On some gauge invariants of Hopf algebras Siu-Hung Ng Iowa State - - PowerPoint PPT Presentation

On some gauge invariants of Hopf algebras

Siu-Hung Ng

Iowa State University, USA

Non-commutative algebraic geometry 2011 Shanghai Workshop September 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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

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 tensor categories?

On some gauge invariants of Hopf algebras

FS-indicators for semisimple Hopf algebras

Tambara-Yamagami [98] answered the question by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H, K are said to be gauge equivalent if H-mod and K-mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f(H) defined for any Hopf 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 by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H, K are said to be gauge equivalent if H-mod and K-mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f(H) defined for any Hopf 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 by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H, K are said to be gauge equivalent if H-mod and K-mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f(H) defined for any Hopf 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 by considering the fusion categories with such fusion rules. For other groups or semisimple quasi-Hopf algebras, more general but computable invariants are needed to be discovered. Two Hopf algebras H, K are said to be gauge equivalent if H-mod and K-mod are equivalent as monoidal categories. Let C be a collection of Hopf algebras which is closed under gauge equivalence. A quantity f(H) defined for any Hopf 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 Hopf algebras 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 invariant because Theorem (Etingof-Gelaki) Let R be the R-matrix of D(H). Then the operator R21R on D(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 Hopf algebras 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 invariant because Theorem (Etingof-Gelaki) Let R be the R-matrix of D(H). Then the operator R21R on D(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 Hopf algebras 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 invariant because Theorem (Etingof-Gelaki) Let R be the R-matrix of D(H). Then the operator R21R on D(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 Hopf algebras 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 invariant because Theorem (Etingof-Gelaki) Let R be the R-matrix of D(H). Then the operator R21R on D(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 Hopf algebras 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 invariant because Theorem (Etingof-Gelaki) Let R be the R-matrix of D(H). Then the operator R21R on D(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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 finite dimensional Hopf algebras. Let H be a f.d. semisimple Hopf algebra over C. Take Λ be the normalized integral, i.e. ǫ(Λ) = 1. [Linchenko-Montgomery] The n-th FS-indicator of a representation 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 the

normalized 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 8 CQ8 1 2 1 8 1 2 1 8 K 1 6 1 4 1 6 1 8 Theorem (Mason, Ng, Schauenburg) Let H be f.d. semisimple Hopf algebra over C. Then the sequence {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 the

normalized 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 8 CQ8 1 2 1 8 1 2 1 8 K 1 6 1 4 1 6 1 8 Theorem (Mason, Ng, Schauenburg) Let H be f.d. semisimple Hopf algebra over C. Then the sequence {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 the

normalized 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 8 CQ8 1 2 1 8 1 2 1 8 K 1 6 1 4 1 6 1 8 Theorem (Mason, Ng, Schauenburg) Let H be f.d. semisimple Hopf algebra over C. Then the sequence {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 the

normalized 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 8 CQ8 1 2 1 8 1 2 1 8 K 1 6 1 4 1 6 1 8 Theorem (Mason, Ng, Schauenburg) Let H be f.d. semisimple Hopf algebra over C. Then the sequence {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

• ver 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

• ver 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

• ver 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

• ver 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

• ver 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 ǫ(Λ) = 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 → 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 ǫ(Λ) = 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 → 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 ǫ(Λ) = 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 → 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)} is periodic 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 of positive 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)} is periodic 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 of positive 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)} is periodic 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 of positive 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−1

x−q

2 if m | 24, (xm − 1)2 if m ∤ 24. [Etingof-Gelaki] qexp(Tm2(q)) = m. Question: Let H be a Hopf algebra over C with qexp(H) = N. Does pH | (xN − 1)l for some integer l? Conjecture: A finite-dimensional complex Hopf algebra H is 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−1

x−q

2 if m | 24, (xm − 1)2 if m ∤ 24. [Etingof-Gelaki] qexp(Tm2(q)) = m. Question: Let H be a Hopf algebra over C with qexp(H) = N. Does pH | (xN − 1)l for some integer l? Conjecture: A finite-dimensional complex Hopf algebra H is 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−1

x−q

2 if m | 24, (xm − 1)2 if m ∤ 24. [Etingof-Gelaki] qexp(Tm2(q)) = m. Question: Let H be a Hopf algebra over C with qexp(H) = N. Does pH | (xN − 1)l for some integer l? Conjecture: A finite-dimensional complex Hopf algebra H is 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−1

x−q

2 if m | 24, (xm − 1)2 if m ∤ 24. [Etingof-Gelaki] qexp(Tm2(q)) = m. Question: Let H be a Hopf algebra over C with qexp(H) = N. Does pH | (xN − 1)l for some integer l? Conjecture: A finite-dimensional complex Hopf algebra H is 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−1

x−q

2 if m | 24, (xm − 1)2 if m ∤ 24. [Etingof-Gelaki] qexp(Tm2(q)) = m. Question: Let H be a Hopf algebra over C with qexp(H) = N. Does pH | (xN − 1)l for some integer l? Conjecture: A finite-dimensional complex Hopf algebra H is 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,

4 1−q

for m even. Corollary Let q, q′ ∈ C be primitive m-th roots of unity. The Taft algebras Tm2(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,

4 1−q

for m even. Corollary Let q, q′ ∈ C be primitive m-th roots of unity. The Taft algebras Tm2(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. Then uq(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. Then uq(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. Then uq(sl2) and uq′(sl2) are gauge equivalent if, and only if, q′ = q±1.

On some gauge invariants of Hopf algebras

The End

Thank you for your attentions.

On some gauge invariants of Hopf algebras