Clifford theory for semisimple Hopf algebras

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Clifford theory for semisimple Hopf algebras

Sebastian Burciu(Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, Romania)

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

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Motivation of the talk Rieffel’s generalization for semisimple artin algebras New results obtained: stabilizers as Hopf subalgebras Applications A counterexample

Clifford theory for semisimple Hopf algebras

Hopf algebras and Tensor categories, University of Almeria (Spain), July 4-8, 2011

Sebastian Burciu

Institute of Mathematics ”Simion Stoilow” of

Romanian Academy

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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 arbitrarynormal

extension of semisimple artin rings. ([8], ’79)

Schneider unified the existent theories in a more general setting, that of Hopf Galois extensions ([2], ’90).

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

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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 arbitrarynormal

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Historical background

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 arbitrarynormal

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Historical background

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 arbitrarynormal extension of semisimple artin rings. ([8], ’79)

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Historical background

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Historical background

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Historical background

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Historical background

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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 Applications

Extensions bykF.

The Drinfeld doubleD(A)

5 A counterexample

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Motivation

LetAbe a semisimple Hopf algebra and letBbe a Hopf subalgebra ofA.

Bis called anormalHopf subalgebra if it is closed under the adjoint action ofAon itself, i.ea₁BS(a₂)⊂Bfor alla∈A. Clifford theory for normal Hopf subalgebras

LetB⊂Abe a normal Hopf subalgebra of the semisimple Hopf algebraA. For a given irreducibleB-moduleM,

find a Hopf subalgebraZ_M ofAwithB⊂Z_M ⊂Asuch that the induction mapind:Irr(Z_M|M)→Irr(A|_M)given by

V 7→A⊗_Z_M V is a bijection.

When such a Hopf sualgebraZ_M exists?

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

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Motivation

Bis called anormalHopf subalgebra if it is closed under the adjoint action ofAon itself, i.ea₁BS(a₂)⊂Bfor alla∈A.

Clifford theory for normal Hopf subalgebras

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Motivation

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Motivation

find a Hopf subalgebraZ_M ofAwithB⊂Z_M ⊂Asuch that the induction map

ind:Irr(Z_M|M)→Irr(A|_M)given by V 7→A⊗_Z_M V is a bijection.

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Motivation

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Motivation

V 7→A⊗_Z_M V

is a bijection.

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Motivation

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Motivation

When such a Hopf sualgebraZ exists?

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Motivation

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Rieffel’s work on semisimple normal extensions

Definition of normal subrings:

LetB⊂Aan extension of semisimple rings. The extension is callednormalifA(I∩B) = (I∩B)Afor any maximal idealIofA.

IfB=kH⊂A=kGfor two finite groupsH⊂Gthen the extensionB⊂Ais normal if and only ifHis a normal subgroup ofG.

Normal extensions of Hopf algebras

More generally the same thing is true for semisimple Hopf algebras.

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Rieffel’s work on semisimple normal extensions

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Rieffel’s work on semisimple normal extensions

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Rieffel’s definition for the stabilizer

Stabilizer of a simple moduleW

LetB⊂Abe a normal extension of semisimple artin rings, andW be an irreducibleB-module. Then a stability subring for W is a semisimple artin subringT, withB⊂T ⊂Aand

(1)Bis a normal subring ofT, (2)J isT-invariant, i.eJT =TJ (3)T +AJ+JA=A, where J :=Ann_B(W).

The stabilizer always exists but is not unique; there might be more then one stabilizers for a given moduleW.

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Rieffel’s definition for the stabilizer

LetB⊂Abe a normal extension of semisimple artin rings,

andW be an irreducibleB-module. Then a stability subring for W is a semisimple artin subringT, withB⊂T ⊂Aand

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Rieffel’s definition for the stabilizer

LetB⊂Abe a normal extension of semisimple artin rings, andW be an irreducibleB-module.

Then a stability subring for W is a semisimple artin subringT, withB⊂T ⊂Aand

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Rieffel’s definition for the stabilizer

(1)Bis a normal subring ofT,

(2)J isT-invariant, i.eJT =TJ (3)T +AJ+JA=A, where J :=Ann_B(W).

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Rieffel’s definition for the stabilizer

(1)Bis a normal subring ofT, (2)J isT-invariant, i.eJT =TJ

(3)T +AJ+JA=A, where J :=Ann_B(W).

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Rieffel’s definition for the stabilizer

(1)Bis a normal subring ofT, (2)J isT-invariant, i.eJT =TJ (3)T +AJ+JA=A,

where J :=Ann_B(W).

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Rieffel’s definition for the stabilizer

(1)Bis a normal subring ofT, (2)J isT-invariant, i.eJT =TJ (3)T +AJ+JA=A, where J :=AnnB(W).

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Rieffel’s definition for the stabilizer

(1)Bis a normal subring ofT, (2)J isT-invariant, i.eJT =TJ (3)T +AJ+JA=A, where J :=AnnB(W).

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Clifford’s correspondence holds in Rieffel’s sense

THEOREM (Rieffel, 1979.)

LetBbe a normal subring of the semi-simple artin ringA, letW be an irreducibleB-module, and letT be a stability subring for W.

Then the process of inducing modules fromT toAgives a bijection between equivalence classes of simpleT-modules havingW as(the) B-constituent and equivalence classes of simpleA-modules havingW as(a)B-constituent.

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Clifford’s correspondence holds in Rieffel’s sense

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Clifford’s correspondence holds in Rieffel’s sense

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Clifford’s correspondence holds in Rieffel’s sense

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

LetB⊂Abe an extension of semisimple rings. Rieffel’s equivalence relation on the set of irreducible A-representationsIrr(A).

WriteM∼ÂN if there is an irreducibleW ∈B−mod, common constituent both forM↓Â_B andN ↓Â_B.

In general∼^A is not an equivalence relation.

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

LetB⊂Abe an extension of semisimple rings.

Rieffel’s equivalence relation on the set of irreducible A-representationsIrr(A).

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

WriteM ∼ÂN if there is an irreducibleW ∈B−mod, common constituent both forM↓Â_B andN ↓Â_B.

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

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More on Rieffel’s work on semisimple normal extensions B ⊂ A

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

Rieffel’s equivalence relation on the set of irreducible B-representationsIrr(B).

WriteW ∼_B W⁰ if there is an irreducibleM ∈A−mod,

common constituent both forW ↑^A_B andW⁰ ↑^A_B. In general∼_B is not an equivalence relation.

Theorem (Rieffel, [8])

An extensionB⊂Aof semisimple artin ring is a normal extension if and only if∼^A is an equivalence relation and Irr(M ↓^A_B)is an entire equivalence class for allM ∈Irr(A).

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

WriteW ∼_B W⁰ if there is an irreducibleM ∈A−mod, common constituent both forW ↑^A_B andW⁰ ↑^A_B.

In general∼_B is not an equivalence relation.

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

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

An extensionB⊂Aof semisimple artin ring is a normal extension if and only if

∼^A is an equivalence relation and Irr(M ↓^A_B)is an entire equivalence class for allM ∈Irr(A).

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

An extensionB⊂Aof semisimple artin ring is a normal extension if and only if∼^Ais an equivalence relation and Irr(M ↓^A_B)is an entire equivalence class for allM ∈Irr(A).

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

An extensionB⊂Aof semisimple artin ring is a normal extension if and only if∼^Ais an equivalence relation and Irr(M ↓^A)is an entire equivalence class for allM ∈Irr(A).

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Characterization of Rieffel’s notion of normal extensions

Theorem (B, Kadison, Kuelshammer, [9])

LetB⊂Abe an extension of semisimple finite dimensional algebras. The relation∼^Ais an equivalence relation if and only ifB⊂Ais a depth three extension.

An extensionB⊂Aof finite dimensional semisimple algebras is a normal extension if and only if it is a depth two extension.

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Characterization of Rieffel’s notion of normal extensions

LetB⊂Abe an extension of semisimple finite dimensional algebras.

The relation∼^Ais an equivalence relation if and only ifB⊂Ais a depth three extension.

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Characterization of Rieffel’s notion of normal extensions

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Characterization of Rieffel’s notion of normal extensions

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Characterization of Rieffel’s notion of normal extensions

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Rieffel’s equivalence relations in our settings

LetB⊂Aan extension of normal ss. Hopf algebras. Some notations

LetB_i be an equivalence class under Rieffel’s equivalence relation forB⊂AonIrr(B).

LetA_i be the corresponding equivalence class onIrr(A). Let

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

LetB⊂Aan extension of normal ss. Hopf algebras.

Some notations

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

Some notations

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

Some notations

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

Some notations

LetA_i be the corresponding equivalence class onIrr(A).

Let

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

Some notations

Let

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ∈A_i

χ(1)χ.

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Rieffel’s equivalence relations in our settings

Some notations

Let

b_i = X

β∈B_i

β(1)β.

Let

a_i = X

χ(1)χ.

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Generalization of the first theorem of Clifford

Conjugate modules

LetMbe an irreducibleB-module with characterα∈C(B).

IfW is anA^∗-module thenW ⊗M becomes aB-module with

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

Here we used that any leftA^∗-moduleW is a right A-comodule viaρ(w) =w0⊗w1.

For any irreducible characterd ∈Irr(A^∗)associated to a simpleA-comoduleW define ^dM :=W ⊗M as a B-module.

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Generalization of the first theorem of Clifford

IfW is anA^∗-module thenW ⊗Mbecomes aB-module with

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

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Generalization of the first theorem of Clifford

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

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Generalization of the first theorem of Clifford

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

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Generalization of the first theorem of Clifford

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

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Generalization of the first theorem of Clifford

b(w⊗m) =w₀⊗(S(w₁)bw₂)m (1)

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Conjugate modules and stabilizers

Generalization of the first theorem of Clifford

Theorem (B. ’10)LetB⊂Abe a normal extension of

semisimple Hopf algebras andM be an irreducibleB-module. ThenM ↑^A_B↓^A_Band⊕_d∈Irr(A∗) dM have the same irreducible B-constituents.

ForA=kGone hasIrr(kG^∗) =G.

IfA=kGandB=kH for a normal subgroupH thend =g∈G and^dMcoincides with the conjugate module^gMintroduced by Clifford in [1].

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Conjugate modules and stabilizers

semisimple Hopf algebras andMbe an irreducibleB-module.

ThenM ↑^A_B↓^A_B and⊕_d∈Irr(A∗) dM have the same irreducible B-constituents.

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Conjugate modules and stabilizers

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Conjugate modules and stabilizers

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Conjugate modules and stabilizers

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Clifford correspondence for normal Hopf algebras

The case when the stabilizer is a Hopf subalgebra

Ifαis the char. ofMthen the char.^dαof^dMis given by

dα(x) =α(Sd₁xd₂) (2) for allx ∈B(see Proposition 5.3 of [10]).

Proposition (B, ’11)

The set{d ∈Irr(A^∗)|^dα=(d)α}is closed under

multiplication and“^∗ ”. Thus it generates a Hopf subalgebra Z_αofAthat containsB.

Z_α is called the stabilizer ofαinA.

IfA=kGandB =kN for a normal subgroupN then the stabilizerZ_α coincides with the stabilizerZ_M ofM

introduced by Clifford in [1].

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Clifford correspondence for normal Hopf algebras

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Clifford correspondence for normal Hopf algebras

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Clifford correspondence for normal Hopf algebras

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

Thus it generates a Hopf subalgebra Z_αofAthat containsB.

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Clifford correspondence for normal Hopf algebras

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Clifford correspondence for normal Hopf algebras

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Clifford correspondence for normal Hopf algebras

IfA=kGandB=kN for a normal subgroupN then the

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Clifford correspondence for normal Hopf algebras

IfA=kGandB=kN for a normal subgroupN then the stabilizerZ_α coincides with the stabilizerZ_M ofM

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Clifford correspondence for normal Hopf algebras

The case of the stabilizerZα

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

LetB⊂Aa normal extension of semisimple Hopf algebras. With the above notations:

1 |Z_α| ≤ ^|A|α(1)_b ²

i(1) whereB_i is the equivalence class ofα.

2 Equality holds if and only ifZ_α is a stabilizer in the sense of Rieffel, see [4].