Cli ord theory for semisimple Hopf algebras · Blattner worked a similar theory for Lie algebras ( [5],’69) Dade generalized Clifford’s results to graded rings ([3], ’80) Rieffel

Clifford theory for semisimple Hopf algebrasSebastian Burciu (Institute of Mathematics ”Simion Stoilow” of the Romanian

Academy, Romania)

[email protected]

The classical Clifford correspondence for normal subgroups is considered in thesetting of semisimple Hopf algebras. We prove that this correspondence still holdsif the extension determined by the normal Hopf subalgebra is cocentral. Otherparticular situations where Clifford theory also works will be discussed. This talk isbased on the paper ”Clifford theory for cocentral extensions” Israel J. Math, 181,2011, (1), 111-123 and some work in progress of the author.

Motivation of the talkRieffel’s generalization for semisimple artin algebras

New results obtained: stabilizers as Hopf subalgebrasApplications

A counterexample

Clifford theory for semisimple Hopf algebrasHopf algebras and Tensor categories,

University of Almeria (Spain), July 4-8, 2011

Sebastian Burciu

Institute of Mathematics ”Simion Stoilow” ofRomanian Academy

Sebastian Burciu Clifford theory for semisimple Hopf algebras



A counterexample

Historical background

A. H. Clifford initiated the theory for groups [[1], ’37].Blattner worked a similar theory for Lie algebras ( [5],’69)Dade generalized Clifford’s results to graded rings ([3], ’80)Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample


A. H. Clifford initiated the theory for groups [[1], ’37].

Blattner worked a similar theory for Lie algebras ( [5],’69)Dade generalized Clifford’s results to graded rings ([3], ’80)Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample


A. H. Clifford initiated the theory for groups [[1], ’37].Blattner worked a similar theory for Lie algebras ( [5],’69)

Dade generalized Clifford’s results to graded rings ([3], ’80)Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample


A. H. Clifford initiated the theory for groups [[1], ’37].Blattner worked a similar theory for Lie algebras ( [5],’69)Dade generalized Clifford’s results to graded rings ([3], ’80)

Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample


A. H. Clifford initiated the theory for groups [[1], ’37].Blattner worked a similar theory for Lie algebras ( [5],’69)Dade generalized Clifford’s results to graded rings ([3], ’80)Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)

Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample


A. H. Clifford initiated the theory for groups [[1], ’37].Blattner worked a similar theory for Lie algebras ( [5],’69)Dade generalized Clifford’s results to graded rings ([3], ’80)Rieffel extended the theory for an arbitrary normalextension of semisimple artin rings. ([8], ’79)Schneider unified the existent theories in a more generalsetting, that of Hopf Galois extensions ([2], ’90).

Witherspoon used Rieffel’s work in the setting of finitedimensional Hopf algebras. ([4], ’99 and [6], ’02).




A counterexample






A counterexample






A counterexample

Plan of the talk

1 Motivation of the talk

2 Rieffel’s generalization for semisimple artin algebras

3 New results obtained: stabilizers as Hopf subalgebras

4 ApplicationsExtensions by kF .The Drinfeld double D(A)

5 A counterexample




A counterexample

Motivation

Let A be a semisimple Hopf algebra and let B be a Hopfsubalgebra of A.

B is called a normal Hopf subalgebra if it is closed under theadjoint action of A on itself, i.e a1BS(a2) ⊂ B for all a ∈ A.

Clifford theory for normal Hopf subalgebrasLet B ⊂ A be a normal Hopf subalgebra of the semisimple Hopfalgebra A. For a given irreducible B-module M,find a Hopf subalgebra ZM of A with B ⊂ ZM ⊂ A such that theinduction map ind : Irr(ZM |M)→ Irr(A|M) given byV 7→ A⊗ZM V is a bijection.

When such a Hopf sualgebra ZM exists?Based on Isr. J. Math, 2011 and some work in progress.




A counterexample

Motivation

Let A be a semisimple Hopf algebra and let B be a Hopfsubalgebra of A.B is called a normal Hopf subalgebra if it is closed under theadjoint action of A on itself, i.e a1BS(a2) ⊂ B for all a ∈ A.






A counterexample

Motivation


Clifford theory for normal Hopf subalgebrasLet B ⊂ A be a normal Hopf subalgebra of the semisimple Hopfalgebra A. For a given irreducible B-module M,

find a Hopf subalgebra ZM of A with B ⊂ ZM ⊂ A such that theinduction map ind : Irr(ZM |M)→ Irr(A|M) given byV 7→ A⊗ZM V is a bijection.





A counterexample

Motivation


Clifford theory for normal Hopf subalgebrasLet B ⊂ A be a normal Hopf subalgebra of the semisimple Hopfalgebra A. For a given irreducible B-module M,find a Hopf subalgebra ZM of A with B ⊂ ZM ⊂ A such that theinduction map

ind : Irr(ZM |M)→ Irr(A|M) given byV 7→ A⊗ZM V is a bijection.





A counterexample

Motivation


Clifford theory for normal Hopf subalgebrasLet B ⊂ A be a normal Hopf subalgebra of the semisimple Hopfalgebra A. For a given irreducible B-module M,find a Hopf subalgebra ZM of A with B ⊂ ZM ⊂ A such that theinduction map ind : Irr(ZM |M)→ Irr(A|M) given by

V 7→ A⊗ZM V is a bijection.





A counterexample

Motivation


Clifford theory for normal Hopf subalgebrasLet B ⊂ A be a normal Hopf subalgebra of the semisimple Hopfalgebra A. For a given irreducible B-module M,find a Hopf subalgebra ZM of A with B ⊂ ZM ⊂ A such that theinduction map ind : Irr(ZM |M)→ Irr(A|M) given byV 7→ A⊗ZM V

is a bijection.





A counterexample

Motivation







A counterexample

Motivation



When such a Hopf sualgebra ZM exists?

Based on Isr. J. Math, 2011 and some work in progress.




A counterexample

Motivation







A counterexample

Rieffel’s work on semisimple normal extensions

Definition of normal subrings:Let B ⊂ A an extension of semisimple rings. The extension iscalled normal if A(I ∩ B) = (I ∩ B)A for any maximal ideal I of A.

If B = kH ⊂ A = kG for two finite groups H ⊂ G then theextension B ⊂ A is normal if and only if H is a normal subgroupof G.

Normal extensions of Hopf algebrasMore generally the same thing is true for semisimple Hopfalgebras.




A counterexample








A counterexample








A counterexample

Rieffel’s definition for the stabilizer

Stabilizer of a simple module W

Let B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).

The stabilizer always exists but is not unique; there might bemore then one stabilizers for a given module W .




A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,

and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module.

Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,

(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ

(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A,

whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample


Stabilizer of a simple module WLet B ⊂ A be a normal extension of semisimple artin rings,and W be an irreducible B-module. Then a stability subring forW is a semisimple artin subring T , with B ⊂ T ⊂ A and(1) B is a normal subring of T ,(2) J is T -invariant, i.e JT = TJ(3) T + AJ + JA = A, whereJ := AnnB(W ).





A counterexample

Clifford’s correspondence holds in Rieffel’s sense

THEOREM (Rieffel, 1979.)Let B be a normal subring of the semi-simple artin ring A, let Wbe an irreducible B-module, and let T be a stability subring forW .

Then the process of inducing modules from T to A gives abijection between equivalence classes of simple T -moduleshavingW as (the) B-constituent and equivalence classes ofsimple A-modules having W as (a) B-constituent.




A counterexample







A counterexample







A counterexample







A counterexample

More on Rieffel’s work on semisimple normalextensions B ⊂ A

Let B ⊂ A be an extension of semisimple rings.

Rieffel’s equivalence relation on the set of irreducibleA-representations Irr(A).

Write M ∼A N if there is an irreducible W ∈ B −mod , commonconstituent both for M ↓AB and N ↓AB.In general ∼A is not an equivalence relation.




A counterexample








A counterexample








A counterexample




Write M ∼A N if there is an irreducible W ∈ B −mod , commonconstituent both for M ↓AB and N ↓AB.

In general ∼A is not an equivalence relation.




A counterexample








A counterexample








A counterexample

More on Rieffel’s work on semisimple normalextensions B ⊂ A, II

Rieffel’s equivalence relation on the set of irreducibleB-representations Irr(B).

Write W ∼B W ′ if there is an irreducible M ∈ A−mod ,common constituent both for W ↑AB and W ′ ↑AB. In general ∼B isnot an equivalence relation.

Theorem (Rieffel, [8])An extension B ⊂ A of semisimple artin ring is a normalextension if and only if ∼A is an equivalence relation andIrr(M ↓AB) is an entire equivalence class for all M ∈ Irr(A).




A counterexample



Write W ∼B W ′ if there is an irreducible M ∈ A−mod ,common constituent both for W ↑AB and W ′ ↑AB.

In general ∼B isnot an equivalence relation.





A counterexample








A counterexample




Theorem (Rieffel, [8])An extension B ⊂ A of semisimple artin ring is a normalextension if and only if

∼A is an equivalence relation andIrr(M ↓AB) is an entire equivalence class for all M ∈ Irr(A).




A counterexample








A counterexample








A counterexample

Characterization of Rieffel’s notion of normalextensions

Theorem (B, Kadison, Kuelshammer, [9])Let B ⊂ A be an extension of semisimple finite dimensionalalgebras. The relation ∼A is an equivalence relation if and onlyif B ⊂ A is a depth three extension.

Theorem (B, Kadison, Kuelshammer, [9])An extension B ⊂ A of finite dimensional semisimple algebrasis a normal extension if and only if it is a depth two extension.




A counterexample


Theorem (B, Kadison, Kuelshammer, [9])Let B ⊂ A be an extension of semisimple finite dimensionalalgebras.

The relation ∼A is an equivalence relation if and onlyif B ⊂ A is a depth three extension.





A counterexample







A counterexample







A counterexample







A counterexample

Rieffel’s equivalence relations in our settings

Let B ⊂ A an extension of normal ss. Hopf algebras.

Some notationsLet Bi be an equivalence class under Rieffel’s equivalencerelation for B ⊂ A on Irr(B) .Let Ai be the corresponding equivalence class on Irr(A).Let

bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample




bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample



Some notations

Let Bi be an equivalence class under Rieffel’s equivalencerelation for B ⊂ A on Irr(B) .Let Ai be the corresponding equivalence class on Irr(A).Let

bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample



Some notationsLet Bi be an equivalence class under Rieffel’s equivalencerelation for B ⊂ A on Irr(B) .

Let Ai be the corresponding equivalence class on Irr(A).Let

bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample



Some notationsLet Bi be an equivalence class under Rieffel’s equivalencerelation for B ⊂ A on Irr(B) .Let Ai be the corresponding equivalence class on Irr(A).

Letbi =

∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample




bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample




bi =∑β∈Bi

β(1)β.

Letai =

∑χ∈Ai

χ(1)χ.




A counterexample

Generalization of the first theorem of CliffordConjugate modules

Let M be an irreducible B-module with character α ∈ C(B).

If W is an A∗-module then W ⊗M becomes a B-modulewith

b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)

Here we used that any left A∗-module W is a rightA-comodule via ρ(w) = w0 ⊗ w1.For any irreducible character d ∈ Irr(A∗) associated to asimple A-comodule W define dM := W ⊗M as aB-module.




A counterexample


Let M be an irreducible B-module with character α ∈ C(B).If W is an A∗-module then W ⊗M becomes a B-modulewith

b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)





A counterexample



b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)





A counterexample



b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)

Here we used that any left A∗-module W is a rightA-comodule via ρ(w) = w0 ⊗ w1.

For any irreducible character d ∈ Irr(A∗) associated to asimple A-comodule W define dM := W ⊗M as aB-module.




A counterexample



b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)





A counterexample



b(w ⊗m) = w0 ⊗ (S(w1)bw2)m (1)





A counterexample

Conjugate modules and stabilizers

Generalization of the first theorem of CliffordTheorem (B. ’10) Let B ⊂ A be a normal extension ofsemisimple Hopf algebras and M be an irreducible B-module.Then M ↑AB↓

AB and ⊕d∈Irr(A∗)

dM have the same irreducibleB-constituents.

For A = kG one has Irr(kG∗) = G.If A = kG and B = kH for a normal subgroup H then d = g ∈ Gand dM coincides with the conjugate module gM introduced byClifford in [1].




A counterexample









A counterexample





For A = kG one has Irr(kG∗) = G.

If A = kG and B = kH for a normal subgroup H then d = g ∈ Gand dM coincides with the conjugate module gM introduced byClifford in [1].




A counterexample









A counterexample









A counterexample

Clifford correspondence for normal Hopf algebrasThe case when the stabilizer is a Hopf subalgebra

If α is the char. of M then the char. dα of dM is given bydα(x) = α(Sd1xd2) (2)

for all x ∈ B (see Proposition 5.3 of [10]).

Proposition (B, ’11)

The set {d ∈ Irr(A∗) | dα = ε(d)α} is closed undermultiplication and “ ∗ ”. Thus it generates a Hopf subalgebraZα of A that contains B.

Zα is called the stabilizer of α in A.If A = kG and B = kN for a normal subgroup N then thestabilizer Zα coincides with the stabilizer ZM of Mintroduced by Clifford in [1].




A counterexample










A counterexample










A counterexample





The set {d ∈ Irr(A∗) | dα = ε(d)α} is closed undermultiplication and “ ∗ ”.

Thus it generates a Hopf subalgebraZα of A that contains B.





A counterexample










A counterexample






Zα is called the stabilizer of α in A.

If A = kG and B = kN for a normal subgroup N then thestabilizer Zα coincides with the stabilizer ZM of Mintroduced by Clifford in [1].




A counterexample










A counterexample










A counterexample

Clifford correspondence for normal Hopf algebrasThe case of the stabilizer Zα

On the dimension of the stabilizer (B, ’11)Let B ⊂ A a normal extension of semisimple Hopf algebras.With the above notations:

1 |Zα| ≤ |A|α(1)2

bi (1)where Bi is the equivalence class of α.

2 Equality holds if and only if Zα is a stabilizer in the sense ofRieffel, see [4].




A counterexample


On the dimension of the stabilizer (B, ’11)

Let B ⊂ A a normal extension of semisimple Hopf algebras.With the above notations:

1 |Zα| ≤ |A|α(1)2






A counterexample



1 |Zα| ≤ |A|α(1)2






A counterexample



1 |Zα| ≤ |A|α(1)2






A counterexample



1 |Zα| ≤ |A|α(1)2






A counterexample



1 |Zα| ≤ |A|α(1)2






A counterexample

Clifford correspondence for normal Hopf algebrasMain result

Theorem (B, ’11)Clifford correspondence holds for Zα if and only if one hasequality in the previous inequality.

Corrolary (B, 11’.)Clifford theory works for the stabilizer Zα if and only if this is astabilizer in Rieffel’s sense.




A counterexample







A counterexample







A counterexample







A counterexample


Back to the group case

If A = kG and B = kN for a normal subgroup N then the aboveinequality is equality.

It states that the number of conjugatemodules of α is the index of the stabilizer of α in G. (Orbitformula)




A counterexample



If A = kG and B = kN for a normal subgroup N then the aboveinequality is equality. It states that the number of conjugatemodules of α is the index of the stabilizer of α in G.

(Orbitformula)




A counterexample



If A = kG and B = kN for a normal subgroup N then the aboveinequality is equality. It states that the number of conjugatemodules of α is the index of the stabilizer of α in G. (Orbitformula)




A counterexample

Extensions by kF .The Drinfeld double D(A)

Outline





5 A counterexample




A counterexample


Applying Schneider’s work to our settingsThe case H = kF .

Let H := A//B and suppose that H = kF for a finite group F .Let Af := ρ−1(A⊗ kf ), for all f ∈ F .

Schneider’s stabilizer when H = kFThe stabilizer Z of M is the set of all f ∈ F such thatAf ⊗B M ∼= M as B-modules. It is a subgroup of F .

Let S := ρ−1(A⊗ Z ). Then S is a subalgebra of AS is not a Hopf subalgebra in general.Clifford correspondence holds for S as a stabilizer of M.




A counterexample



Let H := A//B and suppose that H = kF for a finite group F .

Let Af := ρ−1(A⊗ kf ), for all f ∈ F .






A counterexample




Schneider’s stabilizer when H = kF

The stabilizer Z of M is the set of all f ∈ F such thatAf ⊗B M ∼= M as B-modules. It is a subgroup of F .





A counterexample









A counterexample





Let S := ρ−1(A⊗ Z ). Then S is a subalgebra of A

S is not a Hopf subalgebra in general.Clifford correspondence holds for S as a stabilizer of M.




A counterexample





Let S := ρ−1(A⊗ Z ). Then S is a subalgebra of AS is not a Hopf subalgebra in general.

Clifford correspondence holds for S as a stabilizer of M.




A counterexample









A counterexample


Extensions by kF .

Theorem (B,’11).Suppose that H = kF for some finite group F .

Let M be anirreducible representation of B with character α and let Z ≤ Fbe the stabilizer of M.

1) Then Zα ⊂ S.

2)Clifford correspondence holds for Zα if and only if Zα = S.

3) Clifford correspondence holds for Zα if and only S is a Hopfsubalgebra of A.




A counterexample


Extensions by kF .

Theorem (B,’11).Suppose that H = kF for some finite group F . Let M be anirreducible representation of B with character α and

let Z ≤ Fbe the stabilizer of M.

1) Then Zα ⊂ S.






A counterexample


Extensions by kF .

Theorem (B,’11).Suppose that H = kF for some finite group F . Let M be anirreducible representation of B with character α and let Z ≤ Fbe the stabilizer of M.

1) Then Zα ⊂ S.






A counterexample


Extensions by kF .


1) Then Zα ⊂ S.






A counterexample


Extensions by kF .


1) Then Zα ⊂ S.






A counterexample


Extensions by kF .


1) Then Zα ⊂ S.






A counterexample


Extensions by kF .


1) Then Zα ⊂ S.






A counterexample


A Corollary

Corollary (B, ’11)Suppose that the extension

k −−−−→ B i−−−−→ A π−−−−→ H −−−−→ k

is cocentral. Then the Clifford correspondence holds for ZM forany irreducible B-module M.




A counterexample


A Corollary


k −−−−→ B i−−−−→ A π−−−−→ H −−−−→ k





A counterexample


A Corollary


k −−−−→ B i−−−−→ A π−−−−→ H −−−−→ k

is cocentral.

Then the Clifford correspondence holds for ZM forany irreducible B-module M.




A counterexample


A Corollary


k −−−−→ B i−−−−→ A π−−−−→ H −−−−→ k





A counterexample


Outline





5 A counterexample




A counterexample


Application of Clifford’s correspondence for normalHopf algebrasThe Drinfeld double case

Definition of K (A)

Let K (A) = kG∗ be the largest central Hopf subalgebra of A.Then G is the universal grading group of Rep(A).

Theorem (B, 11’.)

K (A) is a normal Hopf subalgebra of D(A).




A counterexample



Definition of K (A)


Theorem (B, 11’.)





A counterexample



Definition of K (A)


Theorem (B, 11’.)





A counterexample



Definition of K (A)


Theorem (B, 11’.)





A counterexample


Application to D(A)

Theorem (B, 11’.)Clifford correspondence holds for the extensionK (A) ⊂ D(A).

This gives that any irreducible D(A)-module hasthe following form A⊗L(g) V where g ∈ G, L(g) is a Hopfsubalgebra of A containing K (A) and V is an irreducibleL(g)-module.

Drinfeld double D(G) of a group

If A = kG∗ then K (A) = A and the previous theorem gives thewell known description of the irreducible modules over D(G) interms of the centralizers CG(g).




A counterexample


Application to D(A)

Theorem (B, 11’.)Clifford correspondence holds for the extensionK (A) ⊂ D(A).This gives that any irreducible D(A)-module hasthe following form

A⊗L(g) V where g ∈ G, L(g) is a Hopfsubalgebra of A containing K (A) and V is an irreducibleL(g)-module.






A counterexample


Application to D(A)

Theorem (B, 11’.)Clifford correspondence holds for the extensionK (A) ⊂ D(A).This gives that any irreducible D(A)-module hasthe following form A⊗L(g) V where g ∈ G,

L(g) is a Hopfsubalgebra of A containing K (A) and V is an irreducibleL(g)-module.






A counterexample


Application to D(A)

Theorem (B, 11’.)Clifford correspondence holds for the extensionK (A) ⊂ D(A).This gives that any irreducible D(A)-module hasthe following form A⊗L(g) V where g ∈ G, L(g) is a Hopfsubalgebra of A containing K (A) and

V is an irreducibleL(g)-module.






A counterexample


Application to D(A)

Theorem (B, 11’.)Clifford correspondence holds for the extensionK (A) ⊂ D(A).This gives that any irreducible D(A)-module hasthe following form A⊗L(g) V where g ∈ G, L(g) is a Hopfsubalgebra of A containing K (A) and V is an irreducibleL(g)-module.






A counterexample


Application to D(A)







A counterexample


Application to D(A)



If A = kG∗ then

K (A) = A and the previous theorem gives thewell known description of the irreducible modules over D(G) interms of the centralizers CG(g).




A counterexample


Application to D(A)



If A = kG∗ then K (A) = A and

the previous theorem gives thewell known description of the irreducible modules over D(G) interms of the centralizers CG(g).




A counterexample


Application to D(A)







A counterexample

A Counterexample

Exact factorization of groups

Let Σ = FG be an an exact factorization of finite groups. Thisgives a right action C : G × F → G of F on the set G, and a leftaction B : G × F → F of G on the set F subject to the followingtwo conditions:

s B xy = (s B x)((s C x) B y) st C x = (s C (t B x))(t C x)

The actions B and C are determined by the relationsgx = (g B x)(g C x) for all x ∈ F , g ∈ G. Note that 1 B x = xand s C 1 = s.




A counterexample

A Counterexample


Let Σ = FG be an an exact factorization of finite groups.

Thisgives a right action C : G × F → G of F on the set G, and a leftaction B : G × F → F of G on the set F subject to the followingtwo conditions:






A counterexample

A Counterexample


Let Σ = FG be an an exact factorization of finite groups. Thisgives a right action C : G × F → G of F on the set G, and

a leftaction B : G × F → F of G on the set F subject to the followingtwo conditions:






A counterexample

A Counterexample








A counterexample

A Counterexample








A counterexample

A Counterexample








A counterexample

A Counterexample








A counterexample

Definition of the Hopf algebra A

Definition of the smashed product

Consider the Hopf algebra A = kG#kF which is a smashedproduct and coproduct using the above two action. Thestructure of A is given by:

(δgx)(δhy) = δgCx ,hδgxy

∆(δgx) =∑st=g

δs(t B x)⊗ δtx

Then A fits into the abelian extension

k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample



Consider the Hopf algebra A = kG#kF which is a smashedproduct and coproduct using the above two action.

Thestructure of A is given by:


∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample





∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample





∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample





∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample





∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ k




A counterexample





∆(δgx) =∑st=g

δs(t B x)⊗ δtx


k −−−−→ kG i−−−−→ A π−−−−→ kF −−−−→ kSebastian Burciu Clifford theory for semisimple Hopf algebras



A counterexample

A counterexample

As above F acts on Irr(kG) = G. It is easy to see that thisaction is exactly C.Let g ∈ G and Z be the stabilizer of g under C. Using theabove notations it follows that S = A(Z ) = kG#kZ .Remark that the above comultiplication formula implies S is aHopf subalgebra if and only if G B Z ⊂ Z .

Constructing the counterexample

Consider the exact fact factorization S4 = C4S3 where C4 isgenerated by the four cycle g = (1234) and S3 is given by thepermutations that leave 4 fixed. If t = (12) and s = (123) thenthe actions C and B are given in Tables 1 and 2.




A counterexample

A counterexample

As above F acts on Irr(kG) = G. It is easy to see that thisaction is exactly C.

Let g ∈ G and Z be the stabilizer of g under C. Using theabove notations it follows that S = A(Z ) = kG#kZ .Remark that the above comultiplication formula implies S is aHopf subalgebra if and only if G B Z ⊂ Z .






A counterexample

A counterexample

As above F acts on Irr(kG) = G. It is easy to see that thisaction is exactly C.Let g ∈ G and Z be the stabilizer of g under C. Using theabove notations it follows that S = A(Z ) = kG#kZ .

Remark that the above comultiplication formula implies S is aHopf subalgebra if and only if G B Z ⊂ Z .






A counterexample

A counterexample

As above F acts on Irr(kG) = G. It is easy to see that thisaction is exactly C.Let g ∈ G and Z be the stabilizer of g under C. Using theabove notations it follows that S = A(Z ) = kG#kZ .

Remark that the above comultiplication formula implies S is aHopf subalgebra if and only if G B Z ⊂ Z .






A counterexample

A counterexample







A counterexample

A counterexample







A counterexample

A counterexample



Consider the exact fact factorization S4 = C4S3 where C4 isgenerated by the four cycle g = (1234) and S3 is given by thepermutations that leave 4 fixed.

If t = (12) and s = (123) thenthe actions C and B are given in Tables 1 and 2.




A counterexample

A counterexample







A counterexample

A counterexample







A counterexample

C4 C S3 g g2 g3

t g g3 g2

s g2 g3 g

s2 g3 g g2

st g3 g2 g

ts g2 g g3

Table: The right action of S3 on C4

C4 B S3 t s s2 st ts

g ts t s st s2

g2 s2 ts t st s

g3 s s2 ts st t

Table: The left action of C4 on S3




A counterexample

C4 C S3 g g2 g3

t g g3 g2

s g2 g3 g

s2 g3 g g2

st g3 g2 g

ts g2 g g3



g ts t s st s2

g2 s2 ts t st s

g3 s s2 ts st t

Table: The left action of C4 on S3Sebastian Burciu Clifford theory for semisimple Hopf algebras



A counterexample

C4 C S3 g g2 g3

t g g3 g2

s g2 g3 g

s2 g3 g g2

st g3 g2 g

ts g2 g g3



g ts t s st s2

g2 s2 ts t st s

g3 s s2 ts st t

Table: The left action of C4 on S3Sebastian Burciu Clifford theory for semisimple Hopf algebras



A counterexample

The counterexample

The stabilizer of the element g is the subgroup Z = {1, t}

whichis not invariant by the action of C4. Thus the Cliffordcorrespondence does not hold for g.




A counterexample

The counterexample

The stabilizer of the element g is the subgroup Z = {1, t} whichis not invariant by the action of C4.

Thus the Cliffordcorrespondence does not hold for g.




A counterexample

The counterexample

The stabilizer of the element g is the subgroup Z = {1, t} whichis not invariant by the action of C4. Thus the Cliffordcorrespondence does not hold for g.




A counterexample

The counterexample

The stabilizer of the element g is the subgroup Z = {1, t} whichis not invariant by the action of C4. Thus the Cliffordcorrespondence does not hold for g.




A counterexample

References I

C. Clifford.Handbook of Everything.Annals of Math., 38: 533-550, 1937.

H. J. Schneider.,Representation theory of Hopf Galois extensions .Israel J. Math. 1990.@

E. C. Dade,Group-graded rings and modules.Math. Z. 1980.




A counterexample

References II

S. Witherspoon.Clifford correspondence for algebras.Journal of Algbera, 218(1):608–620, 1999.

R. J. Blattner.Induced and produced representation of Lie algebras.Trans. A. M. S. (141):457-474,1969.@

S. Witherspoon.Clifford correspondence for finite-dimensional HopfalgebrasJ. Algebra, 256(2):518–530, 2002.




A counterexample

References III

Y. Zhu.Hopf algebras of prime dimension .Int. Math. Res. Not.. 1,:53-59, 1994.

M. Rieffel.Normal subrings and induced representations.Journal of Algebra, 59(1):364–386, 1979.

S. Burciu, L. Kadison, B. Kuelshammer.On soubgroup depth.Inernat. Electron Journal of Algebra, 9(11):133–166, 2011.




A counterexample

References IV

S. Burciu.Coset decomposition for semisimple Hopf subalgebras.Commun. Algebra, 37(10):3573–3585, 2009.




A counterexample

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A counterexample

Th

ank you!




A counterexample

Tha

nk you!




A counterexample

Than

k you!




A counterexample

Thank

you!




A counterexample

Thank y

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A counterexample

Thank yo

u!




A counterexample

Thank you!


Cli ord theory for semisimple Hopf algebras · Blattner worked a similar theory for Lie algebras ( [5],’69) Dade generalized Clifford’s results to graded rings ([3], ’80) Rieffel

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