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Solvable Hopf algebras and their twists
Journal Pre-proof Solvable Hopf algebras and their twists
Miriam Cohen, Sara Westreich
PII:
S0021-8693(19)30677-5
DOI:
https://doi.org/10.1016/j.jalgebra.2019.12.004
Reference:
YJABR 17479
To appear in:
Journal of Algebra
Received date:
7 July 2019
Please cite this article as: M. Cohen, S. Westreich, Solvable Hopf algebras and their twists, J. Algebra (2020), doi: https://doi.org/10.1016/j.jalgebra.2019.12.004.
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SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS MIRIAM COHEN AND SARA WESTREICH Abstract. We show that for any solvable group G and a Drinfel’d twist J, kGJ is solvable in the sense of the intrinsic definition of solvability given in [2]. More generally, if a Hopf algebra H has a normal solvable series so does H J . Furthermore, while solvable groups are defined as having certain commutative quotients, quasitriangular normally solvable Hopf algebras have appropriate quantum commutative quotients. We end with a detailed example.
Introduction Chains of normal left coideal subalgebras of a Hopf algebra H are analogues of chains of normal subgroups of a group. As a result, replacing normal subgroups with normal left coideal subalgebras gives a satisfactory intrinsic definition of nilpotent Hopf algebras that coincides with the categorical definition of nilpotency when applied to Hopf algebras representations. Generalizing solvability is more complicated, the basic difficulty being that there is no obvious analogue of Hopf quotients of left coideal subalgebras L over N where N ⊂ L is a left coideal subalgebra normal in L but not necessarily in H. It is customary to define solvable Hopf algebras H as those for which Rep(H) is a solvable category. However, this non-intrinsic definition is unsatisfactory as it contradicts our intuition from group theory. Commutative or nilpotent Hopf algebras are not always solvable in that sense. In [2] we suggested an intrinsic definition for solvability for semisimple Hopf algebras, which we refer to as Hopf solvability. In this case we say that the Hopf algebra H is solvable. When H = kG then Hopf solvability is equivalent to G being a solvable group. We proved there that commutative or nilpotent semisimple Hopf algebras are always solvable Hopf algebras. Denote by a, h ∈ H,
˙ ad
the left adjoint action of H on itself, that is, for all had˙ a =
h1 aS(h2 )
2000 Mathematics Subject Classification. 16T05. 1
2
MIRIAM COHEN AND SARA WESTREICH
A left coideal subalgebra of H is (left) normal if it is stable under the (left) adjoint action of H on itself. Left coideal subalgebras N have integrals ΛN (see e.g [6, 7, 11, 12]), which when H is semisimple over a field of characteristic 0 can be chosen to be idempotents. In [2] we observed that a left coideal subalgebra N is normal if and only if its integral ΛN is a central element of H. We then define solvabilty as follows. Definition[2]: Let H be a semisimple Hopf algebra. A chain of left coideal subalgebras of H N0 ⊂ N1 ⊂ · · · ⊂ Nt is a solvable series if for all 0 ≤ i ≤ t − 1, (i) ΛNi ∈ Z(Ni+1 ), the center of Ni+1 , where ΛNi is the integral of Ni , (ii) For all a, b ∈ Ni+1 , (aad˙ b)ΛNi = ε, abΛNi . The Hopf algebra H is solvable if it has a solvable series so that N0 = k and Nt = H. The motivation for this definition is the description of left coideal subalgebras quotients as appropriate subalgebras of H via integrals. This realization is based on [2, Lemma 1.4] where we show the following. Lemma 0.1. [2]: For any left coideal subalgebra N, let N + = N ∩ ker ε ˜ be left coideal subalgebras and let ΛN be the integral of N. Let N ⊂ N of H, Then, ˜ /N ˜ N + as left N ˜ -modules via the natural N ˜ -module pro˜ ΛN ∼ (a) N =N + + ˜ ∼ ˜ /N N ˜ as right N ˜˜ → N ˜ /N ˜ N . Similarly, ΛN N jection π : N = N modules. ˜ then N ˜ N + = N +N ˜ and thus π ˜ (b) If ΛN is central in N |N ΛN is an algebra isomorphism. . (c) If N is normal in H then π|HΛN , where π : H → H/HN + is an algebra isomorphism. Thus, regading solvability for Hopf algebras, the requirement for the subalgebra Ni+1 ΛNi to be trivially ad˙ -acted upon Ni+1 , is the appropriate generalization of solvability for groups, where we require abelian group quotients of the form Gi+1 /Gi . Explicitly, ([2, Ex. 3.7]).
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
Example 0.2. Let G be a finite group, K ⊂ G and ΛK = If ΛK is central in G, then for g ∈ G we have, 1 1 xg = ΛK g = gΛK = gx. |K| x∈K |K| x∈K
3 1 |K|
x∈K
x.
It follows that for x ∈ K, xg = gy for some y ∈ K. Thus K is normal in G. We claim now that G/K is abelian if and only if (aad˙ b)ΛK = bΛK , for all a, b ∈ G. Indeed, assume ab = ba for all a, b ∈ G/K. Then aba−1 = by for some y ∈ K. This implies that x = by x=b x. aba−1 x∈K
−1
Conversely, if aba x∈K x = b y ∈ K. This shows our claim.
x∈K
x∈K
x∈K
x then aba−1 = by for some
One of the equivalent definitions for solvable groups is via the derived series G0 = G, G1 = [G, G], . . . Gi+1 = [Gi , Gi ], Gn = 1. In this case the subgroups {Gi } are all normal in G. This motivates the following definition Definition 0.3. A Hopf algebra is normally solvable if it is solvable with a solvable series {Ni }, so that each Ni normal in H. Equivalently, ΛNi ∈ Z(H) for all i. As was demonstrated in Example 0.2, Property (ii) of solvability is the Hopf version of Gi+1 /Gi being an abelian group. When the Hopf algebra H is normally solvable quasitriangular Hopf algebra, then Property (ii) of solvability takes on the meaning of quantum commutativity. Recall, Definition 0.4. Let (H, R) be a quasitriangular Hopf algebra and let M be a left H-module algebra. Then M is quantum commutative if ab = (R2 · b)(R1 · a) for all a, b ∈ M. Theorem 1.3: Let (H, R) be a semisimple normally solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0, with a series {Ni } of normal left coideal subalgebras. Let N0 = k, Ni+1 = Ni+1 /Ni+1 Ni+ . Then for each i ≥ 0, Ni is a quantum commutative H-module algebra. In paricular N1 is a quantum commutative normal left coideal subalgebra of H.
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MIRIAM COHEN AND SARA WESTREICH
In Section 2 we show that Normal solvablity is a categorical invariant in the following sense: Theorem 2.5: Let H be a finite dimensional Hopf algebra over an algebraically closed field k of characteritic 0. Assume H is normally solvable, then H J is normally solvable for any Drinfel’d twist J for H. Two corollaries result from Theorem 2.5 Corollary 2.6: Let G be a finite solvable group, then kGJ is a normally solvable Hopf algebra. Corollary 2.7: Any triangular Hopf algebra of dimension pa q b is normally solvable. In Section 3, Example 3.1, we explicitly describe a normally solvable twisted group algebra. 1. Solvability and quantum commutativity Throughout this paper, H is a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. We denote by S and s the antipodes of H and H ∗ respectively. When (H, R) is quasitriangular we can relate solvability to quantum commitativity. Following Radford, let HR be the minimal Hopf subalgebra of H, so that R ⊗ R ⊂ H ⊗ H [10]. Recall HR = Rl Rr where Rl = spk {R1 } and Rl = spk {R2 }. Then we have, Proposition 1.1. Let (H, R) be a semisimple solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0 with a series {Ni } of left coideal subalgebras. Assume each Ni is stable under the adjoint action of HR and HR acts trivially by the adjoint action on ΛNi . Then Ni+1 ΛNi is a quantum commutative HR -module algebra. Proof. For all a, b ∈ Ni+1 , abΛNi = (since {a3 } ⊂ Ni+1 commutes with ΛNi ) = (a1 ad˙ b)ΛNi a2 ΛNi = (R2 a2 r2 )ad˙ b ΛNi R1 a1 Sr1 ΛNi (by quasitriangularity of H) 1 2 2 = Rad R a1 Sr1 ΛNi (since HR acts trivially on ΛNI ) ˙ (rad ˙ a2 ad ˙ b)ΛNi 2 = (R2 r2 )ad˙ b R1 aSr1 ΛNi (by solvability, since a2 , (rad ˙ b) ∈ Ni+1 ) 2 1 1 (by properties of R) = Rad ˙ b ΛNi R1 aSR2 ΛNi 2 1 = (Rad ˙ b)(Rad ˙ a)ΛNi
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
5
Proposition 1.1 can be applied to any solvable series {Gi } of a group G. In this case R = 1 ⊗ 1 and quantum commutativity of kGi+1 ΛkGi is equivalent to commutativity and thus to Gi+1 /Gi being an abelian group. Recall the definition of Normal solvability given in Definition 0.3. We show an enlightening observation regarding the connection between normal solvability and quantum commutativity for quasitriangular Hopf algebras. We need first the following lemma. ˜ be normal left coideal subalgebras of H, Lemma 1.2. Let N ⊂ N + ˜ /N ˜ N + is an N = N ∩ ker ε and ΛN the integral of N. Then N = N H-module algebra via had˙ a = had˙ a. ˜ ΛN → N is a left H-module isomrphism The algebra isomorphism π : N ˜ ΛN and the action ˙ on with respect to the adjoint action of H on N ad N. ˜ N + is invariant under the Proof. The action ad˙ is well defined since N adjoint action of H. The rest is trivial. Based on Proposition 1.1 and Lemma 1.2, we conclude. Theorem 1.3. Let (H, R) be a semisimple normally solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0, with a series {Ni } of normal left coideal subalgebras. Let N0 = k, Ni+1 = Ni+1 /Ni+1 Ni+ . Then for each i ≥ 0, Ni is a quantum commutative H-module algebra. In paricular N1 is a quantum commutative normal left coideal subalgebra of H. 2. Twisted Hopf algebra and solvability We wish to thank A. Masuoka for fruitful suggestions concerning twists of Hopf algebras. Let J = J 1 ⊗ J 2 ∈ H ⊗ H be a twist for H. That is, J is an invertible element of H ⊗ H that satisfies (Δ⊗Id)(J)(J ⊗1) = (Id ⊗Δ)(J)(1⊗J) (ε⊗Id)(J) = (Id ⊗ε)(J) = 1. Denote by H J the Hopf algebra with the same product as H and the twisted coproduct and antipode given by: ΔJ (h) = J −1 Δ(h)J where vJ =
S J (h) = vJ−1 S(h)vJ , S(J 1 )J 2
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MIRIAM COHEN AND SARA WESTREICH
Set J −1 = T =
T 1 ⊗ T 2 . Note that T satisfies
(1 ⊗ T )(Id ⊗Δ)(T ) = (T ⊗ 1)(Δ ⊗ Id)(T )
(1) Note that
vJ−1 =
T 1 S(T 2 )
By [8, (2.17)] we have, Δ(vJ ) = (S(T 2 ) ⊗ S(T 1 ))(vJ ⊗ vJ )(T 1 ⊗ T 2 ) S(J 2 )J 1 . Moreover, if H is semisimple then vJ = S(vJ ) =
(2)
Recall, when H is finite dimensional then J ∈ H ⊗ H ∼ = (H ∗ ⊗ H ∗ )∗ ∗ ∗ can be considered as a linear form σ : H ⊗H → k, which is a 2-cocycle for H ∗ . Then Hσ∗ is a Hopf algebra with new multiplication given by: a ·σ b = σ(a1 , b1 )a2 b2 σ −1 (a3 , b3 ) for all a, b ∈ H ∗ . We have (H J )∗ = (H ∗ )σ Given a left H-module V, it is known that it can be considered as a left H J -module (with respect to the same action). Denote the resulting module by J V. We have a tensor equivalence between the categories H Mod and H J Mod, V →J V with the categorical isomorphism JV
⊗J W →J (V ⊗ W ),
v ⊗ w → J −1 · (v ⊗ w).
In particular, consider H as a left H-module under the adjoint action of itself. Let (J H,ad˙ ) denote the corresponding H J -module with H as the underlying vector space and action given by the original adjoint acJ tion endowed from Δ. On the other hand consider (H J , adJ ˙ ), where H is considered as the left H J -module under the adjoint action obtained from ΔJ . Define the map: (3) ι : H → H, ι(a) = T 1 aS(T 2 )vJ for all a ∈ H. Note that for all b ∈ H, ι−1 (b) = J 1 bvJ−1 S(J 2 ) We show,
ε, ι(a) = ε, a
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
7
Lemma 2.1. Let H be any Hopf algebra (not necessarily finite dimensional) over k and let ι be defined as in (3) and consider it as a map ι : (J H, ad˙ H ) → (H J ,ad ˙ J ) Then: H (i) ι : (J H, ad˙ H ) → (H J ,ad ˙ J ) is an H J -module isomorphism. H
(ii) If N is a normal lef coideal subalgebra of H then ι(N ) a normal left coideal of H J . Proof. (i) The proof follows by a direct computation. (ii) We first show that ΔJ (ι(N )) ⊂ H J ⊗ ι(N ). Indeed, let T, t, X, Y be all copies of J −1 , then for all n ∈ N, ΔJ (ι(n)) = = T 1 t11 n1 (St2 )1 (vJ )1 J 1 ⊗ T 2 t12 n2 (St2 )2 (vJ )2 J 2 = T 1 t11 n1 (St2 )1 S(Y 2 )vJ X 1 J 1 ⊗ T 2 t12 n2 (St2 )2 S(Y 1 )vJ X 2 J 2 (by (2)) = T 1 t11 n1 (St2 )1 S(Y 2 )vJ ⊗ T 2 t12 n2 (St2 )2 s(Y 1 )vJ = t1 n1 (S(T 2 t22 ))1 S(Y 2 )vJ ⊗ T 1 t21 n2 (S(T 2 t22 ))2 s(Y 1 )vJ (by (1)) = t1 n1 S(t23 )S(T22 )S(Y 2 )vJ ⊗ T 1 t21 n2 S(t22 )S(T12 )S(Y 1 )vJ = t1 n1 S(t23 )S(T 2 )vJ ⊗ Y 1 T11 t21 n2 S(t22 )S(Y 2 T21 )vJ (by (1)) = t1 n1 S(t22 )S(T 2 )vJ ⊗ Y 1 (T 1 t21 ad˙ n2 )S(Y 2 )vJ = t1 n1 S(t22 )S(T 2 )vJ ⊗ ι(T 1 t21 ad˙ n2 ) Normality of N implies (T 1 t2 )ad˙ n2 ∈ N, hence the result follows. Part (i) implies now that ι(N ) is normal in H J .
Recall[12], for any Hopf algebra H, if π : H → H is a Hopf algebra projection, then H coH = {h ∈ H|h1 ⊗ π(h2 ) = h ⊗ 1} H . Observe is a normal left coideal subalgebra of H of dimension dim dim H coH that H is the unique maximal left coideal N of H, satisfying π(n) = ε(n) for all n ∈ N. We show,
Proposition 2.2. Let N be a normal left coideal subalgebra of a finite dimensional Hopf algebra H and J a twist for H. Then ι(N ) is a normal left coideal subalgebra of H J .
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MIRIAM COHEN AND SARA WESTREICH
Proof. By Lemma 2.1, ι(N ) is a left coideal of H J . It is left to show that it is an algebra. To see that, note first that if I is a Hopf ideal of H then it is a Hopf ideal of H J as well. Indeed, ΔJ (I) = J −1 Δ(I)J ⊂ J −1 (H ⊗ I + I ⊗ H)J ⊂ H ⊗ I + I ⊗ H S J (I) = vJ−1 S(I)vJ ⊂ vJ−1 IvJ = I In particular, HN + is a Hopf ideal of H J , implying that H J /HN + is a Hopf image of H J . We wish to show that ι(N ) = (H J )co(H
J /HN + )
which would imply that ι(N ) is an algebra as well. Now, for each a = ι(n), n ∈ N + , we have by the definition of ι, π(a) = 0. It follows that for all b ∈ ι(N ), π(b) = ε(b). This implies that ι(N ) ⊂ (H J )co(H
J /HN + )
.
Equality of dimensions implies the desired result.
Remark 2.3. Observe that ι(hz) = ι(h)z for all h ∈ H, z ∈ Z(H). It follows in particular that if N is a normal left coideal subalgebra of H with an integral ΛN , (which is central in H), then ΛN is the integral of ι(N ) as well. For the next corollary of Lemma 2.1, recall the following: Let V be an H-module and J V, the corresponding H J -module. One can consider LKerV ⊂ H which is a normal left coideal subalgebra of H, while LKerJ V ⊂ H J is a normal left coideal subalgebra of H J . We have, Corollary 2.4. Let V be an H-module, then ι(LKerV ) = LKerJ V . Proof. Since ι(LKerV ) is a normal left coideal, it is enough to show that ι(LKerV ) acts trivially on J V. Indeed, for all a ∈ LKerV , w ∈J V, we have, ι(a) · w = T 1 aS(T 2 )vJ · w = ε, a T 1 S(T 2 )vJ · w = ε, a · w. The reverse inclusion follows similarly by using ι−1 .
We use the results above to show that normal solvability of H, is invariant under twists. Theorem 2.5. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0. If H is normally solvable so is H J for any twist J of H.
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
9
Proof. Let k = N0 ⊂ N1 ⊂ · · · ⊂ Nt = H be a solvable series consisting of normal left coideal subalgebras of H. We claim that k = ι(N0 ) ⊂ ι(N1 ) ⊂ · · · ⊂ ι(Nt ) = H J is a solvable series consisting of normal left coideal subalgebra of H J . Property (i) of solvability, that is, each ι(Ni ) is a normal left coideal subalgebra of H J , follows from Proposition 2.2. By remark 2.3, {ΛNi } are the corresponding integrals of ι(Ni ). We prove now property (ii) of solvability. We need to show that for all i and for all a, b ∈ Ni , ι(a)ad ˙ J ι(b)ΛNi−1 = ε, a ι(b)ΛNi−1 . H
Set Vi = Ni ΛNi−1 , an H-module under the left adjoint action and Ki = LKerVi . Since H is solvable we have Ni ⊂ Ki . By Corollary 2.4, we have ι(Ni ) ⊂ ι(Ki ) = LKer(J Vi ,adH ) . That is, for all a, b ∈ Ni , ι(a)ad˙ bΛNi−1 = ε, a bΛNi−1 . Apply ι to both sides, we have by Lemma 2.1(1), ι(a)ad ˙ J ι(b)ΛNi−1 = ε, a ι(b)ΛNi−1 . H
This concludes the proof of the theorem.
Since finite solvable groups are normally solvabe, we conclude: Corollary 2.6. Let G be a finite solvable group, then kGJ is a normally solvable Hopf algebra. Another corollary is the following: Corollary 2.7. Let H be a semisimple triangular Hopf algebra of dimension pa q b , then H is normally solvable. Proof. By [3] H = (kG)J where G of order pa q b and J is a twist for kG. The result follows now from Burnside theorem for groups and Theorem 2.5.
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MIRIAM COHEN AND SARA WESTREICH
3. Example In what follows we give an explicit example of a normally solvable Hopf algebra. Example 3.1. Let G = S3 × S3 . We show explicitly how kGJ is a solvable Hopf algebra. For this explicit example it was proved in [5] that kGJ is a simple Hopf algebra in the sense that it has no normal Hopf subalgebra (thus it is not solvable in any sense of [9]). Let S3 = {1, a, a2 , b, ab, a2 b}, a3 = b2 = 1, ab = ba2 }. Then S3 = ZZ3 ZZ2 where ZZ3 = {1, a, a2 } and ZZ2 = {1, b, b2 }. Let S3 be another copy of S3 with generators {a , b }. The group G is solvable with a solvable chain 1 ZZ3 S3 (S3 × ZZ3 ) (S3 × S3 ). Following [5, 2.3], a non-trivial twist on kG can be lifted from a twist J on the abelian, non-normal subgroup ZZ2 × ZZ2 . Calculating the appropriate twist we arrive at 1 J = ((1 + b) ⊗ 1 + (1 − b) ⊗ b ) . 2 Then J 2 = 1 ⊗ 1 and so J = J −1 = T. Also set v = vj , then, 1 v = v −1 = S(J 1 )J 2 = (1 + b + b − bb ) 2 For any g ∈ G we have now, ΔJ (g) = J(g ⊗ g)J = 1 = ((1 + b)g(1 + b) ⊗ g + (1 + b)g(1 − b) ⊗ gb 4 + (1 − b)g(1 + b) ⊗ b g + (1 − b)g(1 − b) ⊗ b gb ) So we have, 1 ΔJ (a) = ((a + a2 ) ⊗ a + (a − a2 ) ⊗ ab ) 2 1 2 ΔJ (a ) = (a (1 + b) ⊗ a + a (1 − b) ⊗ a ) 2 ΔJ (b) = b ⊗ b, ΔJ (b ) = b ⊗ b . Some particular examples of S J , 1 1 S J (a) = (a(1 − b ) + a2 (1 + b ), S J (a ) = (a (1 − b) + a2 (1 + b)) 2 2 S J (a + a2 ) = a + a2 , S J (a2 − a) = (a − a2 )b
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
11
S J (b) = b, S J (b ) = b . Let us start with the minimal normal left coideal subalgebra of kG, N1 = kZZ3 . Then ι(1) = 1 and calculating ι for the other elements yields 1 ι(a) = (a + a2 + (a − a2 )b ), 2 It follows that
1 ι(a2 ) = (a + a2 + (a − a2 )b ) 2
ι(N1 ) = Spk {1, a + a2 , (a − a2 )b } It can be directly checked that ι(N1 ) acts-ad˙ trivially on itself. Regarding quantum commutativity, recall, kGJ is quasitriangular with 1 R = (J −1 )τ J = ((1+b)⊗(1+b)+(1−b)⊗(1+b)b +b (1+b)⊗(1−b)+b (1−b)⊗(1−b)b ). 4 Indeed, Theorem 1.3 is verified, that is, xy = (R2 · y)(R1 · x) for all x, y ∈ ι(N1 ). Our next object in the original chain is the normal left coideal subalgebra N2 = kS3 . Then ι(b) = b and checking the other elements yields ι(N2 ) = spk {1, a + a2 , (a − a2 )b , b, (a + a2 )b, (a − a2 )b b} Since Λι(N1 ) = ΛN1 = 1 + a + a2 we obtain that ι(N2 )Λι(N1 ) = spk {ΛN1 , bΛN1 } Observe that the Hopf subalgebra generated by the right and left tensorands of R is the group algebra (of (kG)J ) - k{1, b, b , bb }. This subgroup commutes with ι(N2 ) hence acts-ad˙ trivially on it. It follows that 1 − b acts on ι(N2 ) as 0, while 1 + b acts as 2, hence for all x, y ∈ ι(N2 )ΛN1 , we have 1 2 1 (Rad ˙ y)((1 + b)ad ˙ x) = yx = xy ˙ y)(Rad ˙ x) = ((1 + b)ad 4 Similarly, let N3 = k(S3 × ZZ3 ), then ι(N3 ) = ι(N2 ) × k{1, a , a2 } Since Λι(N2 ) = ΛN2 = ΛS3 it follows that ι(N3 )Λι(N2 ) = {ΛS3 , ΛS3 a , ΛS3 a2 } Since k{1, b, b , bb } commutes with ι(N3 ) which is (regularly) commutative, the same argument as above works.
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MIRIAM COHEN AND SARA WESTREICH
References [1] S. Burciu, Kernel of representations and coideal subalgebras for Hopf algebras, Glasgow Math. J. 54 (2012) 107-119. [2] M. Cohen and S. Westreich, Solvability for Hopf algebras via integrals, Journal of Algebra 472C (2017) pp. 67-94. [3] P. Etingof and S. Gelaki, The Classification of Triangular Semisimple and Cosemisimple Hopf Algebras Over an Algebraically Closed Field, International Mathematics Research Notices 5 (2000), 223-229. [4] P. Etingof, D. Nikshych and V. Ostrik, Weakly group-theoretical and solvable fusion categories, Adv. Math. 226, 176-205 (2011). [5] C. Galindo and S. Natale, Simple Hopf algebras and deformations of finite groups, Math. Res. Lett. 14 (5-6), 943-954 (2007). [6] M. Koppinnen, Coideal subalgebras in Hopf algebras: freeness, integrals, smash products, Comm. Alg. 21(2), 427-444 (1993). [7] A. Masuoka, Freeness of Hopf algebras over coideal subalgebras, Comm. Algebra 20 (1992), 1353-1373. [8] S. Majid, foundations of quantum group theory, Cambridge University press, Cambridge (1995). [9] S. Montgomery, S.J. Witherspoon, Irreducible representations of crossed products, J. of Pure and Appl. Alg. 129 (1998), 315-326. [10] D. Radford, Minimal quasitriangular Hopf algebras, J. of Alg. 157 (2), 1993, 285-315. [11] S. Skriyabin, Projectivity and freeness over comodule algebras, Trans. of AMS, 359, no 6, (2007), 2597-2623 [12] M. Takeuchi, Quotient Spaces for Hopf Algebras. Comm. Alg., 22(7):25032523, 1995. Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel Email address: [email protected] Department of Management, Bar-Ilan University, Ramat-Gan, Israel Email address: [email protected]
Miriam Cohen, Sara Westreich
PII:
S0021-8693(19)30677-5
DOI:
https://doi.org/10.1016/j.jalgebra.2019.12.004
Reference:
YJABR 17479
To appear in:
Journal of Algebra
Received date:
7 July 2019
Please cite this article as: M. Cohen, S. Westreich, Solvable Hopf algebras and their twists, J. Algebra (2020), doi: https://doi.org/10.1016/j.jalgebra.2019.12.004.
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SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS MIRIAM COHEN AND SARA WESTREICH Abstract. We show that for any solvable group G and a Drinfel’d twist J, kGJ is solvable in the sense of the intrinsic definition of solvability given in [2]. More generally, if a Hopf algebra H has a normal solvable series so does H J . Furthermore, while solvable groups are defined as having certain commutative quotients, quasitriangular normally solvable Hopf algebras have appropriate quantum commutative quotients. We end with a detailed example.
Introduction Chains of normal left coideal subalgebras of a Hopf algebra H are analogues of chains of normal subgroups of a group. As a result, replacing normal subgroups with normal left coideal subalgebras gives a satisfactory intrinsic definition of nilpotent Hopf algebras that coincides with the categorical definition of nilpotency when applied to Hopf algebras representations. Generalizing solvability is more complicated, the basic difficulty being that there is no obvious analogue of Hopf quotients of left coideal subalgebras L over N where N ⊂ L is a left coideal subalgebra normal in L but not necessarily in H. It is customary to define solvable Hopf algebras H as those for which Rep(H) is a solvable category. However, this non-intrinsic definition is unsatisfactory as it contradicts our intuition from group theory. Commutative or nilpotent Hopf algebras are not always solvable in that sense. In [2] we suggested an intrinsic definition for solvability for semisimple Hopf algebras, which we refer to as Hopf solvability. In this case we say that the Hopf algebra H is solvable. When H = kG then Hopf solvability is equivalent to G being a solvable group. We proved there that commutative or nilpotent semisimple Hopf algebras are always solvable Hopf algebras. Denote by a, h ∈ H,
˙ ad
the left adjoint action of H on itself, that is, for all had˙ a =
h1 aS(h2 )
2000 Mathematics Subject Classification. 16T05. 1
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MIRIAM COHEN AND SARA WESTREICH
A left coideal subalgebra of H is (left) normal if it is stable under the (left) adjoint action of H on itself. Left coideal subalgebras N have integrals ΛN (see e.g [6, 7, 11, 12]), which when H is semisimple over a field of characteristic 0 can be chosen to be idempotents. In [2] we observed that a left coideal subalgebra N is normal if and only if its integral ΛN is a central element of H. We then define solvabilty as follows. Definition[2]: Let H be a semisimple Hopf algebra. A chain of left coideal subalgebras of H N0 ⊂ N1 ⊂ · · · ⊂ Nt is a solvable series if for all 0 ≤ i ≤ t − 1, (i) ΛNi ∈ Z(Ni+1 ), the center of Ni+1 , where ΛNi is the integral of Ni , (ii) For all a, b ∈ Ni+1 , (aad˙ b)ΛNi = ε, abΛNi . The Hopf algebra H is solvable if it has a solvable series so that N0 = k and Nt = H. The motivation for this definition is the description of left coideal subalgebras quotients as appropriate subalgebras of H via integrals. This realization is based on [2, Lemma 1.4] where we show the following. Lemma 0.1. [2]: For any left coideal subalgebra N, let N + = N ∩ ker ε ˜ be left coideal subalgebras and let ΛN be the integral of N. Let N ⊂ N of H, Then, ˜ /N ˜ N + as left N ˜ -modules via the natural N ˜ -module pro˜ ΛN ∼ (a) N =N + + ˜ ∼ ˜ /N N ˜ as right N ˜˜ → N ˜ /N ˜ N . Similarly, ΛN N jection π : N = N modules. ˜ then N ˜ N + = N +N ˜ and thus π ˜ (b) If ΛN is central in N |N ΛN is an algebra isomorphism. . (c) If N is normal in H then π|HΛN , where π : H → H/HN + is an algebra isomorphism. Thus, regading solvability for Hopf algebras, the requirement for the subalgebra Ni+1 ΛNi to be trivially ad˙ -acted upon Ni+1 , is the appropriate generalization of solvability for groups, where we require abelian group quotients of the form Gi+1 /Gi . Explicitly, ([2, Ex. 3.7]).
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
Example 0.2. Let G be a finite group, K ⊂ G and ΛK = If ΛK is central in G, then for g ∈ G we have, 1 1 xg = ΛK g = gΛK = gx. |K| x∈K |K| x∈K
3 1 |K|
x∈K
x.
It follows that for x ∈ K, xg = gy for some y ∈ K. Thus K is normal in G. We claim now that G/K is abelian if and only if (aad˙ b)ΛK = bΛK , for all a, b ∈ G. Indeed, assume ab = ba for all a, b ∈ G/K. Then aba−1 = by for some y ∈ K. This implies that x = by x=b x. aba−1 x∈K
−1
Conversely, if aba x∈K x = b y ∈ K. This shows our claim.
x∈K
x∈K
x∈K
x then aba−1 = by for some
One of the equivalent definitions for solvable groups is via the derived series G0 = G, G1 = [G, G], . . . Gi+1 = [Gi , Gi ], Gn = 1. In this case the subgroups {Gi } are all normal in G. This motivates the following definition Definition 0.3. A Hopf algebra is normally solvable if it is solvable with a solvable series {Ni }, so that each Ni normal in H. Equivalently, ΛNi ∈ Z(H) for all i. As was demonstrated in Example 0.2, Property (ii) of solvability is the Hopf version of Gi+1 /Gi being an abelian group. When the Hopf algebra H is normally solvable quasitriangular Hopf algebra, then Property (ii) of solvability takes on the meaning of quantum commutativity. Recall, Definition 0.4. Let (H, R) be a quasitriangular Hopf algebra and let M be a left H-module algebra. Then M is quantum commutative if ab = (R2 · b)(R1 · a) for all a, b ∈ M. Theorem 1.3: Let (H, R) be a semisimple normally solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0, with a series {Ni } of normal left coideal subalgebras. Let N0 = k, Ni+1 = Ni+1 /Ni+1 Ni+ . Then for each i ≥ 0, Ni is a quantum commutative H-module algebra. In paricular N1 is a quantum commutative normal left coideal subalgebra of H.
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MIRIAM COHEN AND SARA WESTREICH
In Section 2 we show that Normal solvablity is a categorical invariant in the following sense: Theorem 2.5: Let H be a finite dimensional Hopf algebra over an algebraically closed field k of characteritic 0. Assume H is normally solvable, then H J is normally solvable for any Drinfel’d twist J for H. Two corollaries result from Theorem 2.5 Corollary 2.6: Let G be a finite solvable group, then kGJ is a normally solvable Hopf algebra. Corollary 2.7: Any triangular Hopf algebra of dimension pa q b is normally solvable. In Section 3, Example 3.1, we explicitly describe a normally solvable twisted group algebra. 1. Solvability and quantum commutativity Throughout this paper, H is a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. We denote by S and s the antipodes of H and H ∗ respectively. When (H, R) is quasitriangular we can relate solvability to quantum commitativity. Following Radford, let HR be the minimal Hopf subalgebra of H, so that R ⊗ R ⊂ H ⊗ H [10]. Recall HR = Rl Rr where Rl = spk {R1 } and Rl = spk {R2 }. Then we have, Proposition 1.1. Let (H, R) be a semisimple solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0 with a series {Ni } of left coideal subalgebras. Assume each Ni is stable under the adjoint action of HR and HR acts trivially by the adjoint action on ΛNi . Then Ni+1 ΛNi is a quantum commutative HR -module algebra. Proof. For all a, b ∈ Ni+1 , abΛNi = (since {a3 } ⊂ Ni+1 commutes with ΛNi ) = (a1 ad˙ b)ΛNi a2 ΛNi = (R2 a2 r2 )ad˙ b ΛNi R1 a1 Sr1 ΛNi (by quasitriangularity of H) 1 2 2 = Rad R a1 Sr1 ΛNi (since HR acts trivially on ΛNI ) ˙ (rad ˙ a2 ad ˙ b)ΛNi 2 = (R2 r2 )ad˙ b R1 aSr1 ΛNi (by solvability, since a2 , (rad ˙ b) ∈ Ni+1 ) 2 1 1 (by properties of R) = Rad ˙ b ΛNi R1 aSR2 ΛNi 2 1 = (Rad ˙ b)(Rad ˙ a)ΛNi
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
5
Proposition 1.1 can be applied to any solvable series {Gi } of a group G. In this case R = 1 ⊗ 1 and quantum commutativity of kGi+1 ΛkGi is equivalent to commutativity and thus to Gi+1 /Gi being an abelian group. Recall the definition of Normal solvability given in Definition 0.3. We show an enlightening observation regarding the connection between normal solvability and quantum commutativity for quasitriangular Hopf algebras. We need first the following lemma. ˜ be normal left coideal subalgebras of H, Lemma 1.2. Let N ⊂ N + ˜ /N ˜ N + is an N = N ∩ ker ε and ΛN the integral of N. Then N = N H-module algebra via had˙ a = had˙ a. ˜ ΛN → N is a left H-module isomrphism The algebra isomorphism π : N ˜ ΛN and the action ˙ on with respect to the adjoint action of H on N ad N. ˜ N + is invariant under the Proof. The action ad˙ is well defined since N adjoint action of H. The rest is trivial. Based on Proposition 1.1 and Lemma 1.2, we conclude. Theorem 1.3. Let (H, R) be a semisimple normally solvable quasitriangular Hopf algebra over an algebraically closed field of characteristic 0, with a series {Ni } of normal left coideal subalgebras. Let N0 = k, Ni+1 = Ni+1 /Ni+1 Ni+ . Then for each i ≥ 0, Ni is a quantum commutative H-module algebra. In paricular N1 is a quantum commutative normal left coideal subalgebra of H. 2. Twisted Hopf algebra and solvability We wish to thank A. Masuoka for fruitful suggestions concerning twists of Hopf algebras. Let J = J 1 ⊗ J 2 ∈ H ⊗ H be a twist for H. That is, J is an invertible element of H ⊗ H that satisfies (Δ⊗Id)(J)(J ⊗1) = (Id ⊗Δ)(J)(1⊗J) (ε⊗Id)(J) = (Id ⊗ε)(J) = 1. Denote by H J the Hopf algebra with the same product as H and the twisted coproduct and antipode given by: ΔJ (h) = J −1 Δ(h)J where vJ =
S J (h) = vJ−1 S(h)vJ , S(J 1 )J 2
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MIRIAM COHEN AND SARA WESTREICH
Set J −1 = T =
T 1 ⊗ T 2 . Note that T satisfies
(1 ⊗ T )(Id ⊗Δ)(T ) = (T ⊗ 1)(Δ ⊗ Id)(T )
(1) Note that
vJ−1 =
T 1 S(T 2 )
By [8, (2.17)] we have, Δ(vJ ) = (S(T 2 ) ⊗ S(T 1 ))(vJ ⊗ vJ )(T 1 ⊗ T 2 ) S(J 2 )J 1 . Moreover, if H is semisimple then vJ = S(vJ ) =
(2)
Recall, when H is finite dimensional then J ∈ H ⊗ H ∼ = (H ∗ ⊗ H ∗ )∗ ∗ ∗ can be considered as a linear form σ : H ⊗H → k, which is a 2-cocycle for H ∗ . Then Hσ∗ is a Hopf algebra with new multiplication given by: a ·σ b = σ(a1 , b1 )a2 b2 σ −1 (a3 , b3 ) for all a, b ∈ H ∗ . We have (H J )∗ = (H ∗ )σ Given a left H-module V, it is known that it can be considered as a left H J -module (with respect to the same action). Denote the resulting module by J V. We have a tensor equivalence between the categories H Mod and H J Mod, V →J V with the categorical isomorphism JV
⊗J W →J (V ⊗ W ),
v ⊗ w → J −1 · (v ⊗ w).
In particular, consider H as a left H-module under the adjoint action of itself. Let (J H,ad˙ ) denote the corresponding H J -module with H as the underlying vector space and action given by the original adjoint acJ tion endowed from Δ. On the other hand consider (H J , adJ ˙ ), where H is considered as the left H J -module under the adjoint action obtained from ΔJ . Define the map: (3) ι : H → H, ι(a) = T 1 aS(T 2 )vJ for all a ∈ H. Note that for all b ∈ H, ι−1 (b) = J 1 bvJ−1 S(J 2 ) We show,
ε, ι(a) = ε, a
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
7
Lemma 2.1. Let H be any Hopf algebra (not necessarily finite dimensional) over k and let ι be defined as in (3) and consider it as a map ι : (J H, ad˙ H ) → (H J ,ad ˙ J ) Then: H (i) ι : (J H, ad˙ H ) → (H J ,ad ˙ J ) is an H J -module isomorphism. H
(ii) If N is a normal lef coideal subalgebra of H then ι(N ) a normal left coideal of H J . Proof. (i) The proof follows by a direct computation. (ii) We first show that ΔJ (ι(N )) ⊂ H J ⊗ ι(N ). Indeed, let T, t, X, Y be all copies of J −1 , then for all n ∈ N, ΔJ (ι(n)) = = T 1 t11 n1 (St2 )1 (vJ )1 J 1 ⊗ T 2 t12 n2 (St2 )2 (vJ )2 J 2 = T 1 t11 n1 (St2 )1 S(Y 2 )vJ X 1 J 1 ⊗ T 2 t12 n2 (St2 )2 S(Y 1 )vJ X 2 J 2 (by (2)) = T 1 t11 n1 (St2 )1 S(Y 2 )vJ ⊗ T 2 t12 n2 (St2 )2 s(Y 1 )vJ = t1 n1 (S(T 2 t22 ))1 S(Y 2 )vJ ⊗ T 1 t21 n2 (S(T 2 t22 ))2 s(Y 1 )vJ (by (1)) = t1 n1 S(t23 )S(T22 )S(Y 2 )vJ ⊗ T 1 t21 n2 S(t22 )S(T12 )S(Y 1 )vJ = t1 n1 S(t23 )S(T 2 )vJ ⊗ Y 1 T11 t21 n2 S(t22 )S(Y 2 T21 )vJ (by (1)) = t1 n1 S(t22 )S(T 2 )vJ ⊗ Y 1 (T 1 t21 ad˙ n2 )S(Y 2 )vJ = t1 n1 S(t22 )S(T 2 )vJ ⊗ ι(T 1 t21 ad˙ n2 ) Normality of N implies (T 1 t2 )ad˙ n2 ∈ N, hence the result follows. Part (i) implies now that ι(N ) is normal in H J .
Recall[12], for any Hopf algebra H, if π : H → H is a Hopf algebra projection, then H coH = {h ∈ H|h1 ⊗ π(h2 ) = h ⊗ 1} H . Observe is a normal left coideal subalgebra of H of dimension dim dim H coH that H is the unique maximal left coideal N of H, satisfying π(n) = ε(n) for all n ∈ N. We show,
Proposition 2.2. Let N be a normal left coideal subalgebra of a finite dimensional Hopf algebra H and J a twist for H. Then ι(N ) is a normal left coideal subalgebra of H J .
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MIRIAM COHEN AND SARA WESTREICH
Proof. By Lemma 2.1, ι(N ) is a left coideal of H J . It is left to show that it is an algebra. To see that, note first that if I is a Hopf ideal of H then it is a Hopf ideal of H J as well. Indeed, ΔJ (I) = J −1 Δ(I)J ⊂ J −1 (H ⊗ I + I ⊗ H)J ⊂ H ⊗ I + I ⊗ H S J (I) = vJ−1 S(I)vJ ⊂ vJ−1 IvJ = I In particular, HN + is a Hopf ideal of H J , implying that H J /HN + is a Hopf image of H J . We wish to show that ι(N ) = (H J )co(H
J /HN + )
which would imply that ι(N ) is an algebra as well. Now, for each a = ι(n), n ∈ N + , we have by the definition of ι, π(a) = 0. It follows that for all b ∈ ι(N ), π(b) = ε(b). This implies that ι(N ) ⊂ (H J )co(H
J /HN + )
.
Equality of dimensions implies the desired result.
Remark 2.3. Observe that ι(hz) = ι(h)z for all h ∈ H, z ∈ Z(H). It follows in particular that if N is a normal left coideal subalgebra of H with an integral ΛN , (which is central in H), then ΛN is the integral of ι(N ) as well. For the next corollary of Lemma 2.1, recall the following: Let V be an H-module and J V, the corresponding H J -module. One can consider LKerV ⊂ H which is a normal left coideal subalgebra of H, while LKerJ V ⊂ H J is a normal left coideal subalgebra of H J . We have, Corollary 2.4. Let V be an H-module, then ι(LKerV ) = LKerJ V . Proof. Since ι(LKerV ) is a normal left coideal, it is enough to show that ι(LKerV ) acts trivially on J V. Indeed, for all a ∈ LKerV , w ∈J V, we have, ι(a) · w = T 1 aS(T 2 )vJ · w = ε, a T 1 S(T 2 )vJ · w = ε, a · w. The reverse inclusion follows similarly by using ι−1 .
We use the results above to show that normal solvability of H, is invariant under twists. Theorem 2.5. Let H be a semisimple Hopf algebra over an algebraically closed field of characteristic 0. If H is normally solvable so is H J for any twist J of H.
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
9
Proof. Let k = N0 ⊂ N1 ⊂ · · · ⊂ Nt = H be a solvable series consisting of normal left coideal subalgebras of H. We claim that k = ι(N0 ) ⊂ ι(N1 ) ⊂ · · · ⊂ ι(Nt ) = H J is a solvable series consisting of normal left coideal subalgebra of H J . Property (i) of solvability, that is, each ι(Ni ) is a normal left coideal subalgebra of H J , follows from Proposition 2.2. By remark 2.3, {ΛNi } are the corresponding integrals of ι(Ni ). We prove now property (ii) of solvability. We need to show that for all i and for all a, b ∈ Ni , ι(a)ad ˙ J ι(b)ΛNi−1 = ε, a ι(b)ΛNi−1 . H
Set Vi = Ni ΛNi−1 , an H-module under the left adjoint action and Ki = LKerVi . Since H is solvable we have Ni ⊂ Ki . By Corollary 2.4, we have ι(Ni ) ⊂ ι(Ki ) = LKer(J Vi ,adH ) . That is, for all a, b ∈ Ni , ι(a)ad˙ bΛNi−1 = ε, a bΛNi−1 . Apply ι to both sides, we have by Lemma 2.1(1), ι(a)ad ˙ J ι(b)ΛNi−1 = ε, a ι(b)ΛNi−1 . H
This concludes the proof of the theorem.
Since finite solvable groups are normally solvabe, we conclude: Corollary 2.6. Let G be a finite solvable group, then kGJ is a normally solvable Hopf algebra. Another corollary is the following: Corollary 2.7. Let H be a semisimple triangular Hopf algebra of dimension pa q b , then H is normally solvable. Proof. By [3] H = (kG)J where G of order pa q b and J is a twist for kG. The result follows now from Burnside theorem for groups and Theorem 2.5.
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3. Example In what follows we give an explicit example of a normally solvable Hopf algebra. Example 3.1. Let G = S3 × S3 . We show explicitly how kGJ is a solvable Hopf algebra. For this explicit example it was proved in [5] that kGJ is a simple Hopf algebra in the sense that it has no normal Hopf subalgebra (thus it is not solvable in any sense of [9]). Let S3 = {1, a, a2 , b, ab, a2 b}, a3 = b2 = 1, ab = ba2 }. Then S3 = ZZ3 ZZ2 where ZZ3 = {1, a, a2 } and ZZ2 = {1, b, b2 }. Let S3 be another copy of S3 with generators {a , b }. The group G is solvable with a solvable chain 1 ZZ3 S3 (S3 × ZZ3 ) (S3 × S3 ). Following [5, 2.3], a non-trivial twist on kG can be lifted from a twist J on the abelian, non-normal subgroup ZZ2 × ZZ2 . Calculating the appropriate twist we arrive at 1 J = ((1 + b) ⊗ 1 + (1 − b) ⊗ b ) . 2 Then J 2 = 1 ⊗ 1 and so J = J −1 = T. Also set v = vj , then, 1 v = v −1 = S(J 1 )J 2 = (1 + b + b − bb ) 2 For any g ∈ G we have now, ΔJ (g) = J(g ⊗ g)J = 1 = ((1 + b)g(1 + b) ⊗ g + (1 + b)g(1 − b) ⊗ gb 4 + (1 − b)g(1 + b) ⊗ b g + (1 − b)g(1 − b) ⊗ b gb ) So we have, 1 ΔJ (a) = ((a + a2 ) ⊗ a + (a − a2 ) ⊗ ab ) 2 1 2 ΔJ (a ) = (a (1 + b) ⊗ a + a (1 − b) ⊗ a ) 2 ΔJ (b) = b ⊗ b, ΔJ (b ) = b ⊗ b . Some particular examples of S J , 1 1 S J (a) = (a(1 − b ) + a2 (1 + b ), S J (a ) = (a (1 − b) + a2 (1 + b)) 2 2 S J (a + a2 ) = a + a2 , S J (a2 − a) = (a − a2 )b
SOLVABLE HOPF ALGEBRAS AND THEIR TWISTS
11
S J (b) = b, S J (b ) = b . Let us start with the minimal normal left coideal subalgebra of kG, N1 = kZZ3 . Then ι(1) = 1 and calculating ι for the other elements yields 1 ι(a) = (a + a2 + (a − a2 )b ), 2 It follows that
1 ι(a2 ) = (a + a2 + (a − a2 )b ) 2
ι(N1 ) = Spk {1, a + a2 , (a − a2 )b } It can be directly checked that ι(N1 ) acts-ad˙ trivially on itself. Regarding quantum commutativity, recall, kGJ is quasitriangular with 1 R = (J −1 )τ J = ((1+b)⊗(1+b)+(1−b)⊗(1+b)b +b (1+b)⊗(1−b)+b (1−b)⊗(1−b)b ). 4 Indeed, Theorem 1.3 is verified, that is, xy = (R2 · y)(R1 · x) for all x, y ∈ ι(N1 ). Our next object in the original chain is the normal left coideal subalgebra N2 = kS3 . Then ι(b) = b and checking the other elements yields ι(N2 ) = spk {1, a + a2 , (a − a2 )b , b, (a + a2 )b, (a − a2 )b b} Since Λι(N1 ) = ΛN1 = 1 + a + a2 we obtain that ι(N2 )Λι(N1 ) = spk {ΛN1 , bΛN1 } Observe that the Hopf subalgebra generated by the right and left tensorands of R is the group algebra (of (kG)J ) - k{1, b, b , bb }. This subgroup commutes with ι(N2 ) hence acts-ad˙ trivially on it. It follows that 1 − b acts on ι(N2 ) as 0, while 1 + b acts as 2, hence for all x, y ∈ ι(N2 )ΛN1 , we have 1 2 1 (Rad ˙ y)((1 + b)ad ˙ x) = yx = xy ˙ y)(Rad ˙ x) = ((1 + b)ad 4 Similarly, let N3 = k(S3 × ZZ3 ), then ι(N3 ) = ι(N2 ) × k{1, a , a2 } Since Λι(N2 ) = ΛN2 = ΛS3 it follows that ι(N3 )Λι(N2 ) = {ΛS3 , ΛS3 a , ΛS3 a2 } Since k{1, b, b , bb } commutes with ι(N3 ) which is (regularly) commutative, the same argument as above works.
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MIRIAM COHEN AND SARA WESTREICH
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