Consider a field k of characteristic zero with algebraic closure k and absolute Galois group Gal(k/k), and assume that we have a classification of Hopf algebras of ...
Forms of pointed Hopf algebras S. Caenepeel Faculty of Applied Sciences, Free University of Brussels, VUB Pleinlaan 2 B-1050 Brussels, Belgium
S. Dˇascˇalescu Faculty of Mathematics University of Bucharest Strada Academiei 14 RO-70109 Bucharest 1, Romania
L. Le Bruyn∗ Department of Mathematics University of Antwerp, UIA Universiteitsplein 1 B-2610 Wilrijk, Belgium
Abstract Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F , or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p2 and p3 , with p odd.
0
Introduction
Consider a field k of characteristic zero with algebraic closure k and absolute Galois group Gal(k/k), and assume that we have a classification of Hopf algebras of dimension d over k of a specific type. How can we use this to describe the corresponding Hopf algebras over k? Since the pioneering work of J.P. Serre [10] it is well known that the answer to this type of problem is given by Galois descent and Galois cohomology. Whereas these methods have been extensively used in the study of (central simple) algebras and varieties, applications to finite dimensional Hopf algebras are rare in the literature. The only papers known to us are [9], [5] and [8]. In this note, descent techniques will be applied to classify forms of certain pointed Hopf algebras, namely the pointed Hopf algebras that have been constructed recently using Ore extensions [2], and restricted quantum groups. The main result in [5] is that we have a one-to-one correspondence between twisted forms of the groupalgebra kC, C being a finitely generated abelian group, and the Galois extensions l of k with Galois group Gal(l/k) = Aut(C). We will see in Section 2 that a similar result holds for pointed Hopf algebras constructed by Ore extensions: Aut(C) has to be replaced by a suitable subgroup F = AutH (C), where C is the finite abelian group determining the coradical, and twisted forms of ∗
This author is a research director at the FWO (Belgium). Research supported by the project “Hopf algebras and (co)Galois theory” supported by the Flemish and Romanian governments.
1
H correspond to the product of the set of F -Galois extensions, and a finite product of sets of the form k ∗ /(k ∗ )ni . In Section 3, we give some classification results in dimension p 2 and p3 (p odd) as application. The theory is different in the “quadratic” case, this is the situation where we construct H by adding indeterminates xi to the coradical that satisfy an equation of the type x 2i = ai (g − 1). We discuss a particular example in Section 4, and we prove that Hopf algebra forms now correspond to the set of isometry classes of m-ary quadratic forms. Finally, in Section 5, we consider restricted quantum groups. These are finite dimensional pointed Hopf algebras constructed starting from a semisimple Lie algebra. The Hopf algebra forms are now in one-to-one correspondence with the Galois extensions of k with the graph symmetry group of the Dynkin diagram of the Lie algebra as Galois group.
1
Descent theory
For the reader’s convenience, we refresh some basic results about descent theory. Our main references are [7] and [10]. We will present the theory in such a way that it can be applied easily to Hopf algebras. Let k be a commutative ring, and l a commutative faithfully flat k-algebra. Consider an “object” N (a module, an algebra, a Hopf algebra,. . . ) defined over l. Descent theory gives an answer to the following two problems: 1. existence of forms: does there exist an object M defined over k such that M ⊗ k l ∼ = N? 2. classification: if such a k-form of N exists, classify all forms upto isomorphism. The answer to the first question is basically the following (cf. [7, II.3.2]): M exists (as a k-module) if and only if there exists a descent datum for N , this is an l ⊗ k l-isomorphism u : l ⊗k N → N ⊗k l such that u2 = u3 ◦ u1 . If N is an l-algebra and u is an l ⊗k l-algebra map, then M is a k algebra. Similar properties hold for other structures like coalgebras, Hopf algebras or F -module algebras (F being a group). In the Hopf algebra case, it suffices to prove that u is a bialgebra map, since bialgebra automorphisms between Hopf algebras preserve the antipode (cf. [12, Lemma 4.0.4]). In principle M can be recovered from N , but this construction turns out to be too complicated in practical computations. However, if l is a Galois extension of k (in the sense of [4]), then we have a more explicit construction. Write G = Gal(l/k), and take σ ∈ G. A map f ∈ End k (N ) is called σ-semilinear if f (an) = σ(a)f (n) for all a ∈ l and n ∈ N . With these notations, we have the following result (cf. [7, III.5.1]): Proposition 1.1 (Galois descent for modules and algebras) Let l be a Galois extension of k with group G. N ∈ l-mod descends to M ∈ k-mod if and only if there exists a Galois descent datum for N , that is a group homomorphism φ : G → Autk (N ) such that φ(σ) is σ-semilinear for all σ ∈ G. The descended module M is the fixed module M = N G , with the G-action on N induced by φ. Observe that similar statements hold for algebras (see [7, III.5.3]) and Hopf algebras (see [9, Prop. 1]). For example, if N = L is an l-Hopf algebra, then the descended module M = H is a k-Hopf 2
algebra if µL ◦ (φ(σ) ⊗ φ(σ)) = φ(σ) ◦ µL
(φ(σ) ⊗ φ(σ)) ◦ ∆L = ∆L ◦ ◦φ(σ) σ ◦ εL = εL ◦ φ(σ)
φ(σ)(1L ) = 1L
for all σ ∈ G. µL is the multiplication map on L, and ∆ L and εL are the comultiplication and counit. Let us now look at the second problem. Assume that the l-module N descends to M ∈ k-mod. It is well-known (cf. [7, II.8]) that we have a bijection between the (pointed) set of isomorphism classes of k-modules M 0 that are isomorphic to l ⊗k M ∼ = N after base extension, and the Amitsur cohomology set H 1 (l/k, Autl (• ⊗k M )): Mod(l/k, M ) ∼ = H 1 (l/k, Aut• (• ⊗k M ))
(1)
We refer to [7] for the definition of Amitsur cohomology. We obtain similar statements about isomorphism classes of algebras, Hopf algebras,. . . after we replace the automorphism group on the right hand sight of (1) by the appropriate automorphism group, for example, the set of Hopf algebra forms of l ⊗k H becomes isomorphic to Hopf(l/k, H) ∼ (• ⊗k H)) = H 1 (l/k, Aut Hopf •
(2)
Taking inductive limits over all faithfully flat extensions l of k, we obtain a formula for all twisted forms of H as a k-Hopf algebra Hopf(k, H) ∼ (• ⊗k H)) = H 1 (kfl , AutHopf •
(3)
Here are some well-known explicit variations of (3), that will be used in the sequel. Taking M = k n in (1), we obtain the (generalized) Hilbert Theorem 90, after taking inductive limits: Mod (l/k, k n ) ∼ = H 1 (l/k, GLn )
(4)
If k is a field, then both sides of (4) are trivial, since there exists only one vector space of dimension n. For n = 1, we find (5) Pic(k) ∼ = H1 (l/k, m ) Now let F be a finite group. The dual F k = ⊕ γ∈F F vγ of the group algebra kF is an F -module algebra, the F -action is given by δ · vγ = vγδ−1 (6) for all γ, δ ∈ F . F k is an F -Galois extension of k, and an F -module algebra form of F k is again an F -Galois extension. Furthermore Autalg F (F k) = F (k) the group of all continuous maps from Spec(k) to F , this is F to the power the number of connected components of k. We find the following formula for the set of commutative F -Galois extensions of k: (7) Gal(k, F ) ∼ = H 1 (kfl , F ) 3
There exists a kind of “dual” version of (7). Consider the group ring kF . Then kF is a strongly F -graded k-algebra with k as part of degree 1. The F -graded algebra forms of kF have the same property, and are in one-to-one correspondence with H 1 (kfl , F ∗ ), with F ∗ (k) = Hom(F, m (k)), the character group. If F = Cn , then F ∗ = µn , and, by Kummer theory, H 1 (kfl , µn ) = m (k)/m (k)n
(8)
If l is a Galois extension of k with group G, then all Amitsur cohomology can be replaced by group cohomology. For example (1) takes the form Mod(l/k, M ) ∼ = H 1 (l/k, Aut• (• ⊗k M )) ∼ = H 1 (G, Autl (l ⊗k M ))
(9)
The G-action on Autl (l ⊗k M ) is given by the formula σ · f = (σ ⊗ IM ) ◦ f ◦ (σ −1 ⊗ IM )
(10)
We also obtain the following formula (if l is connected): Gal(l/k, F ) ∼ = Hom(G, F ) = H 1 (l/k, F ) ∼
(11)
since the G-action on F is trivial. If k is a field of characteristic zero, then the flat cohomology can be rewritten as absolute Galois cohomology, for example H 1 (kfl , Aut• (• ⊗ M )) ∼ = H 1 (Gal(k/k), Autk (M ))
(12)
with k the algebraic closure of k, and M = k ⊗ k M . In this particular situation, we give one more example, which will be used in Section 5. Over k, all m-ary quadratic forms are trivial, and the group of isometries of the trivial m-ary quadratic form over k is Om (k). Thus the set of isometry classes of m-ary quadratic forms over k is classified by (cf. [10, III.1.2, Prop. 4]) (13) H 1 (Gal(k/k), Om (k)) The correspondence (1) is known in principle, but the construction is not very useful in practical situations. If l/k is Galois, then we have the following more explicit and practical description. Proposition 1.2 Let l be a Galois extension of the commutative ring k, with Galois group G, and consider a k-module M . The twisted form of l ⊗ k M corresponding to a cocycle ψ ∈ Z 1 (G, Autl (l ⊗k M )) under the isomorphism Mod (l/k, M ) ∼ = H 1 (G, Autl (l ⊗k M ))
(14)
is equal to (l ⊗ M )G , where the G-action on l ⊗k M is induced by the Galois descent datum φ : G → Endk (l ⊗k M ), with φ(σ)(a ⊗ m) = ψ(σ)(σ(a) ⊗ m) = σ(a)(ψ(σ)(1 ⊗ m))
(15)
for all a ∈ l and m ∈ M . Similar results hold for the variations of (14) (for algebras, coalgebras, Hopf algebras,. . . ).
4
Proof We have to show that there is a one-to-one correspondence between the cocycle classes [ψ] and isomorphism classes of descent data. We will only prove that φ is multiplicative, leaving the other details to the reader. Take σ, τ ∈ G, m ∈ M and a ∈ l, and write ψ(τ )(1 ⊗ m) =
X i
ai ⊗ m i
Using the cocycle relation ψ(στ ) = ψ(σ) ◦ (σ · ψ(τ )) we find φ(στ )(a ⊗ m) = (στ )(a)ψ(στ )(1 ⊗ m)
= (στ )(a)ψ(σ)((σ · ψ(τ ))(1 ⊗ m))
= (στ )(a)ψ(σ)(
X i
=
X i
σ(ai ) ⊗ mi )
σ(τ (a)ai )ψ(σ)(1 ⊗ mi )
= φ(σ) τ (a)(
X i
ai ⊗ m i )
= φ(σ) τ (a)(ψ(τ )(1 ⊗ m))
= φ(σ) φ(τ )(a ⊗ m)) and it follows that φ(στ ) = φ(σ) ◦ φ(τ ).
2
The Ore extension construction and Galois extensions
Let C be a finite abelian group and C ∗ its group of characters. We assume that the field k is of characteristic zero and has enough roots of unity such that C ∗ ∼ = C. Let t be a positive integer and g = (g1 , . . . , gt ) ∈ C t ; g ∗ = (g1∗ , . . . , gt∗ ) ∈ (C ∗ )t n = (n1 , . . . , nt ) ∈ t ; a = (a1 , . . . , at ) ∈ {0, 1}t ; b = (bi,j )1≤i,j≤t ∈ Mt,t (k) such that ni ≥ 2 for any 1 ≤ i ≤ t. We also ask the following conditions to be satisfied - < gi∗ , gi > is a primitive ni -th root of unity for any i. - < gj∗ , gi >=< gi∗ , gj >−1 for any i 6= j. - If ai = 1, then (gi∗ )ni = 1. - If gini = 1, then ai = 0. - bij = − < gi∗ , gj > bji for any i, j. - If bij 6= 0, then gi∗ gj∗ = 1. Then we can define the Hopf algebra H = H(C, n, g, g ∗ , a, b) generated by kC and t indeterminates x1 , . . . , xt subject to the relations xi c =< gi∗ , c > cxi , xni i = ai (gini − 1) for any 1 ≤ i ≤ t and c ∈ C xj xi = hgj∗ , gi ixi xj + bij (gi gj − 1) for any i 6= j The coalgebra structure is given by ∆(c) = c ⊗ c ; ε(c) = 1 5
∆(xi ) = gi ⊗ xi + xi ⊗ 1 ; ε(xi ) = 0 for any c ∈ C and i = 1, . . . , t. This is a particular case of the construction in [2], where Hopf algebras are constructed by starting with group algebras with the usual Hopf algebra structure, adding indeterminates as skew primitives by Ore extensions, then factoring by a certain Hopf ideal. We remind from [2] that H is pointed of dimension n 1 . . . nt |C|, with coradical kC. In this Section, we will determine the Hopf algebra forms of H in the case where n i > 2 for all i. In Section 4, we will consider the case n i = 2. Keeping (3) in mind, we first have to compute the Hopf algebra automorphisms of H. Consider the following subgroup of Aut(C): ∗ AutH (C) = {f ∈ Aut(C) | ∃s ∈ St with f (gi ) = gs(i) , gi∗ = gs(i) ◦ f, ai = as(i) for 1 ≤ i ≤ t}
Without loss of generality, we can assume that a i = 1 for 1 ≤ i ≤ u and ai = 0 for u < i ≤ t. Theorem 2.1 Let k be a field of characteristic zero, and H = H(C, n, g, g ∗ , a, b) with ni > 2 for any 1 ≤ i ≤ t. Then AutHopf (H) ∼ = AutH (C) × (k ∗ )t−u × µn1 (k) × . . . × µnu (k) Proof (inspired by [2, Theorem 2.1]) Let f ∈ Aut Hopf (H). It induces a group automorphism of C. Fix some i, and let J = {j | gj = f (gi )}. Since xi is a (1, gi )-skew primitive, f (xi ) must be a (1, f (gi ))-skew primitive, hence by the description of the coradical filtration of H (see [2]) we have f (xi ) = α(f (gi ) − 1) +
X
βj xj
j∈J
for some scalars α, βj . Now apply f to the relation xi c = hgi∗ , cicxi , and obtain that α(f (gi ) − 1)f (c) +
X
j∈J
βj hgj∗ , f (c)if (c)xj = αhgi∗ , ci(f (gi ) − 1)f (c) +
X
j∈J
βj hgi∗ , cif (c)xj
for any c ∈ C, showing that α = 0 and βj = 0 for any j ∈ J such that gj∗ f 6= gi∗ . Obviously J is not empty, since f (xi ) 6= 0. On the other hand |J| < 2. Indeed, if j, t ∈ J, j 6= t, then g j = gt = f (gi ) and gj∗ = gt∗ . Then hgj∗ , gj i = hgj∗ , gt i = hgt∗ , gj i−1 = hgj∗ , gj i−1 showing that nj = 2, a contradiction. Therefore J is a singleton, thus there is precisely one j such that gj = f (gi ) and gj∗ ◦ f = gi∗ , and we have f (xi ) = βj xj for some βj ∈ k ∗ . It is clear that ai = 1 if and only if aj = 1, and in this case βj ∈ µni (k). Thus we have associated to f a group automorphism f|C ∈ AutH (C) and a t-tuple (βj )1≤j≤t ∈ µn1 (k) × · · · × µnu (k) × (k ∗ )t−u . Conversely, for any f ∈ AutH (C) (with the corresponding permutation denoted by s), and any (βj )1≤j≤t ∈ µn1 (k) × · · · × µnu (k) × (k ∗ )t−u , it is straightforward to check that f can be extended to a Hopf automorphism of H by setting f (x i ) = βs(i) xs(i) for any i. Let k be the algebraic closure of k, and write H = k ⊗ k H. We will establish the action of the absolute Galois group Gal(k/k) on Aut Hopf (H). Take σ ∈ Gal(k/k), and f ∈ AutHopf (H), with corresponding permutation s and (β 1 , . . . , βt ) ∈ µn1 (k) × · · · × µnu (k) × (k ∗ )t−u . Using (10), we easily find that (σ · f )(c) = f (c), for all c ∈ C, and (σ · f )(xi ) =
(σ ⊗ 1) ◦ f ◦ (σ −1 ⊗ 1) (xi )
= (σ ⊗ 1)(βs(i) xs(i) ) = σ(βs(i) )xs(i) 6
and we conclude that σ acts trivially on F = Aut H (C), and in the natural way on µn1 (k) × · · · × µnu (k) × (k ∗ )t−u . Using the results of Section 1, we find ∗ Hopf(k, H) ∼ = H 1 (Gal(k/k), F ) × H 1 (Gal(k/k), k )t−u × H 1 (Gal(k/k), µn1 (k)) × · · · × H 1 (Gal(k/k), µn1 (k))
Comparing this formula to (5), (7) and (8), we obtain the following result: Theorem 2.2 Let k and H be as in Theorem 2.1. Then we have a one-to-one correspondence between the isomorphism classes of Hopf algebra forms of H and the product set of the isomorphism classes of AutH (C)-Galois extensions of k and k ∗ /(k ∗ )n1 × · · · × k ∗ /(k ∗ )nu . We remark at this point that the Galois extensions in Theorem 2.2 are not necessarily field extensions: they are commutative k-algebras that are Galois in the sense of [4]. Let us try to make our results more explicit. First, we take an n-tuple α = (α 1 , . . . , αu ) representing an element of k ∗ /(k ∗ )n1 × · · · × k ∗ /(k ∗ )nu . Without going into full detail, we mention that the Hopf algebra form H 0 of H corresponding to α has the same description as H, but with x ni i = gini − 1 replaced by xni i = αi (gini − 1) for 1 ≤ i ≤ u. The Hopf algebra form corresponding to an F -Galois extension K of k is more complicated to describe. We apply the procedure described in Proposition 1.2. As we have seen in (11), the elements in H 1 (Gal(K/k), F ) = Hom(F, F ) represent cocycles in H 1 (Gal(k/k), F ). Take the cocycle c represented by the identity map I F : F → F . We will first show that the corresponding F -Galois extension of k is just K itself. We have to apply one of the variations of Proposition 1.2. Using (15), we find that the map φ : F → Endk (K ⊗ F k) is given by φ(δ)(a ⊗ vγ ) = δ(a) ⊗ vγδ−1 for all a ∈ K and γ, δ ∈ F . A little computation shows that if aγ = δ(aγδ ) for all γ, δ ∈ F . Thus
(K ⊗ F k)F = {
X
γ∈F
P
γ∈F
aγ ⊗ vγ ∈ (K ⊗ F k)F if and only
γ −1 (a) ⊗ vγ | δ ∈ F }
and the map K → sending a to γ∈F γ −1 (a) ⊗ vγ is an F -module algebra isomorphism. This means that K corresponds to the cocycle c, as needed. The corresponding Hopf algebra form H 0 of H is constructed in a similar way. For any σ ∈ F = AutH (C), we find an automorphism f = φ(σ) of K ⊗ H determined by the formulas (see Theorem 2.1 and Proposition 1.2): (K ⊗ F k)F
P
f (a) = σ(a) ; f (c) = σ(c) ; f (xi ) = xs(i) for all a ∈ K and c ∈ C, with s the permutation corresponding to σ. φ meets the conditions of Proposition 1.1, and H 0 = (K ⊗k H)F . The explicit formula for the elements of H 0 looks weird, but several particular cases can be discussed in a more elegant way. 7
Group algebras Take t = 0. Then H = kC, and AutH (C) = Aut(C). We now have a one-to-one correspondence between forms of H and Galois extensions K of k with Galois group F = Aut(C). This correspondence still holds in the situation where k is replaced by an arbitrary commutative ring. The Hopf algebra form H 0 corresponding to K is {
X
γ∈C
aγ uγ ∈ KC | aσ(γ) = σ(aγ ), for all γ ∈ Aut(C)}
(16)
(16) is originally due to Hagem¨ uller and Pareigis (see Theorem 5 in [5]). As a consequence, we have the following classification result: Theorem 2.3 Let k be a field of characteristic zero. Then there is a one-to-one correspondence between k-Hopf algebras of dimension p, and cyclic Galois field extensions of dimension a divisor of p − 1. In particular, if k contains a primitive p − 1-th root of 1, then there is a bijection between isomorphism classes Hopf algebras of dimension p and the set k ∗ /(k ∗ )p−1 . Proof By Zhu’s Theorem [13], the only k-Hopf algebra of dimension p is kC p . By our previous results, its Hopf algebra forms are classified by Galois extensions (in the sense of [4]) with Galois group Cp−1 . These are in one-to-one correspondence with Galois field extensions with a subgroup of Cp−1 as Galois group. The second statement follows from Kummer theory. Example 2.4 We give an easy example, which can also be found in [5], in a slightly different form. Let G = C3 . Then Aut(G) = C2 . Let K = k[X]/(X 2 − a) be a quadratic extension of k. The √ √ nontrivial element σ of the Galois group is given by σ( a) = − a, and the corresponding Hopf algebra is Ha = {b + cu + σ(c)u2 ∈ KC3 | b ∈ k, c ∈ K} √ A k-basis for Ha is given by {1, v = u + u2 , w = a(u − u2 )} or {1, w, w 2 }, and Ha can also be described as follows: Ha = k[w]/(w 3 + 3aw) with comultiplication, counit and antipode given by ∆(w) = 1 ⊗ w + w ⊗ 1 +
1 (w ⊗ w2 + w2 ⊗ w) 2a
ε(w) = 0 ; S(w) = −w
In particular, if k = , then we have two Hopf algebras of dimension 3, namely H 1 = C3 (corresponding to the trivial Galois extension ) and H −1 = C3 (corresponding to ). If k = , then there are infinitely Hopf algebras of dimension p, since ∗ /(∗ )2 is infinite. This shows that, over a field that is not algebraically closed, the Kaplansky 10 conjecture already fails in dimension 3. Rigid pointed Hopf algebras We return to the case t > 0. If AutH (C) = 1, then H is the only Hopf algebra form of H, and then we say that the pointed Hopf algebra H is rigid. This happens in the following situations: 1) gi 6= gj for any i 6= j, and g1 , . . . , gt generate C; 2) gi∗ 6= gj∗ for any i 6= j, and g1∗ , . . . , gt∗ generate C ∗ . In particular, the Taft Hopf algebra T p2 = H(Cp , p, c, c∗ , 0, 0) (with t = 1, c a generator of Cp and hc∗ , ci a primitive p-th root of unity) is rigid. Observe that there are p − 1 such Hopf algebras (corresponding to the p − 1 possible choices of a primitive p-th root of 1). 8
Remark 2.5 Let k be a commutative ring, and assume that the order of the group C and the numbers ni are invertible in k. Also assume that k has enough roots of unity, such that C ∼ = C ∗. Then the Ore extension construction can still be applied, and Theorem 2.2 can be generalized, replacing Galois cohomology by flat derived cohomology. The results have to be modified as follows: H 1 (kfl , m ) ∼ = Pic(k) (see (5)) and we have a Kummer theory exact sequence 1−→m (k)/m (k)ni −→H 1 (kfl , µni )−→Picni (k)−→1 In fact an element of H 1 (kfl , µni ) is represented by a k-module I of rank one together with an isomorphism f : I ⊗ni → k (these isomorphism are in correspondence with m (k)/m (k)ni . Consider ((I1 , f1 ), . . . , (Iu , fu ), Iu+1 , . . . , It ) representing an element of H 1 (kfl , µn1 ) × · · · × H 1 (kfl , µnu ) × Pic(k)t−u . The corresponding Hopf algebra form H 0 of H can be described as follows. For 1 ≤ i ≤ u, let Ai be the tensor algebra T (Ii ) modulo the ideal generated by elements of the form f (a1 ⊗ · · · ⊗ ani ) − a1 ⊗ · · · ⊗ ani For u + 1 ≤ i ≤ t, let Ai = T (Ii )/Ii⊗ni . Now put
H 0 = kC ⊗ A1 ⊗ · · · ⊗ At as a k-module. The multiplication and comultiplication rules are the obvious ones.
3
Classification results
Let k be a field of characteristic zero. In Theorem 2.3, we have classified all k-Hopf algebras of dimension p. In this Section, we will give some (partial) classification results in dimension p 2 and p3 , for odd prime numbers p. Classification of Hopf algebras of dimension p 2 We do not have a full classification of k-Hopf algebras of dimension p 2 , but it is conjectured that they are all pointed, and then it is well-known that they are either the group ring kC p2 or k(Cp ×Cp ), or one of the p − 1 Taft Hopf algebras. We have seen above that the Taft algebras are rigid, and we have described the Hopf algebra forms of the group rings. We therefore have the following classification result. Theorem 3.1 Let k be a field of characteristic zero containing a primitive p-th root of unity. Then there is one-to-one correspondence between the isomorphism classes of Hopf algebras of dimension p2 that become pointed after a base extension, and the disjoint union of the following three sets: 1) isomorphism classes of Galois extensions with Galois group m (/p2 ); 2) isomorphism classes of Galois extensions with Galois group Gl 2 (/p); 3) the primitive p-th roots of unity in k.
Classification of Hopf algebras of dimension p 3 If k is an algebraically closed field of characteristic zero, then we can classify all pointed Hopf algebras of dimension p3 (see [1], [3] or [11]). Using the results of Section 2, we will classify all the Hopf algebras that are forms of pointed Hopf algebras. For the sake of simplicity, we assume that 9
k contains a (fixed) primitive p2 -th root of unity η, and we write λ = η p . Let us introduce the following notation. Cp =< c > ; Cp∗ =< c∗ > ; hc∗ , ci = λ Cp2 =< d >
;
Cp∗2 =< d∗ >
Cp × Cp =< c1 > × < c2 > hc∗i , cj i
; =
;
hd∗ , di = η
(Cp × Cp )∗ =< c∗1 > × < c∗2 >
λ 1
if i = j if i = 6 j
We follow the classification of [3], and discuss all the forms in each separate case. type 1: H = H(Cp , (p, p), (c, c), ((c∗ )i , (c∗ )−i , (0, 0), 0). Over k there are p − 1 non isomorphic Hopf algebras of this type (corresponding to i = 1, . . . , p − 1. If f ∈ AutH (Cp ), then f (c) = c, so f is trivial. Thus AutH (Cp ) is trivial and our p − 1 Hopf algebras are rigid. type 2: H = H(Cp , (p, p), (c, ci ), ((c∗ )j , (c∗ )−ij , (0, 0), 0), with 1 ≤ i, j ≤ p − 1. This gives (p − 1) 2 /2 non-isomorphic Hopf algebras over k (the Hopf algebra defined by i and j is isomorphic to the one defined by i−1 and j −1 , see [2]). Take f ∈ AutH (Cp ). If the corresponding permutation s of {1, 2} is trivial, then f is the identity. If s permutes 1 and 2, then it follows from the definition of AutH (Cp ) that f (c) = ci and (c∗ )j ◦ f = (c∗ )−ij Applying both sides of the second formula to c, we obtain that h(c∗ )j , f (c)i = λij
= h(c∗ )−ij , ci = λ−ij
and it follows that 2ij ≡ 0 mod p, contradicting the fact that p 6= 2 and our assumptions on i and j. Thus our (p − 1)2 /2 Hopf algebras are rigid. type 3: H = H(Cp × Cp , p, c, (c∗ )i , 0, 0), with 1 ≤ i ≤ p − 1. This means that H is the tensor product of one of the p − 1 Taft Hopf algebras and kC p . Now AutH (Cp × Cp ) ∼ = Cp−1 = Aut(Cp ) ∼ and we see that the forms of each of our p − 1 Hopf algebras are classified by the set of isomorphism classes of Cp−1 -Galois extensions of k. type 4: H = H(Cp2 , p, d, (d∗ )ip , 0, 0), with 1 ≤ i ≤ p − 1. If f ∈ AutH (Cp2 ), then f (d) = d, and f is the identity. Once more, we have p − 1 rigid Hopf algebras. type 5: H = H(Cp2 , p, dip , d∗ , 0, 0), with 1 ≤ i ≤ p−1. Again, AutH (Cp2 ) is trivial, since d∗ ◦f = d∗ , and we have p − 1 rigid Hopf algebras. type 6: H = H(Cp2 , p, d, (d∗ )ip , 1, 0), with 1 ≤ i ≤ p − 1. AutH (Cp2 ) is trivial, and the forms of H are classified by k ∗ /(k ∗ )p , by Theorem 2.2. type 7: H is a group ring. Then H = kG, with G = C p × Cp × Cp , Cp2 × Cp , Cp3 , G1 or G2 , where G1 and G2 are the two nonabelian groups of order p 3 . The forms of H are classified by the set of Galois extensions of k with Galois group respectively GL n (/p), Aut(Cp2 × Cp ), m (/p3 ), Aut(G1 ) and Aut(G2 ). We summarize our results as follows. Theorem 3.2 Let k be a field of characteristic zero containing a primitive p 2 -th root of unity. Then there is one-to-one correspondence between the isomorphism classes of Hopf algebras of dimension 10
p3 that become pointed after a base extension, and the disjoint union of the following sets: 1) The set of 3(p−1)+(p−1)2 /2 first nonnegative integers. The corresponding (rigid) Hopf algebras are the ones of type 1,2,4 and 5; 2) p − 1 copies of the set of isomorphism classes of C p−1 -Galois extensions of k (type 3); 3) p − 1 copies of k ∗ /(k ∗ )p (type 6); 4) the disjoint union of the sets of Galois extensions of k with Galois group GL 3 (/p), Aut(Cp2 ×Cp ), 3 m (/p ), Aut(G1 ) and Aut(G2 ).
4
The Ore extension construction and quadratic forms
Up till now all examples of k-forms of Hopf algebras we have given were determined by Galois cohomology of a finite Gal(k/k)-set. In this Section we will give a class of pointed Hopf algebras having an (infinite) algebraic group of automorphisms with non-trivial H 1 . Consider the Hopf algebra
H = H C4 , (2, . . . , 2), (c, . . . , c), (c∗ , . . . , c∗ ), (1, . . . , 1, 0, . . . , 0)
with c a generator of C4 , and hc∗ , ci = 1. H is the k-Hopf algebra with generators c, x 1 , . . . , xt and with relations c4 = 1 ; xi c + cxi = 0 ; xi xj + xj xi = 0 for i 6= j x2i = c2 − 1 for 1 ≤ i ≤ u x2i = 0 for u < i ≤ t
The coalgebra structure on H is given by ∆(c) = c ⊗ c ; ∆(xi ) = c ⊗ xi + xi ⊗ 1 ; ε(c) = 1 ; ε(xi ) = 0 The antipode sends c to c−1 and xi to −xi c−1 . As we have seen in Section 2, H is a pointed Hopf algebra of dimension 2t+2 with coradical kC4 . In order to compute the twisted forms of H = k⊗ k H, we first have to compute the Hopf algebra automorphism group of H. Proposition 4.1 The Hopf algebra automorphism group of H is equal to Aut
Hopf
(H) =
Ou (k) 0
Mu×(t−u) (k) GLt−u (k)
⊂ GLt (k)
Proof Let f ∈ AutHopf (H). Then, f induces a group automorphism on G(H) = C 4 . The only g ∈ P (H) such that dim (P1,g ) > 1 is g = c and therefore we must have that f (c) = c and so f |G(H) is the identity. As f (P1,c ) = P1,c and P1,c = k(1 − c) + kx1 + · · · + kxt , we have for all 1 ≤ i ≤ t f (xi ) = αi (c − 1) +
t X
αij xj
j=1
Applying f to xi c + cxi = 0 we obtain that αi = 0. Moreover, for 1 ≤ i ≤ t we have f (xi )2 =
u X
j=1
α2ij (c2 − 1)
11
Thus f (x2i ) if and only if
u X
=
α2ij
j=1
c2 − 1 0
=
1 0
for 1 ≤ i ≤ u for u < i ≤ t for 1 ≤ i ≤ u for u < i ≤ t
If i 6= j we have f (xi )f (xj ) = (
t X
αik xk )(
k=1 u X
=
k=1
and similarly f (xj )f (xi ) =
u X
k=1
t X
αjl xl )
l=1
αik αjk (c2 − 1) +
αjk αik (c2 − 1) +
X
1≤k= 1 for 1 ≤ i ≤ u < Bi , Bi >= 0 for u < i ≤ t < Bi , Bj >= 0 for i 6= j
It follows that B1 , · · · , Bu is an orthonormal basis for k u . The third statement then tells us that Bi = 0 for i > u, finishing our proof. Theorem 4.2 Let H be as above. Then we have a one-to-one correspondence between 1) isomorphism classes of k-forms of H, and 2) equivalence classes of m-ary quadratic forms defined over k. Proof As H is defined over k, we know that the isomorphism classes of twisted forms of H are in one-to-one correspondence with H 1 (Gal(k/k), AutHopf (H)). We have exact sequences of Gal(k/k)groups 1−→Mm×(n−m) (k)−→G(k)−→Om (k)−→1 (17) 1−→G(k)−→AutHopf (H)−→GLn−m (k)−→1 where G=
Om (k) 0
Mm,n−m (k) In−m
12
⊂ AutHopf (H)
(18)
(18) gives rise to a long exact sequence 1
−→ −→
G(k) 1 H (Gal(k/k), G(k))
−→ AutHopf (H) −→ H 1 (Gal(k/k), AutHopf (H))
π
1 −→ GLn−m (k) 1 −→ H (Gal(k/k), GL n−m (k))
It follows from Proposition 4.1 that π 1 is surjective, and the last term in the sequence vanishes because of the (generalized) Hilbert 90 formula (see (4)). We therefore have an isomorphism H 1 (Gal(k/k), G(k)) ∼ = H 1 (Gal(k/k), Aut Hopf (H)) (17) results in a long exact sequence π
2 1 · · · −→H 1 (Gal(k/k), Mm,n−m (k))−→H 1 (Gal(k/k), G(k))−→H (Gal(k/k), Om (k))−→ · · ·
The first term vanishes by [10, II.1.2] (remark that, as an abelian group, M m,n−m (k) is isomorphic to k
m×(n−m)
). Also π2 is surjective, since we have group section s : O m (k) → G(k) given by s(M ) =
M 0
0 In−m
s preserves the Gal(k/k)-action, and a cocycle c σ ∈ Z 1 (Gal(k/k), Om (k)) lifts to a cocycle s(cσ ) ∈ Z 1 (Gal(k/k), G(k)). We have now shown that the set of isomorphism classes of forms of H is isomorphic to H 1 (Gal(k/k), AutHopf (H)) ∼ = H 1 (Gal(k/k), Om (k)) and this set is isomorphic to the set set of equivalence classes of m-ary quadratic forms over k, by (13). Remark 4.3 Let q = ha1 , · · · , am i be an m-ary quadratic form over k with a i ∈ k ∗ /(k ∗ )2 . The corresponding k-form Hq of H is easily seen to be the k-Hopf algebra generated by c, x 1 , · · · , xn with defining relations c4 = 1, xi c + cxi = 0, xi xj + xj xi = 0 for i 6= j x2i = ai (c2 − 1) for 1 ≤ i ≤ m x2i = 0 for m < i ≤ n
and costructure as H.
5
Restricted quantum groups
The foregoing discussion may lead to believe that it is unusual for pointed Hopf algebras to have descent forms. In this section we treat the most popular class, that of restricted quantum groups. Whereas these pointed Hopf algebras seem to have a very rigid structure, they often do have nontrivial descent forms. Let g be a semisimple Lie algebra over k of rank n with Cartan matrix C = (a ij ) ∈ Mn () and vector d = (d1 , . . . , dn ) ∈ n+ of relative prime integers such that d.C is symmetric. The quantum group Uq (g) is the algebra generated by Ei , Fi , Ki±1 for i = 1, . . . , r satisfying the following relations Ki Kj = Kj Ki ; Ki Ki−1 = 1 = Ki−1 Ki 13
Ki Ej = q di aij Ej Ki
Ki Fj = q −di aij Fj Ki
;
Ki − Ki−1 q di − q −di
Ei Fj − Fj Ei = δij 1−aij
X
(−1)
s
s=0
1 − aij s
!
1 − aij s
!
1−aij
X
(−1)s
s=0
1−aij −s
Ei
Ej Eis = 0
di 1−aij −s
Fi
Fj Fis = 0
di
It is given a Hopf algebra structure by defining ∆(Ei ) = Ei ⊗ 1 + Ki ⊗ Ei S(Ei ) = −Ki−1 Ei ε(Ei ) = 0
;
∆(F ) = Fi ⊗ Ki−1 + 1 ⊗ Fi
∆(Ki ) = Ki ⊗ Ki ;
S(F ) = −Fi Ki ;
ε(Fi ) = 0
;
;
S(Ki ) = Ki−1
ε(Ki ) = 1
Let now q be a primitive l-th root of unity (where l is odd and if g contains components of type G2 , l is not divisible by 3), then one verifies that k[E il , Fil , Ki±l ; 1 ≤ i ≤ r] is a central subHopf algebra with augmentation ideal I = (E il , Fil , Kil − 1 : 1 ≤ i ≤ n). The restricted quantum group associated to g is the finite dimensional k-Hopf algebra Uq (g) = Uq (g)/Uq (g)I It is a pointed Hopf algebra of dimension l 2N +n where N is the number of positive roots of g. Our main objective here is to describe the Hopf algebra forms of U q (g). Let Dg be the Dynkindiagram associated to g and AutGraph (Dg ) the group of graph-symmetries of D g . Recall that if g is simple this group has the following description: 1 for types : A1 , Bn , Cn , E7 , E8 , F4 , G2 µ2 for types : An (n ≥ 2), Dn (n ≥ 5), E6
S3
for type :
D4
Observe that when g has several components of the same type, then corresponding symmetric group is a subgroup of AutGraph (Dg ). Following Section 1, we have to compute the group of kHopf algebra isomorphisms of the restricted quantum group. The following Proposition can also be deduced from the computation of the Hopf algebra automorphisms of U q (g) given in [6, 10.4.7]. We present a “pointed Hopf” proof. Proposition 5.1 For g semisimple we have an exact sequence ∗
1−→(k )n −→AutHopf Uq (g)−→AutGraph (Dg )−→1
14
(19)
Proof Let H = Uq (g), then H0 = k[×ni=1 /l] generated by the monomials in the K i which are the only grouplike elements of H. For g, h grouplike let P g,h = {x ∈ H | δ(x) = x ⊗ g + h ⊗ x}. Then we have P1,Ki = kEi + kFi Ki and if M is a monomial in the Ki of degree ≥ 2 then we have P1,M = 0 Let φ be a Hopf algebra automorphism of H, then grouplikes are send to grouplikes whence φ(K i ) = Mi for a monomial Mi in the Kj . As 1 is fixed any P1,g is sent to some P1,h . Therefore we must have that φ(Ki ) = Ks(i) for some permutation s ∈ Sn . Clearly, φ(P1,Ki ) = P1,Ks(i) whence φ(Ei ) = αi Es(i) + βi Fs(i) Ks(i) However, applying φ to the equation K i Ei = q 2di Ei Ki and observing that all dj are positive integers, it follows that βi = 0 and that s ∈ Sn must be such that di = ds(i) . By symmetry (or by looking at the sets PK −1 ,1 and repeating the argument) we find i
φ(Ei ) = αi Eσ(i) and φ(Fi ) = α−1 i Fσ(i) ∗
for some αi ∈ k . Finally, apply φ to the equation Ki Ej = q di aij Ej Ki , and deduce that s ∈ Sn must be such that for all i, j we have di aij = ds(i) as(i)s(j) whence aij = as(i)s(j) Therefore, s must be a graph automorphism of the Dynkin diagram of g. Conversely, given the defining relations of the algebra and coalgebra structure it is easy to verify ∗ that any n + 1-tuple of elements (α1 , . . . , αn , s) from ×ni=1 k × AutGraph (Dg ) determines as above an Hopf algebra automorphism. Observe that there is a non-trivial action of the Dynkin graph automorphisms on the torus. Theorem 5.2 Assume that q ∈ k. Then there is a one-to-one correspondence between 1) isomorphism classes of Hopf algebra forms of U q (g), and 2) Galois extensions of k with Galois group Aut Graph (Dg ). Proof The exact sequence (19) results in a long exact sequence in Galois cohomology ∗
π
· · · −→H 1 (Gal(k/k), (k )n )−→H 1 (Gal(k/k), AutHopf (Uq (g)))−→H 1 (Gal(k/k), AutGraph (Dg )) The first term is trivial by Hilbert 90 and our result will follow from (7), if we prove that π is surjective. We will use [10, I.5.6, Prop. 41]. ∗ Consider a cocycle c ∈ Z 1 (Gal(k/k), Aut Graph (Dg )). Because T = (k )n is an Abelian group, there is an action of AutGraph (Dg ) on T (by permutation) and the skew-group T c is well defined, as in [10, I.5.6]. In [10, I.5.6] a cohomology class ∆(c) ∈ H 2 (Gal(k/k), Tc ) is defined as follows. As Gal(k/k) acts trivially on Aut Graph (Dg ), c : Gal(k/k)−→AutGraph (Dg ) is a groupmorphism, and we can lift c to a groupmorphism c˜ : Gal(k/k)−→AutHopf (Uq (g)) 15
determined by g˜.[Ki , Ei , Fi ] = [Kc(g)i , Ec(g)i , Fc(g)i ] But then the cocycle ∆(c) determined by ˜ 0 −1 ag,g0 = g˜.(g g˜0 ).gg with values in Tc is trivial. Whence, by [10, I.5.6, Prop. 41] c lies in the image of π.
References [1] N. Andruskiewitsch, H.-J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order p3 , J. Algebra, in press. [2] M. Beattie, S. D˘asc˘alescu, and L. Gr¨ unenfelder, Constructing pointed Hopf algebras by Ore extensions, preprint. [3] S. Caenepeel and S. D˘asc˘alescu, Pointed Hopf algebras of dimension p 3 , J. Algebra, in press. [4] F. DeMeyer, E. Ingraham, “Separable algebras over commutative rings”, Lecture Notes in Math. 181, Springer Verlag, Berlin, 1971. [5] R. Hagem¨ uller and B. Pareigis, Hopf algebra forms of the multiplicative group and other groups, Manuscripta Math. 55 (1986), 121-136. [6] A. Joseph, “Quantum groups and their primitive ideals” Ergebnisse der Mathematik 29, Springer-Verlag, Berlin, 1995. [7] M. Knus and M. Ojanguren, “Th´eorie de la Descente et Alg`ebres d’Azumaya”, Lect. Notes in Math. 389, Springer Verlag, Berlin, 1974. [8] B. Pareigis, Forms of Hopf algebras and Galois theory, Banach Center Publ. 26 (1990), 75-93. [9] D.E. Radford, E.J. Taft and R.L. Wilson, Forms of certain Hopf algebras, Manuscripta Math. 17 (1975), 333-338. [10] J.P. Serre, “Cohomologie Galoisienne”, Lect. Notes in Math. 5, Springer Verlag, Berlin, 1965. [11] D. S ¸ tefan, F. Van Oystaeyen, Hochschild cohomology and coradical filtration of pointed coalgebras. Applications, J. Algebra, in press. [12] M. E. Sweedler, “Hopf algebras”, Benjamin, New York, 1969. [13] Y. Zhu, Hopf algebras of prime dimension, Int. Math. Research Notices 1 (1994), 53-59.
16
