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### Transcript of On some gauge invariants of Hopf algebras · 2011-09-14 · If H and K are gauge equivalent then H...

On some gauge invariants of Hopf algebras

Siu-Hung Ng

Iowa State University, USA

Non-commutative algebraic geometry2011 Shanghai WorkshopSeptember 12-16, 2011

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

Question:

Let Q8, D8 be the quaternion and dihedral groups.

Complex irreducible representations of the two groups:degree 1: χ1, χ2, χ3, χ4

degree 2: ψ.

CD8∼= CQ8 as C-algebras but not isomorphic as Hopf

algebras

CQ8-mod ∼= CD8-mod as C-linear categories.

They have the same fusion rules or Grothendieck ring :{χ1, χ2, χ3, χ4} ∼= Z2 × Z2,χiψ = ψ = ψχi , ψψ = χ1 + χ2 + χ3 + χ4.

Are CD8-mod and CQ8-mod equivalent as tensorcategories?

On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras

Tambara-Yamagami [98] answered the question byconsidering the fusion categories with such fusion rules.

For other groups or semisimple quasi-Hopf algebras, moregeneral but computable invariants are needed to bediscovered.

Two Hopf algebras H,K are said to be gauge equivalent ifH-mod and K -mod are equivalent as monoidal categories.

Let C be a collection of Hopf algebras which is closedunder gauge equivalence. A quantity f (H) defined for anyHopf algebra H in C is called a gauge invariant if

f (H) = f (K )

for all Hopf algebras K gauge equivalent to H.

On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras

Tambara-Yamagami [98] answered the question byconsidering the fusion categories with such fusion rules.

For other groups or semisimple quasi-Hopf algebras, moregeneral but computable invariants are needed to bediscovered.

Two Hopf algebras H,K are said to be gauge equivalent ifH-mod and K -mod are equivalent as monoidal categories.

Let C be a collection of Hopf algebras which is closedunder gauge equivalence. A quantity f (H) defined for anyHopf algebra H in C is called a gauge invariant if

f (H) = f (K )

for all Hopf algebras K gauge equivalent to H.

On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras

Tambara-Yamagami [98] answered the question byconsidering the fusion categories with such fusion rules.

For other groups or semisimple quasi-Hopf algebras, moregeneral but computable invariants are needed to bediscovered.

Two Hopf algebras H,K are said to be gauge equivalent ifH-mod and K -mod are equivalent as monoidal categories.

Let C be a collection of Hopf algebras which is closedunder gauge equivalence. A quantity f (H) defined for anyHopf algebra H in C is called a gauge invariant if

f (H) = f (K )

for all Hopf algebras K gauge equivalent to H.

On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras

Tambara-Yamagami [98] answered the question byconsidering the fusion categories with such fusion rules.

For other groups or semisimple quasi-Hopf algebras, moregeneral but computable invariants are needed to bediscovered.

Two Hopf algebras H,K are said to be gauge equivalent ifH-mod and K -mod are equivalent as monoidal categories.

Let C be a collection of Hopf algebras which is closedunder gauge equivalence. A quantity f (H) defined for anyHopf algebra H in C is called a gauge invariant if

f (H) = f (K )

for all Hopf algebras K gauge equivalent to H.

On some gauge invariants of Hopf algebras

Some well-known gauge invariants

Consider the collection Cf of all finite-dimensional Hopfalgebras over C.

dim H is a gauge invariant because

Theorem (Schauenburg)

If H and K are gauge equivalent then H ∼= K as C-algebras.

The exponent of a Hopf algebra is a gauge invariantbecause

Theorem (Etingof-Gelaki)

Let R be the R-matrix of D(H). Then the operator R21R onD(H)⊗ D(H) has finite unipotent index qexp(H). Moreover,qexp(H) is a gauge invariant.

On some gauge invariants of Hopf algebras

Some well-known gauge invariants

Consider the collection Cf of all finite-dimensional Hopfalgebras over C.

dim H is a gauge invariant because

Theorem (Schauenburg)

If H and K are gauge equivalent then H ∼= K as C-algebras.

The exponent of a Hopf algebra is a gauge invariantbecause

Theorem (Etingof-Gelaki)

Let R be the R-matrix of D(H). Then the operator R21R onD(H)⊗ D(H) has finite unipotent index qexp(H). Moreover,qexp(H) is a gauge invariant.

On some gauge invariants of Hopf algebras

Some well-known gauge invariants

Consider the collection Cf of all finite-dimensional Hopfalgebras over C.

dim H is a gauge invariant because

Theorem (Schauenburg)

If H and K are gauge equivalent then H ∼= K as C-algebras.

The exponent of a Hopf algebra is a gauge invariantbecause

Theorem (Etingof-Gelaki)

Let R be the R-matrix of D(H). Then the operator R21R onD(H)⊗ D(H) has finite unipotent index qexp(H). Moreover,qexp(H) is a gauge invariant.

On some gauge invariants of Hopf algebras

Some well-known gauge invariants

Consider the collection Cf of all finite-dimensional Hopfalgebras over C.

dim H is a gauge invariant because

Theorem (Schauenburg)

If H and K are gauge equivalent then H ∼= K as C-algebras.

The exponent of a Hopf algebra is a gauge invariantbecause

Theorem (Etingof-Gelaki)

Let R be the R-matrix of D(H). Then the operator R21R onD(H)⊗ D(H) has finite unipotent index qexp(H). Moreover,qexp(H) is a gauge invariant.

On some gauge invariants of Hopf algebras

Some well-known gauge invariants

Consider the collection Cf of all finite-dimensional Hopfalgebras over C.

dim H is a gauge invariant because

Theorem (Schauenburg)

If H and K are gauge equivalent then H ∼= K as C-algebras.

The exponent of a Hopf algebra is a gauge invariantbecause

Theorem (Etingof-Gelaki)

Let R be the R-matrix of D(H). Then the operator R21R onD(H)⊗ D(H) has finite unipotent index qexp(H). Moreover,qexp(H) is a gauge invariant.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Frobenius-Schur indicators

If H = CG where G is a finite group, qexp(H) = exp(G).

We would like to find more gauge invariants for finitedimensional Hopf algebras.

Let H be a f.d. semisimple Hopf algebra over C.

Take Λ be the normalized integral, i.e. ε(Λ) = 1.

[Linchenko-Montgomery] The n-th FS-indicator of arepresentation V of H is defined as

νn(V ) = χV (Λ[n]) .

In particular, we consider regular representation V of H.

On some gauge invariants of Hopf algebras

Invariance of FS-indicators

For a finite group algebra CG, Λ = 1|G|∑

g∈G g is thenormalized integral.

Λ[n] = 1|G|∑

g∈G gn and so

νn(CG) = χreg(Λ[n]) = #{x ∈ G | xn = 1} .

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

CD8 1 6 1 8 1 6 1 8CQ8 1 2 1 8 1 2 1 8K 1 6 1 4 1 6 1 8

Theorem (Mason, Ng, Schauenburg)

Let H be f.d. semisimple Hopf algebra over C. Then thesequence {νn(H)} is a gauge invariant of H.

On some gauge invariants of Hopf algebras

Invariance of FS-indicators

For a finite group algebra CG, Λ = 1|G|∑

g∈G g is thenormalized integral.

Λ[n] = 1|G|∑

g∈G gn and so

νn(CG) = χreg(Λ[n]) = #{x ∈ G | xn = 1} .

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

CD8 1 6 1 8 1 6 1 8CQ8 1 2 1 8 1 2 1 8K 1 6 1 4 1 6 1 8

Theorem (Mason, Ng, Schauenburg)

Let H be f.d. semisimple Hopf algebra over C. Then thesequence {νn(H)} is a gauge invariant of H.

On some gauge invariants of Hopf algebras

Invariance of FS-indicators

For a finite group algebra CG, Λ = 1|G|∑

g∈G g is thenormalized integral.

Λ[n] = 1|G|∑

g∈G gn and so

νn(CG) = χreg(Λ[n]) = #{x ∈ G | xn = 1} .

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

CD8 1 6 1 8 1 6 1 8CQ8 1 2 1 8 1 2 1 8K 1 6 1 4 1 6 1 8

Theorem (Mason, Ng, Schauenburg)

Let H be f.d. semisimple Hopf algebra over C. Then thesequence {νn(H)} is a gauge invariant of H.

On some gauge invariants of Hopf algebras

Invariance of FS-indicators

For a finite group algebra CG, Λ = 1|G|∑

g∈G g is thenormalized integral.

Λ[n] = 1|G|∑

g∈G gn and so

νn(CG) = χreg(Λ[n]) = #{x ∈ G | xn = 1} .

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

CD8 1 6 1 8 1 6 1 8CQ8 1 2 1 8 1 2 1 8K 1 6 1 4 1 6 1 8

Theorem (Mason, Ng, Schauenburg)

Let H be f.d. semisimple Hopf algebra over C. Then thesequence {νn(H)} is a gauge invariant of H.

On some gauge invariants of Hopf algebras

Exponent, dimension and periodicity of indicators

Theorem (Ng-Schauenburg)Let H be a semisimple (quasi-)Hopf algebra

over C.

1 Then {νn(H)}n≥1 is a periodic sequence in

O(QN) of period N, where N = qexp(H).

2 qexp(H) = least positive integer n such that

νn(H) = dim H

In particular, the sequence {νn(H)}n≥1 is linearly

recursive and satisfies the polynomial xN − 1.

On some gauge invariants of Hopf algebras

Exponent, dimension and periodicity of indicators

Theorem (Ng-Schauenburg)Let H be a semisimple (quasi-)Hopf algebra

over C.

1 Then {νn(H)}n≥1 is a periodic sequence in

O(QN) of period N, where N = qexp(H).

2 qexp(H) = least positive integer n such that

νn(H) = dim H

In particular, the sequence {νn(H)}n≥1 is linearly

recursive and satisfies the polynomial xN − 1.

On some gauge invariants of Hopf algebras

Exponent, dimension and periodicity of indicators

Theorem (Ng-Schauenburg)Let H be a semisimple (quasi-)Hopf algebra

over C.

1 Then {νn(H)}n≥1 is a periodic sequence in

O(QN) of period N, where N = qexp(H).

2 qexp(H) = least positive integer n such that

νn(H) = dim H

In particular, the sequence {νn(H)}n≥1 is linearly

recursive and satisfies the polynomial xN − 1.

On some gauge invariants of Hopf algebras

Exponent, dimension and periodicity of indicators

Theorem (Ng-Schauenburg)Let H be a semisimple (quasi-)Hopf algebra

over C.

1 Then {νn(H)}n≥1 is a periodic sequence in

O(QN) of period N, where N = qexp(H).

2 qexp(H) = least positive integer n such that

νn(H) = dim H

In particular, the sequence {νn(H)}n≥1 is linearly

recursive and satisfies the polynomial xN − 1.

On some gauge invariants of Hopf algebras

Exponent, dimension and periodicity of indicators

Theorem (Ng-Schauenburg)Let H be a semisimple (quasi-)Hopf algebra

over C.

1 Then {νn(H)}n≥1 is a periodic sequence in

O(QN) of period N, where N = qexp(H).

2 qexp(H) = least positive integer n such that

νn(H) = dim H

In particular, the sequence {νn(H)}n≥1 is linearly

recursive and satisfies the polynomial xN − 1.

On some gauge invariants of Hopf algebras

Non-semisimple case

Two-sided integral Λ such that ε(Λ) 6= 0 does not exist in a

non-semisimple Hopf algebra.

[Kashina-Sommerhäuser-Zhu] For any semisimple

complex Hopf algebra H, νn(H) = Tr(S ◦ Pn−1) where S

is the antipode and Pn : H → H, x 7→ x [n] is the Sweedler

power map.

However, the antipode S and the Sweedler power maps Pn

always exist.

On some gauge invariants of Hopf algebras

Non-semisimple case

Two-sided integral Λ such that ε(Λ) 6= 0 does not exist in a

non-semisimple Hopf algebra.

[Kashina-Sommerhäuser-Zhu] For any semisimple

complex Hopf algebra H, νn(H) = Tr(S ◦ Pn−1) where S

is the antipode and Pn : H → H, x 7→ x [n] is the Sweedler

power map.

However, the antipode S and the Sweedler power maps Pn

always exist.

On some gauge invariants of Hopf algebras

Non-semisimple case

Two-sided integral Λ such that ε(Λ) 6= 0 does not exist in a

non-semisimple Hopf algebra.

[Kashina-Sommerhäuser-Zhu] For any semisimple

complex Hopf algebra H, νn(H) = Tr(S ◦ Pn−1) where S

is the antipode and Pn : H → H, x 7→ x [n] is the Sweedler

power map.

However, the antipode S and the Sweedler power maps Pn

always exist.

On some gauge invariants of Hopf algebras

Indicators for finite dimensional Hopf algebras

Theorem (Kashina-Montgomery-Ng)

Let H be a finite-dimensional Hopf algebra over any field k, and

n a positive integer.

1 Then, the scalar νn(H) = Tr(S ◦ Pn−1) is a gauge invariant.

Moreover, the sequence {νn(H)}n≥1 is linearly recursive.

2 Let λ ∈ H∗ and Λ ∈ H be left integrals such that λ(Λ) = 1.

Then

νn(H) = λ(Λ[n]) .

In particular, the minimal polynomial pH of this recursive

sequence is also a gauge invariant.

On some gauge invariants of Hopf algebras

Indicators for finite dimensional Hopf algebras

Theorem (Kashina-Montgomery-Ng)

Let H be a finite-dimensional Hopf algebra over any field k, and

n a positive integer.

1 Then, the scalar νn(H) = Tr(S ◦ Pn−1) is a gauge invariant.

Moreover, the sequence {νn(H)}n≥1 is linearly recursive.

2 Let λ ∈ H∗ and Λ ∈ H be left integrals such that λ(Λ) = 1.

Then

νn(H) = λ(Λ[n]) .

In particular, the minimal polynomial pH of this recursive

sequence is also a gauge invariant.

On some gauge invariants of Hopf algebras

Indicators for finite dimensional Hopf algebras

Theorem (Kashina-Montgomery-Ng)

Let H be a finite-dimensional Hopf algebra over any field k, and

n a positive integer.

1 Then, the scalar νn(H) = Tr(S ◦ Pn−1) is a gauge invariant.

Moreover, the sequence {νn(H)}n≥1 is linearly recursive.

2 Let λ ∈ H∗ and Λ ∈ H be left integrals such that λ(Λ) = 1.

Then

νn(H) = λ(Λ[n]) .

In particular, the minimal polynomial pH of this recursive

sequence is also a gauge invariant.

On some gauge invariants of Hopf algebras

Indicators for finite dimensional Hopf algebras

Theorem (Kashina-Montgomery-Ng)

Let H be a finite-dimensional Hopf algebra over any field k, and

n a positive integer.

1 Then, the scalar νn(H) = Tr(S ◦ Pn−1) is a gauge invariant.

Moreover, the sequence {νn(H)}n≥1 is linearly recursive.

2 Let λ ∈ H∗ and Λ ∈ H be left integrals such that λ(Λ) = 1.

Then

νn(H) = λ(Λ[n]) .

In particular, the minimal polynomial pH of this recursive

sequence is also a gauge invariant.

On some gauge invariants of Hopf algebras

Some examples

If H is a semisimple complex Hopf algebra, then {νn(H)} isperiodic and the period is N = qexp(H). In particular,pH | xN − 1.

The sequence can be unbounded if H is not semisimple.

For H = T4(−1), the sequence {νn(H)} is the sequence ofpositive integers:

1,2,3,4, ...

The recursive relation pH = (x − 1)2.

On some gauge invariants of Hopf algebras

Some examples

If H is a semisimple complex Hopf algebra, then {νn(H)} isperiodic and the period is N = qexp(H). In particular,pH | xN − 1.

The sequence can be unbounded if H is not semisimple.

For H = T4(−1), the sequence {νn(H)} is the sequence ofpositive integers:

1,2,3,4, ...

The recursive relation pH = (x − 1)2.

On some gauge invariants of Hopf algebras

Some examples

If H is a semisimple complex Hopf algebra, then {νn(H)} isperiodic and the period is N = qexp(H). In particular,pH | xN − 1.

The sequence can be unbounded if H is not semisimple.

For H = T4(−1), the sequence {νn(H)} is the sequence ofpositive integers:

1,2,3,4, ...

The recursive relation pH = (x − 1)2.

On some gauge invariants of Hopf algebras

More examples

Using GAP, we compute the indicator sequence of Tm2(q)

for m = 2, ...,24, where q is a primitive m-th root unity.

The minimal polynomial pm of the sequence{νn(Tm2(q)}n≥1 can be summarized as follow:

pm =

(

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

[Etingof-Gelaki] qexp(Tm2(q)) = m.

Question: Let H be a Hopf algebra over C withqexp(H) = N. Does pH | (xN − 1)l for some integer l?

Conjecture: A finite-dimensional complex Hopf algebra His semisimple if, and only if pH has no multiple root.

On some gauge invariants of Hopf algebras

More examples

Using GAP, we compute the indicator sequence of Tm2(q)

for m = 2, ...,24, where q is a primitive m-th root unity.

The minimal polynomial pm of the sequence{νn(Tm2(q)}n≥1 can be summarized as follow:

pm =

(

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

[Etingof-Gelaki] qexp(Tm2(q)) = m.

Question: Let H be a Hopf algebra over C withqexp(H) = N. Does pH | (xN − 1)l for some integer l?

Conjecture: A finite-dimensional complex Hopf algebra His semisimple if, and only if pH has no multiple root.

On some gauge invariants of Hopf algebras

More examples

Using GAP, we compute the indicator sequence of Tm2(q)

for m = 2, ...,24, where q is a primitive m-th root unity.

The minimal polynomial pm of the sequence{νn(Tm2(q)}n≥1 can be summarized as follow:

pm =

(

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

[Etingof-Gelaki] qexp(Tm2(q)) = m.

Question: Let H be a Hopf algebra over C withqexp(H) = N. Does pH | (xN − 1)l for some integer l?

Conjecture: A finite-dimensional complex Hopf algebra His semisimple if, and only if pH has no multiple root.

On some gauge invariants of Hopf algebras

More examples

Using GAP, we compute the indicator sequence of Tm2(q)

for m = 2, ...,24, where q is a primitive m-th root unity.

The minimal polynomial pm of the sequence{νn(Tm2(q)}n≥1 can be summarized as follow:

pm =

(

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

[Etingof-Gelaki] qexp(Tm2(q)) = m.

Question: Let H be a Hopf algebra over C withqexp(H) = N. Does pH | (xN − 1)l for some integer l?

Conjecture: A finite-dimensional complex Hopf algebra His semisimple if, and only if pH has no multiple root.

On some gauge invariants of Hopf algebras

More examples

Using GAP, we compute the indicator sequence of Tm2(q)

for m = 2, ...,24, where q is a primitive m-th root unity.

The minimal polynomial pm of the sequence{νn(Tm2(q)}n≥1 can be summarized as follow:

pm =

(

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

[Etingof-Gelaki] qexp(Tm2(q)) = m.

Question: Let H be a Hopf algebra over C withqexp(H) = N. Does pH | (xN − 1)l for some integer l?

Conjecture: A finite-dimensional complex Hopf algebra His semisimple if, and only if pH has no multiple root.

On some gauge invariants of Hopf algebras

More examples

[Shimizu] The roots of pH are roots of unity.

For H = uq(sl2) where q is a primitive 3rd root of unity.Then

νn(H) = n2

for n ≥ 1. The minimal polynomial pH is (x − 1)3.

By [EG], qexp(uq(sl2)) = 3 and so pH | (x3 − 1)3.

On some gauge invariants of Hopf algebras

More examples

[Shimizu] The roots of pH are roots of unity.

For H = uq(sl2) where q is a primitive 3rd root of unity.Then

νn(H) = n2

for n ≥ 1. The minimal polynomial pH is (x − 1)3.

By [EG], qexp(uq(sl2)) = 3 and so pH | (x3 − 1)3.

On some gauge invariants of Hopf algebras

More examples

[Shimizu] The roots of pH are roots of unity.

For H = uq(sl2) where q is a primitive 3rd root of unity.Then

νn(H) = n2

for n ≥ 1. The minimal polynomial pH is (x − 1)3.

By [EG], qexp(uq(sl2)) = 3 and so pH | (x3 − 1)3.

On some gauge invariants of Hopf algebras

Application

For H = Tm2(q), where q is a primitive m-th root of unity.

ν2(H) = Tr(S) =

2

1+q(m+1)/2 for m odd,

41−q for m even.

Corollary

Let q,q′ ∈ C be primitive m-th roots of unity. The Taft algebrasTm2(q) and Tm2(q′) are gauge equivalent if, and only if, q = q′.

On some gauge invariants of Hopf algebras

Application

For H = Tm2(q), where q is a primitive m-th root of unity.

ν2(H) = Tr(S) =

2

1+q(m+1)/2 for m odd,

41−q for m even.

Corollary

Let q,q′ ∈ C be primitive m-th roots of unity. The Taft algebrasTm2(q) and Tm2(q′) are gauge equivalent if, and only if, q = q′.

On some gauge invariants of Hopf algebras

Application (Shimizu)

Let q ∈ C be a root of unity of odd order. Then

ν2(uq(sl2)) = 4|1 + q|−2.

Corollary

Let q,q′ ∈ C be roots of unity of the same odd order. Thenuq(sl2) and uq′(sl2) are gauge equivalent if, and only if,q′ = q±1.

On some gauge invariants of Hopf algebras

Application (Shimizu)

Let q ∈ C be a root of unity of odd order. Then

ν2(uq(sl2)) = 4|1 + q|−2.

Corollary

Let q,q′ ∈ C be roots of unity of the same odd order. Thenuq(sl2) and uq′(sl2) are gauge equivalent if, and only if,q′ = q±1.

On some gauge invariants of Hopf algebras

Application (Shimizu)

Let q ∈ C be a root of unity of odd order. Then

ν2(uq(sl2)) = 4|1 + q|−2.

Corollary

Let q,q′ ∈ C be roots of unity of the same odd order. Thenuq(sl2) and uq′(sl2) are gauge equivalent if, and only if,q′ = q±1.

On some gauge invariants of Hopf algebras

The End