ( A ) is known to hold if the coradical of A is cocommutative(in particular, if A is pointed) by [42, Theorem]. Lemma 5.4.
Keep the notation from Lemma 5.2 and Remark 5.3(2). Define B := A co A/P , a left coideal subalgebra of A with C ⊆ B , over which A is flat. (1) B + N ( A ) /N ( A ) = O ( G/G ◦ ) , the coordinate ring of the discrete part of G ;(2) B + N ( A ) /N ( A ) is a finite-dimensional semisimple Hopf subalgebra of A/N ( A ) with P = B + A + N ( A ) ;(3) A/N ( A ) is a free B + N ( A ) /N ( A ) -module.(4) Suppose that A is semiprime or that the coradical of A is cocommutative. Then B is a semilocal (finite dimensional) Frobenius subalgebra of A with P = B + A ,and A is a free B -module and B is a free C -module.Proof. Flatness of A over B follows from [45, Theorem 3.4]. (1) is a basic part of thetheory of algebraic groups, see for example [74, § G ◦ is a normal subgroup of G [74, § P/N ( A ) is a conormal Hopfideal of A/N ( A ) and B + A + N ( A ) = P by [72, § A is cocommutative. Then C + A = N ( A ) by Remark5.3(3). Thus C + B = N ( B ) = N ( A ) ∩ B, and so B/C + B is semisimple by (2). Since C + B is a nilpotent finitely generated ideal of B , it follows that B is finite dimensional. A commutative Hopf algebra is free over anyfinite-dimensional left (or right) coideal subalgebra [44, Theorem 3.5(iii)], which proves A is B -free. In the semiprime case this is (2) and (3).Moreover, B = ⊕ ti =1 B i , where each B i ∼ = C is a finite dimensional commutative localalgebra. By Lemma 5.2(2), this proves B is Frobenius, semilocal and a free C -module. (cid:3) Remark 5.5.
Presumably (4) is true without the additional hypothesis on the coradical,but we have not been able to prove this. It is not true in general that B is a Hopfsubalgebra of A , as is shown by the example in Remark 5.3(1), in which C = B .5.2. The nilradical and primeness: commutative-by-finite case.
We carry theobservations and notation of the previous subsection into the next result, which providesparallel results for commutative-by-finite Hopf algebras. These are in part motivated byspeculations of Lu, Wu and Zhang [41, §
6, Theorem 6.5, Remark 6.6], proposing that anexact sequence of Hopf algebras similar to that recalled in Lemma 5.4 for the coordinatering A of an affine algebraic group G , namely0 −→ B = A co A/P = O ( G/G ◦ ) −→ A = O ( G ) −→ A/P = O ( G ◦ ) −→ , should be valid more widely. They in fact prove a partial version of this suggestionfor noetherian affine regular Hopf k -algebras of GK-dimension 1, with k of arbitrarycharacteristic and not necessarily algebraically closed, [41, Theorem 6.5]. Proposition 5.6.
Let H be an affine commutative-by-finite Hopf algebra, with commu-tative normal Hopf subalgebra A . Let P and C , G and B be as in Lemma 5.2, Remark5.3 and Lemma 5.4 respectively.(1) N ( A ) is left H -stable if and only if it is right H -stable.Assume for the rest of the proposition that N ( A ) is H -stable.(2) The minimal prime ideal P of A is an H -stable Hopf ideal, so N ( A ) H and P H are Hopf ideals of H . Moreover, (5.7) N ( A ) H ∩ A = N ( A ) = N ( H ) ∩ A, I HOPF ALGEBRAS 19 where N ( H ) denotes the nilradical of H , and P H ∩ A = P. (3) C is invariant under the left adjoint action of H .(4) Let Q , . . . , Q r be the minimal prime ideals of H which are contained in H + , so r ≥ , with r = 1 if H/N ( A ) H is smooth. For all i = 1 , . . . , r , Q i ∩ A = P, and N ( A ) H ⊆ P H ⊆ r \ i =1 Q i . (5) If H is prime, then A is a domain.(6) Assume that either A is semiprime or H is pointed.(i) The right coinvariants of the coaction induced by the Hopf surjection H ։ H/N ( A ) H are H co H/N ( A ) H = A co A/N ( A ) = C, a local Frobenius left coideal subalgebra of A over which H is a free moduleand C + A = N ( A ) .(ii) The right coinvariants induced by the Hopf quotient b π : H → H/P H are H co H/P H = A co A/P = B, a semilocal Frobenius left coideal subalgebra of A over which H is a faithfullyflat projective module and B + H = P H . Moreover,
B/C + B = O ( G/G ◦ ) isa semisimple normal Hopf subalgebra of H/N ( A ) H over which H/N ( A ) H is a free module.Proof. (1) Since S | A is an automorphism, S ( N ( A )) = N ( A ). The statement followsfrom Lemma 4.11(3).(2) By Lemma 5.2 P is left and right H -stable, hence P H and N ( A ) H are ideals byLemma 4.10. Since both P and N ( A ) are Hopf ideals of A by Lemma 5.2, P H and N ( A ) H are Hopf ideals of H . By (1) and Lemma 4.10, N ( A ) H = HN ( A ). Hence N ( A ) H is nilpotent and so N ( A ) H ∩ A = N ( A ). Moreover, N ( A ) H ⊆ N ( H ), and so N ( A ) ⊆ N ( H ) ∩ A . The reverse inclusion is obvious. The minimal prime ideal P is theannihilator of some nonzero ideal I of A [32, Theorem 86], hence I ( P H ∩ A ) = 0 andso P H ∩ A ⊆ P . The reverse inclusion is clear.(3) Consider the Hopf surjection π ′ : A → A/N ( A ) and denote the corresponding right A/N ( A )-coaction of A by ρ . Take h ∈ H and c ∈ C . Thus, ρ ( h · c ) = X h c S ( h ) ⊗ π ′ ( h c S ( h )) = X h c S ( h ) ⊗ π ′ ( h · c ) . Since by Lemma 5.2(2) C is a left coideal with C + ⊆ N ( A ) and N ( A ) is left H -stable,we have ρ ( h · c ) = P h c S ( h ) ⊗ ǫ ( h ) ǫ ( c ) = ( h · c ) ⊗
1. Therefore, C is invariant underleft adjoint action of H .(4) Let Q , . . . , Q r be as stated, so that clearly r ≥
1, with r = 1 if H/N ( A ) H issmooth since in this case H/N ( A ) H is a finite direct sum of prime rings by Theorem2.12(3). Everything to be proved concerns objects which contain N ( A ) H and by (5.7) N ( A ) H ∩ A = N ( A ). Thus we can factor by the Hopf ideal N ( A ) H of H , and henceassume in proving (4) that A is semiprime. Let P = P , . . . , P s be the minimal primes of A , so that I := s \ j =2 P j is an ideal of A which is not contained in A + (noting that A , being semiprime, is smoothand hence a direct sum of domains). Now IP = { } . Hence, by the left H -stability of P and Lemma 4.10, for each i = 1 , . . . , r ,(5.8) IHP = { } ⊆ Q i . But I is not contained in Q i since Q i ∩ A ⊆ A + , so (5.8) implies that(5.9) P ⊆ Q i ∩ A. On the other hand, comparing GK-dimensions,GK dim( A ) ≥ GK dim(
A/P ) ≥ GK dim(
A/Q i ∩ A )= GK dim( H/Q i ) = GK dim( H ) , using (5.9) for the second inequality and [34, Proposition 5.5] for the first equality. Forthe final equality, observe first that, since it is a minimal prime, Q i survives as a properideal Q ( H ) Q i in the artinian quotient ring Q ( H ) of H , which exists by Theorem 2.9(5).Hence the (left, say) annihilator J of Q i in H is non-zero by Theorem 2.9(7), since thisis true for all proper ideals in a quasi-Frobenius ring, [71, Proposition XVI.3.1]. Thus,GK dim( J ) ≤ GK dim(
H/Q i ) ≤ GK dim( H ) by [34, Proposition 5.1, Lemma 3.1], andso the desired equality follows from Theorem 2.9(3). Since GK dim( A ) = GK dim( H )by Theorem 2.3(6), equality holds throughout the above chain of inequalities, so thatGK dim( A/P ) = GK dim(
A/Q i ∩ A ). Since any proper quotient of the affine domain A/P must have a strictly lower GK-dimension [34, Proposition 3.15], this proves (4).(5) This is a special case of (4), with r = 1 and Q = { } , which forces P = { } .(6) Assume A is semiprime or H is pointed. Recall the Hopf epimorphism π : H → H .By definition, H co H/N ( A ) H ⊆ H co H/P H ⊆ H co π = A , the last equality following fromTheorem 2.3(7).(i) By (5.7) H co H/N ( A ) H = C . When H is pointed, C + A = N ( A ) by [42, Theorem] (thisis trivial when A is semiprime). By Theorem 2.3(7) and Lemma 5.2(2) H is C -projectiveand, since C is local, H is C -free [58, Theorem 4.44].(ii) By (2), H co H/P H = B . By Theorem 2.3(7) and Lemma 5.4(4) H is a faithfullyflat projective B -module. By Lemma 5.4(2) B/C + B = O ( G/G ◦ ) is a finite dimensionalsemisimple Hopf subalgebra of A/N ( A ) and P = B + A . By (2) and Lemma 4.10, B + H = HB + . Since H/N ( A ) H is faithfully flat over B/C + B , B/C + B is normal in H/N ( A ) H by [48, Proposition 3.4.3]. A Hopf algebra is free over any finite-dimensional normalHopf subalgebra [61, Theorem 2.1(2)], thus H/N ( A ) H is a free B/C + B -module. (cid:3) Remarks 5.7. (1) An unsatisfactory aspect of Proposition 5.6 is the need to assume N ( A ) is H -stable. It seems unlikely that this will always hold, and indeed the structureof the nilradical N ( H ) is a delicate question. It is clear that if A is semiprime and H is semisimple then H is semiprime (and is even a direct sum of prime algebras byProposition 2.10(1) and Theorem 2.12(3)), but the converse is easily seen to be false.And even the question as to when a smash product A H of a commutative H -modulealgebra A by a finite dimensional Hopf algebra H is semiprime has been the subjectof much research and remains currently unresolved - see for example [68]. Notice that I HOPF ALGEBRAS 21 even if one considers a finite dimensional commutative T -module algebra R , with T afinite dimensional Hopf algebra, then N ( R ) may not be T -stable - for instance, considerExample 4.7 and in its notation take T to be the n -dimensional Taft algebra and R tobe A/ m ( T ) a for some a ∈ k × .(2) Even when N ( A ) is H -stable, the exact sequence of Hopf algebras0 −→ B −→ H −→ H/P H −→ P H is in general not a prime ideal of H . Inthe notation of the proposition consider, for instance, the trivial case where A = k , so H = H , P = { } , r = 1 and Q = H + .Notwithstanding Remark 5.7(1), there is no problem when A is orbitally semisimple(Definition 4.5 and discussion in § Proposition 5.8.
Let H be an affine commutative-by-finite Hopf algebra with normalcommutative Hopf subalgebra A . Suppose A is orbitally semisimple. Then, the nilradical N ( A ) is a left and right H -stable Hopf ideal of A .Proof. Since A is orbitally semisimple, N ( A ) = \ m ∈ Maxspec( A ) m = \ m ∈ Maxspec( A ) m ( H ) = \ m ∈ Maxspec( A ) ( H ) m by Lemma 4.11(2), and since each ideal m ( H ) is right H -stable and each ( H ) m is left H -stable, N ( A ) is left and right H -stable. (cid:3) Representation theory: simple modules
Background facts.
Let H be an affine commutative-by-finite Hopf algebra, andrecall that, throughout this paper, the field k is assumed to be algebraically closed.(The results of this section could be recast without this last hypothesis, but they wouldbe significantly more complicated.) Since H is a finite module over its affine centre byCorollary 2.8, its simple modules are finite dimensional over k , by Kaplansky’s theorem[9, I.13.3]. To be more precise, let Q , . . . , Q t be the minimal prime ideals of H , soevery simple H -module is annihilated by at least one Q i . Recall Posner’s theorem, forexample from [9, I.13.3], stating that each algebra H/Q i has a central simple quotientring Q ( H/Q i ), and the PI-degree of H/Q i is defined to be the square root n i of thedimension of Q ( H/Q i ) over its centre. So n i is a positive integer, andmax { dim k ( V ) : V a simple H/Q i -module } = n i , by [9, Theorem I.13.5, Lemma III.1.2(2)]. Indeed most simple H/Q i -modules havedimension n i , in that the intersection of the annihilators of these topmost-dimensionsimple H/Q i -modules is Q i , whereas the intersection of the annihilators of the smallersimple modules strictly contains Q i , [9, Lemma III.1.2].Given the above we define the representation theoretic PI-degree of H to berep . PI . deg( H ) := max { n i : 1 ≤ i ≤ t } . Thus rep . PI . deg( H ) = max { dim k ( V ) : V a simple H -module } . Since the minimal degree min . deg( R ) of a ring R satisfying a polynomial identity isby definition the minimal degree of a monic multilinear polynomial satisfied by R ,(6.10) rep . PI . deg( H ) ≤
12 min . deg( H )by [9, I.13.3], where this is an equality if H is semiprime, but in general is strict.6.2. Bounds on dimensions of simple modules.Theorem 6.1.
Let H be an affine commutative-by-finite Hopf algebra, finite over thenormal commutative Hopf subalgebra A , and V a simple left H -module. Keep the nota-tion of § n ( V ) := min { n i : Q i · V = 0 , ≤ i ≤ t } . Let d A ( H ) denote the minimal number of generators of H as an A -module.(1) There exists m ∈ Maxspec( A ) such that Ann A ( V ) = m ( H ) .(2) There is an embedding A/ m ( H ) ֒ → V of A -modules. Hence, dim k ( A/ m ( H ) ) ≤ dim k ( V ) ≤ n ( V ) ≤ rep . PI . deg( H ) ≤
12 min . deg( H ) ≤ d A ( H ) , where the final inequality requires A H to be projective, as ensured by any of hypotheses(i)-(iv) of Theorem 2.3(7).Proof. (1) As discussed in § k ( V ) ≤ n ( V ) ≤ rep . PI . deg( H ) . In particular, dim k ( V ) < ∞ , so that A V contains a simple A -submodule V with anni-hilator m ∈ Maxspec( A ). Since H m ( H ) is a 2-sided ideal of H , { v ∈ V : ( H m ( H ) ) v = 0 } is a non-zero H -submodule of V . By simplicity, ( H m ( H ) ) V = 0. In particular, m ( H ) ⊆ Ann A ( V ). Conversely, Ann A ( V ) is contained in m and is easily seen to be H -stable, sothat Ann A ( V ) ⊆ m ( H ) by Lemma 4.2, proving (1).(2) Let { v , . . . , v t } be a k -basis of V . Then A/ m ( H ) embeds in V ⊕ t via ι : A → V ⊕ t : a ( av , . . . , av t ) , because ker ι = m ( H ) by (1). Since A/ m ( H ) is a Frobenius algebra by Proposition4.4(2), it is self-injective. Therefore, A/ m ( H ) is (isomorphic to) a direct summand ofthe A/ m ( H ) -module V ⊕ t , thanks to its inclusion in V ⊕ t via ι . Now A/ m ( H ) is finite-dimensional, hence commutative artinian, so it is a (finite) direct sum of non-isomorphicindecomposable submodules, each of which must be a summand of A V . Therefore, A/ m ( H ) embeds in A V .The first four inequalities in the displayed chain now follow from this embeddingtogether with (6.11) and (6.10). To prove the final inequality, assume one of the fourhypotheses (i)-(iv) of Theorem 2.3(7), so that H is a projective A -module by Theorem2.3(7)(c). Thus, for every maximal ideal n of A , A n ⊗ A H is a free left A n -module ofrank at most d A ( H ). Via the right action of H , this yields a homomorphism from H tothe algebra of d A ( H ) × d A ( H ) matrices over A n . Since the intersection of the kernels ofthese maps as n ranges through Maxspec( A ) is { } , it follows from the Amitsur-Levitskitheorem, [46, Theorem 13.3.3(iii)], that min . deg( H ) ≤ d A ( H ), as required. (cid:3) I HOPF ALGEBRAS 23
Typically one expects the upper bound d A ( H ) in the above to be replaceable bydim k ( H ). This is certainly the case when A is a domain. This and other simplificationsyield the following. Corollary 6.2.
Retain the notation of Theorem 6.1. Suppose that H is a prime affinecommutative-by-finite Hopf algebra such that A is semiprime. Let V be a simple H -module and m the annihilator of a simple A -submodule of V . Then (6.12) dim k ( A/ m ( H ) ) ≤ dim k ( V ) ≤ rep . PI . deg( H ) = 12 min . deg( H ) ≤ dim k ( H ) . Proof.
The equality rep . PI . deg( H ) = min . deg( H ) when H is prime follows from [9,Lemma III.1.2]. Since A is semiprime the hypothesis of Proposition 5.6 holds, and A isa domain by part (5) of that result. Therefore, A H is projective by Theorem 2.3(7)(c),so all of Theorem 6.1(2) is valid. Moreover, A H is a locally free A -module of constantrank, and this rank r must be given by r = rank A A + ( A A + ⊗ A H ) = dim k ( H ) , where the equality follows by Nakayama’s lemma. Localising further, to the quotientfield Q ( A ) of A , we see that H embeds via right multiplication operators in r × r matricesover Q ( A ). Hence min . deg( H ) ≤ r by Amitsur-Levitski, [46, Theorem 13.3.3(iii)]. (cid:3) Commutative-by-(semisimple & cosemisimple) Hopf algebras
Preliminaries.
In moving towards a deeper understanding of affine commutative-by-finite Hopf algebras, an obvious strategy is to impose restrictions on the finite dimen-sional Hopf quotient H of such an algebra H . Adopting this approach, a natural firstclass to study are those Hopf k -algebras H which are affine and commutative-by-finitewith H = H/A + H semisimple and cosemisimple for some choice of normal commutativeHopf subalgebra A . For brevity, we shall write in this case A ⊆ H ∈ CSC ( k ) . Here are two obvious constructions of such Hopf algebras. First, take a coordinatering A = O ( T ) of an algebraic group T over a field k and a finite group Γ whoseorder is a unit in k , with a homomorphism α from Γ to Aut( T ). So Γ acts on A by( γ · f )( t ) = f ( γ − ( t )), for γ ∈ Γ , f ∈ O ( T ) , t ∈ T , and we can form the smash product H = A . This is a Hopf algebra with the given coproduct of A and with Γ consistingof group-likes; and clearly A ⊆ H ∈ CSC ( k ). More generally, k Γ can be replaced byany semisimple and cosemisimple Hopf algebra H , with a Hopf algebra homomorphism α from H to k Aut( T ). At one extremity α ( H ) could be k Aut( T ) , yielding the tensorproduct H = A ⊗ k H. A second large collection of examples is provided in § H = kG where G has a finitely generated abelian normal subgroup N of finite index in G , such that G/N has no elements of order char k . The main resultof this section, Theorem 7.2, suggests that these examples may go some way towardsexhausting all the possibilities for such H .Before proving Theorem 7.2 we need to recall a concept from noncommutative ringtheory, and a basic result on finite dimensional Hopf algebras [65], extending [43]. Anideal I of a noetherian ring R is polycentral if there are elements x , . . . , x t ∈ I with I = P ti =1 x i R , such that x is in the centre Z ( R ) of R and, for j = 2 , . . . , t , x j + P j − i =1 x i R ∈ Z ( R/ P j − i =1 x i R ). Polycentral ideals share many of the properties of idealsof commutative noetherian rings, [46, Chapter 4, § § Proposition 7.1. (Skryabin, [65])
Every coideal subalgebra of a finite-dimensionalsemisimple Hopf k -algebra is semisimple.Proof. Every left or right coideal subalgebra of a finite-dimensional Hopf algebra isFrobenius [65, Theorem 6.1]. Since k is algebraically closed, a coideal subalgebra of asemisimple Hopf algebra is Frobenius if and only if it is semisimple by [43, Theorem2.1]. (cid:3) Given the length of the statement and proof of Theorem 7.2, the reader may find ithelpful to look first at the discussion immediately following its proof, including Corollary7.5. The latter records the simplifications which occur if A ⊆ H ∈ CSC and H is prime .7.2. Structure theorem.Theorem 7.2.
Let A ⊆ H ∈ CSC ( k ) with GK dim( H ) = n . Let N ( A ) denote thenilradical of A and let P be the unique minimal prime ideal of A with P ⊆ A + , as inProposition 5.6. Let G ◦ ⊆ G be the algebraic groups such that O ( G ) = A/N ( A ) and O ( G ◦ ) = A/P , as in Remark 5.3.(1) N ( A ) and P are (left and right) H -stable Hopf ideals of A , and N ( A ) H = N ( H ) and P H are semiprime Hopf ideals of H .(2) A/N ( A ) ⊆ H/N ( A ) H and A/P ⊆ H/P H are in
CSC ( k ) , with Gelfand-Kirillovand global dimensions n . They are faithfully flat H -Galois extensions of A/N ( A ) and A/P respectively.(3) Let Q , . . . , Q t be the minimal prime ideals of H . Precisely one minimal prime of H , say Q , is contained in H + , and this minimal prime contains P H . Reorderthe remaining Q i and fix r , ≤ r ≤ t , so that P ⊆ Q i if and only if i ≤ r . Then N ( A ) H = t \ i =1 Q i , P H = r \ i =1 Q i , Q j ∩ A = P, (1 ≤ j ≤ r ) . Moreover
H/N ( A ) H ∼ = t M i =1 H/Q i and H/P H ∼ = r M i =1 H/Q i , direct sums of prime algebras of Gelfand-Kirillov and global dimensions n .(4) There are subalgebras (7.13) C ⊆ B ⊆ A ⊆ D ⊆ H, such that:(i) C := A co A/N ( A ) is a local Frobenius left coideal subalgebra of A with C + A ⊆ N ( A ) , and A is a free left and right C -module. Moreover, C is invariantunder the left adjoint action of H .(ii) B := A co A/P is a left coideal subalgebra of A , over which A is flat and suchthat (7.14) P = B + A + N ( A ) . (iii) There is a factor group algebra k Γ of H with Hopf epimorphism α : H → k Γ ,such that the left and right adjoint actions of H on A/P both factor throughan inner faithful k Γ -action. Thus, D := H co k Γ is a left coideal subalgebraof H with H left and right faithfully flat over D . Moreover, D is invariant I HOPF ALGEBRAS 25 under the left adjoint action of H , D + H = HD + = ker α is a Hopf ideal of H and (7.15) H/D + H ∼ = k Γ . (5) D/A + D is semisimple.(6) For all a ∈ A and d ∈ D , ad − da ∈ P D , so that
A/P ⊆ Z ( D/P D ) , where Z ( R ) denotes the centre of the ring R .(7) D/N ( A ) D has global dimension n , so it is homologically homogeneous and is adirect sum of prime algebras, each of GK-dimension and global dimension n .(8) There is a unique minimal prime ideal L of D with L ⊆ D + . Moreover, (7.16) L ∩ A = P and (7.17) L = \ i ≥ ( D + ) i + N ( A ) D. (9) D/L is an affine commutative domain of global dimension n .(10) L is a left H -stable ideal of D , so LH is a Hopf ideal of H . Thus,(i) D/L is a left coideal subalgebra of
H/LH and a finite module over the Hopfsubalgebra
A/P of H/LH ;(ii) the left adjoint action of H on D induces a left adjoint action of H/LH on D/L ; and this action factors through an inner faithful group action, forsome group Λ which maps surjectively onto Γ .(11) There are inclusions N ( A ) H ⊆ P H ⊆ LH ⊆ Q , with Q ∩ D = L. (12) Let E := H co H/LH , a left coideal subalgebra of H with B ⊆ E ⊆ D. Then E = D co D/L and it is invariant under the left adjoint action of H .(13) Assume further that H is pointed. Then, H is a faithfully flat left and right E -module, Q = LH and H and H/LH are crossed products: H ∼ = A σ H and H/Q ∼ = ( D/L ) τ Γ , for cocycles σ and τ .Proof. (1),(2) Since H is cosemisimple, A is H -orbitally semisimple by Theorem 4.8(3),hence N ( A ) is H -stable by Proposition 5.8. In particular, N ( A ) H is a nilpotent Hopfideal of H by Proposition 5.6(2). It follows from Proposition 2.10(1) that H/N ( A ) H issmooth, so that N ( A ) H is semiprime by Theorem 2.12(3). Since N ( A ) H is nilpotent,it is thus the nilradical of H .Since N ( A ) is H -stable, P is also H -stable and P H is a Hopf ideal of H by Proposition5.6(2). That P H is semiprime follows for the same reasons as applied to N ( A ) H . Thefact that the global and Gelfand-Kirillov dimensions of H/N ( A ) H and H/P H equal n isa consequence of Theorem 2.9(3) coupled with Theorem 2.12(1). The relevant extensionsare faithfully flat H -Galois by Theorem 2.3(7)(ii)(a).(3) As already noted, Theorem 2.12(3) applies to both H/N ( A ) H and to H/P H , sothat both of these algebras are finite direct sums of prime algebras of GK-dimension n . Therefore GK dim( H/Q i ) = n for all the minimal primes Q , . . . , Q t of H , and the displayed intersections and direct sum decompositions are clear. The remainder of (3)follows from Proposition 5.6(4).(4)(i) This was proved in Lemma 5.2(2) and Proposition 5.6(3).(ii) This was proved in Lemma 5.4.(iii) Since P is left and right H -stable by Proposition 5.6(2), the H -actions on A restrict to H -actions on A/P . Consider first the right adjoint action of H on A/P . By[67, Theorem 2] it factors through a group algebra k Γ, via a Hopf epimorphism H ։ k Γ,with k Γ acting inner faithfully on
A/P . Let α be the Hopf epimorphism H ։ H ։ k Γand D := H co α , so that D is a left coideal subalgebra of H , and is invariant under theleft adjoint action of H by [48, Lemma 3.4.2(2)]. Since α factors through π : H → H , A ⊆ H co π ⊆ H co α = D. By [50, Corollary 1.5], H is left and right faithfully flat over D , and ker α = D + H ,whence D + H is a Hopf ideal and (7.15) follows. By Koppinen’s lemma [50, Lemma 1.4]and the fact that D + H is an ideal of H , S ( D + H ) = HD + ⊆ D + H . Since S ( A + H ) = HA + , S induces a bijection on the finite dimensional space H , so that HD + = D + H .Now repeat the above argument for the left adjoint action, yielding an epimorphismof Hopf algebras β : H ։ H ։ k Λ, for a finite group Λ with k Λ acting inner faithfullyon
A/P . We claim that(7.18) ker α = ker β, so that Γ = Λ . Let h ∈ ker α and v ∈ A/P . By the proof of Lemma 4.11(1),0 = ad r ( h )( Sv ) = S ( ad ℓ ( S − h )( v )) . Since S is an automorphism of A/P , this implies that ad ℓ ( S − h )( A/P ) = 0 , so that S − (ker α ) ⊆ ker β. A similar argument yields the reverse inclusion. But S ( D + H ) = D + H as above, henceker β = S − ( D + H ) = D + H = ker α , proving (7.18).(5) Note that D/A + D = D/ ( A + H ∩ D ) ∼ = ( D + A + H ) /A + H ⊆ H , where the equalityfollows by faithful flatness of H over D . Therefore (5) is a consequence of Proposition7.1, since H is semisimple.(6) Since D + ⊆ ker α , D acts by ǫ in the right adjoint action of H on A/P . Thus, for a ∈ A, d ∈ D , ad = X d S ( d ) ad = X d ( ad r )( d )( a ) ≡ X d ǫ ( d ) a = da mod P H, since D is a left coideal. By (4)(iii), D ⊆ H is faithfully flat, so P H ∩ D = P D .(7) Since H is right faithfully D -flat, N ( A ) H ∩ D = N ( A ) D . Let A ′ = A/N ( A ) , D ′ = D/N ( A ) D and H ′ = H/N ( A ) H .First, gl . dim( D ′ ) ≤ n . To see this, note that gl . dim( H ′ ) = n by (1) and H ′ / ( D ′ ) + H ′ is cosemisimple by (7.15), so the inequality follows by [33, Lemma 9], (in which the keypoint is that the cosemisimplicity of H ′ / ( D ′ ) + H ′ ensures that the left D ′ -module directsum decomposition H ′ = D ′ ⊕ U of [45, Corollary 2.9] can be achieved as D ′ -bimodules).As just explained, D ′ is a left D ′ -direct summand of H ′ , so D ′ is a projective A ′ -module by Theorem 2.3(7)(c). Let V be an irreducible left D ′ -module, and supposethat pr . dim D ′ ( V ) = t . Restricting a D ′ -projective resolution of V to A ′ , it follows thatpr . dim A ′ ( V ) ≤ t . However, dim k ( V ) < ∞ and A ′ is a finite direct sum of commutativeaffine domains, each of global dimension n , by (the commutative case of) Theorem I HOPF ALGEBRAS 27 A ′ -modules, and so all the finite dimensional A ′ -modules, have projective dimension n . Thus n ≤ t , so n = t and D ′ is homologicallyhomogeneous. The direct sum decomposition of D ′ now follows from [11, Theorem 5.3].(8) There is a unique minimal prime ideal L of D with L ⊆ D + by the decompositionof D/N ( A ) D into a direct sum of prime rings in (7). If P = N ( A ) then clearly P ⊆ L since N ( A ) D = N ( A ) H ∩ D is a nilpotent ideal of D by (1). Suppose on the other handthat N ( A ) ( P . Since P is the unique minimal prime of A with P ⊆ A + , there exists y ∈ A \ A + with yP ⊆ N ( A ). By stability of P , yDP ⊆ yP D ⊆ N ( A ) D ⊆ L. Hence, since y / ∈ L ,(7.19) P ⊆ L ∩ A. For the reverse inclusion, note that
D/L is a finite (left, say)
A/L ∩ A -module, so thatGK dim( A/L ∩ A ) = GK dim( D/L ) = n by [34, Proposition 5.5] and (7). This forces equality to hold in (7.19), as GK dim( A/P ) = n by (2), and a proper factor of an affine commutative domain has strictly lower GK-dimension [34, Proposition 3.15].We now prove (7.17). Since L ⊆ D + and L/N ( A ) D is generated by an idempotentin D/N ( A ) D by (7), L/N ( A ) D ⊆ ( \ i ≥ ( D + ) i + N ( A ) D ) /N ( A ) D. Hence L ⊆ T i ≥ ( D + ) i + N ( A ) D . We prove the reverse inclusion. Since D/A + D issemisimple by (5) and the image of A is central in D/L by (6) and (7.16), D + /L isa polycentral ideal of D/L . Hence, by the version of Krull’s Intersection Theorem forpolycentral ideals, [51, Theorems 11.2.8 and 11.2.13], \ i ≥ ( D + ) i ⊆ L, since D/L is a prime ring. This proves the required equality.(9) We first prove that
D/L is commutative. By (7.17), it suffices to prove that D/ ( D + ) i is commutative for each i ≥
1. Choose elements a , . . . , a m of A + whose images forma k -basis of A + / ( A + ) . Let e ∈ D + be such that e + A + D is the central idempotentgenerator of D + /A + D guaranteed by (5). Then D + / ( D + ) = ( De + X j Da j + ( D + ) ) / ( D + ) = ( ke + X j ka j + ( D + ) ) / ( D + ) . Since the quotient D + / ( P j ka j + ( D + ) ) is a factor of D + /A + D , it is idempotent and,being also a factor of D + / ( D + ) , it must be zero, so(7.20) D + = X j ka j + ( D + ) . Therefore, for each i ≥ D + / ( D + ) i is spanned by monomials of length at most i − in a , . . . , a m . Hence D/ ( D + ) i is commutative, as required. Thus D/L is an affine commutative do-main, since L is prime and D is a finite A -module. That the global dimension of D/L is n follows from (7) and the fact that L is a minimal prime.(10) First note that N ( A ) and D + are left H -stable by (1) and (4)(iii) respectively, hence L is left H -stable by (7.17). Moreover, N ( A ) H and D + H are Hopf ideals of H againby parts (1) and (4)(iii). Thus ∩ i ( D + H ) i = ( ∩ i ( D + ) i ) H is a Hopf ideal by Lemma 5.1.Therefore, by (7.17), LH = N ( A ) H + \ i ( D + H ) i is a Hopf ideal of H .(i) By faithful flatness of H over D , as ensured by (4)(iii),(7.22) LH ∩ D = L. Therefore,
D/L is a left coideal subalgebra of
H/LH . By (7.16),
A/P embeds in
H/LH ,and this map is a homomorphism of Hopf algebras whose image is contained in
D/L .For the rest of the proof of (10), we write A ′ := A/P , D ′ := D/L and H ′ := H/LH , sothat A ′ ⊆ D ′ ⊆ H ′ by (7.16) and (7.22), with A ′ a Hopf subalgebra and D ′ a left coidealsubalgebra of H ′ , and with D ′ an affine commutative domain by (9).(ii) By (4)(iii) the left adjoint action of H on itself preserves D and, since L is left H -stable, this induces a left adjoint action of H ′ on D ′ . Let a ∈ A ′ + and d ∈ D ′ . Then,by commutativity of D ′ and the fact that A ′ is a Hopf subalgebra,(7.23) ad ℓ ( a ′ )( d ′ ) = X a ′ d ′ S ( a ′ ) = X a ′ S ( a ′ ) d ′ = ǫ ( a ′ ) d ′ = 0for all a ′ ∈ A ′ + and d ′ ∈ D ′ . Thus this left adjoint action on D ′ factors through H ′ A ′ + = A ′ + H ′ .Since H ′ /A ′ + H ′ is semisimple and cosemisimple and D ′ is a commutative domain by(9), this action in turn factors through an inner faithful group action by [67, Theorem2], say H ′ /I ∼ = k Λ for some finite group Λ and some Hopf ideal I of H ′ that annihilates D ′ under the left adjoint action. However, by (4)(iii) the left adjoint action of H on A ′ factors through D + H with H/D + H ∼ = k Γ acting inner faithfully. And, since A ′ ⊆ D ′ ,it follows that(7.24) I ⊆ D ′ + H ′ and we have a Hopf epimorphism k Λ ։ k Γ. This proves (ii).(11) Only the parts involving Q remain to be proved. That LH ⊆ Q follows byessentially the same argument as applied to P in (8) - namely, there exists z ∈ D \ D + with zL = { } , so z ( HL ) = z ( LH ) = { } ⊆ Q , with z / ∈ Q , so that L ⊆ Q ∩ D . Andextending the argument along the lines of (8) yields Q ∩ D = L .(12) Since LH is a Hopf ideal by (10), we may consider the coinvariants E of the Hopfsurjection H ։ H/LH . On one hand, it follows from (7.17) that LH ⊆ D + H , hence wemust have E = H co H/LH ⊆ H co k Γ = D by (7.15). On the other hand, since D ⊆ H isfaithfully flat, it follows from (7.16) that LH ∩ A = P , hence B = A co A/P ⊆ H H/LH = E .Furthermore, E is invariant under the left adjoint action of H by [48, Lemma 3.4.2(2)].(13) Suppose that H is pointed. By [42, Theorem] H is faithfully flat over its left coidealsubalgebra E . As was pointed out in Remark 2.4(4), by [60, Corollary 4.3], also stated I HOPF ALGEBRAS 29 as [48, Theorem 8.4.8], H ∼ = A σ H , for some cocycle σ . Similarly, H/LH is pointed,so the inclusion
D/L ⊆ H/LH yields the decomposition
H/LH ∼ = ( D/L ) τ Γ . Moreover, this crossed product by a finite group acting faithfully on the commutativedomain
D/L is prime by [52, Corollary 12.6]. (cid:3)
Remark 7.3.
Consider again, in the light of the above theorem, the exact sequence ofalgebras suggested in [41, § § −→ E −→ D −→ H −→ k Γ −→ , where E ⊆ D ⊆ H and k Γ ∼ = H/D + H , with H and k Γ Hopf algebras, E and D leftadjoint invariant left coideal subalgebras of H with E finite dimensional, with D/E + D an affine commutative domain, left coideal subalgebra of H/E + H, and H/E + H ∼ = ( D/E + D ) τ k Γ , a prime crossed product.That is, in crude terms, such a Hopf algebra H in CSC ( k ) would be (finite dimensional)-by-(prime crossed product) where the prime crossed product is of a commutative domain acted on by a finite groupalgebra. Obstructions to confirming this description include the absence of positiveanswers to the following, where we retain the notation of Theorem 7.2: Questions 7.4. (1) Is E a finite dimensional algebra?(2) Is H/LH a prime crossed product, with LH = Q and H/LH ∼ = ( D/L ) τ Γ, if H isnot pointed?When H is prime, the first of these issues disappears, yielding: Corollary 7.5.
Let A ⊆ H ∈ CSC ( k ) , with H prime. Then, after replacing A bya larger smooth commutative affine domain D which is a left H -invariant left coidealsubalgebra of H ,(1) H/D + H ∼ = k Γ for a finite group Γ whose order is a unit in k ;(2) There exists a group Λ which acts faithfully on D via the left adjoint action; andthe group algebra k Λ maps surjectively onto k Γ .(3) Suppose in addition that H is pointed. Then H is a crossed product of D by k Γ ,that is, H ∼ = D σ k Γ for some cocycle σ .Proof. The hypothesis that H is prime implies that, in the notation of Theorem 7.2, Q = { } . Thus, P = L = { } by Theorem 7.2(11) and (7.16). Thus D is a commutativeaffine domain containing A , and the corollary is a special case of the theorem. (cid:3) Examples and consequences.
The first of the following two simple examplesshows that the inclusions of (7.13) can all be strict. The second shows that even when H is prime D may strictly contain A . Examples 7.6.
Let k be algebraically closed of characteristic 0.(1) Let G = ( h x i× S ) ⋊ C , where S is the symmetric group on 3 symbols and h x i is theinfinite cyclic group. Let σ and β be respectively a 3-cycle and a 2-cycle in S and let a be a generator of C , with C acting trivially on S and acting on h x i by a · x = x − .Let H = kG and A = k ( h x i × h σ i ). Thus A ⊆ H ∈ CSC ( k ), with H = k ( h β i × C ).Then, P = ( σ − A, B = k h σ i . The H -action on A/P ∼ = k h x i factors through Γ ∼ = C , so D = k ( h x i × S ) , L = P D + ( β − D. Moreover,
D/L ∼ = k h x i and E = kS . Thus in this case C ( B ( A ( D ( H, B ( E ( D and P H ( LH = Q = E + H, with H ∼ = A H ∼ = D A/P ∼ = D/L ∼ = k h x i . (2) Let H = k [ x ± ] with x grouplike, and let A = k [ x ± ]. Then A ⊆ H ∈ CSC ( k ), with H = kC . In this case D = H and Γ = { } , but A ( D .Bringing together the description of prime algebras in CSC ( k ) from Corollary 7.5 withthe Clifford-theoretic analysis of Theorem 6.1 yields Theorem 7.7.
Let A ⊆ H ∈ CSC ( k ) , with H prime, and let D and Γ be as in Corollary7.5. Let V be a simple left H -module, and choose m ∈ Maxspec( D ) with Ann V ( m ) = { } .(1) | Γ : C Γ ( m ) | ≤ dim k ( V ) ≤ | Γ | , where C Γ ( m ) := { γ ∈ Γ : m γ = m } .(2) PI − degree( H ) = | Γ | . In particular, the maximum dimension of simple H -modules is Γ , and the simple H -modules of dimension | Γ | are induced from simple D -modules.(3) If H is pointed then (1) can be strengthened: there exists ℓ ≤ | C Γ ( m ) such that dim k ( V ) = ℓ | Γ : C Γ ( m ) | , with V a free A/ m Γ -module of rank ℓ .Proof. (1) The existence of m follows as in the proof of Theorem 6.1(1), with A replacedby D , noting that in the present situation H is replaced by k Γ, and m ( H ) = m H = \ { m γ : γ ∈ Γ } . Since D/ m Γ is a direct sum of copies of k , an easier version of the proof of Theorem6.1(2) shows that D/ m Γ embeds in V as a D -submodule, proving the first inequality in(1).By Theorem 7.2(4)(iii) and Corollary 7.5(1), H is a locally free module of rank | Γ | over the affine commmutative domain D , so thatdim k ( H/H m ) = | Γ | . Since the H -module V is a factor of H/H m , (1) is proved.(2) Since Γ acts on D by k -algebra automorphisms, the extension Q ( D Γ ) ⊆ Q ( D ) isGalois with Galois group Γ. Hence dim Q ( D Γ ) Q ( D ) = | Γ | by Galois theory. By genericfreeness, [24, Theorem 14.4], there is a non-empty open subset P of Maxspec( D Γ ) suchthat, for p ∈ P , D p is D Γ p -free of rank | Γ | . Because the field extension Q ( D Γ ⊆ D is Galois, it is separable. Moreover D is smooth by Theorem 7.2(9). Therefore, by I HOPF ALGEBRAS 31 [62, Theorem 7 of Ch.II, § D Γ ⊆ D is generically unramified. Thatis, there exists p ∈ P such that p D is semimaximal with dim k ( D/ p D ) = | Γ | . Let b m ∈ Maxspec( D ) with p ⊆ b m . By classical invariant theory, [4, Chap.1],Γ − orbit( b m ) = { m ∈ Maxspec( D ) : p ⊆ m } . Thus | Γ − orbit( b m ) | = | Γ | . Therefore (1) implies (2).(3) Suppose that H is pointed. Then H is a crossed product by Corollary 7.5(3), sothere are units b γ i in H such that H = ⊕ i D b γ i , and it follows easily that, for all i ,Ann V ( m b γ i ) = b γ i Ann V ( m ) . Moreover, V is a semisimple A/ m Γ -module and hence V = M i Ann V ( m b γ i ) . Setting ℓ = dim k (Ann V ( m )) therefore gives (3). (cid:3) Acknowledgements
Some of this research will form part of the PhD thesis of the second author at theUniversity of Glasgow. His PhD is funded by the Portuguese Foundation for Scienceand Technology Fellowship SFRH/BD/102119/2014, to whom we are very grateful. Theresearch of the first author was supported in part by Leverhulme Emeritus FellowshipEM-2017-081.We thank Stefan Kolb (Newcastle), Uli Kr¨ahmer (Dresden), Christian Lomp (Porto)and Xingting Wang (Howard) for very helpful comments.
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School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, Scot-land
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