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.

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

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.

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

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.

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

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.

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

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∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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 ∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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 ∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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 ∼^AN if there is an irreducibleW ∈B−mod, common constituent both forM↓^A_B andN ↓^A_B.

In general∼^A is not an equivalence relation.

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

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).

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

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).

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

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).

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

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).

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

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

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

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

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

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.

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

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

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

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.

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

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

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

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.

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

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

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

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.

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

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

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

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.

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

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

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

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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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(1)χ.

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

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.

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

Generalization of the first theorem of Clifford

Conjugate modules

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

IfW is anA^∗-module thenW ⊗Mbecomes 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.

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

Generalization of the first theorem of Clifford

Conjugate modules

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

IfW is anA^∗-module thenW ⊗Mbecomes 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.

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

Generalization of the first theorem of Clifford

Conjugate modules

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

IfW is anA^∗-module thenW ⊗Mbecomes 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.

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

Generalization of the first theorem of Clifford

Conjugate modules

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

IfW is anA^∗-module thenW ⊗Mbecomes 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.

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

Generalization of the first theorem of Clifford

Conjugate modules

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

IfW is anA^∗-module thenW ⊗Mbecomes 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.

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

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].

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

Conjugate modules and stabilizers

Generalization of the first theorem of Clifford

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

semisimple Hopf algebras andMbe an irreducibleB-module.

ThenM ↑^A_B↓^A_B and⊕_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].

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

Conjugate modules and stabilizers

Generalization of the first theorem of Clifford

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

semisimple Hopf algebras andMbe an irreducibleB-module.

ThenM ↑^A_B↓^A_B and⊕_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].

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

Conjugate modules and stabilizers

Generalization of the first theorem of Clifford

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

semisimple Hopf algebras andMbe an irreducibleB-module.

ThenM ↑^A_B↓^A_B and⊕_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].

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

Conjugate modules and stabilizers

Generalization of the first theorem of Clifford

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

semisimple Hopf algebras andMbe an irreducibleB-module.

ThenM ↑^A_B↓^A_B and⊕_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].

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

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].

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

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].

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

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].

In document Clifford theory for semisimple Hopf algebras (página 32-166)