On some gauge invariants of Hopf algebras

Siu-Hung Ng

Iowa State University, USA

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

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?

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

algebras

Question:

degree 2: ψ.

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.

f (H) = f (K )

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.

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.

νn(V ) = χV (Λ[n]) .

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.

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

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

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

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.

over C.

νn(H) = dim H

over C.

νn(H) = dim H

over C.

νn(H) = dim H

over C.

νn(H) = dim H

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.

Non-semisimple case

power map.

always exist.

Non-semisimple case

power map.

always exist.

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.

νn(H) = λ(Λ[n]) .

In particular, the minimal polynomial pH of this recursive

sequence is also a gauge invariant.

νn(H) = λ(Λ[n]) .

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.

Some examples

1,2,3,4, ...

Some examples

1,2,3,4, ...

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:

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.

More examples

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

More examples

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

More examples

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

More examples

xm−1x−q

)2if m | 24,

(xm − 1)2 if m - 24.

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.

More examples

νn(H) = n2

More examples

νn(H) = n2

Application

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

ν2(H) = Tr(S) =

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′.

Application

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

ν2(H) = Tr(S) =

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′.

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.

ν2(uq(sl2)) = 4|1 + q|−2.

Corollary

ν2(uq(sl2)) = 4|1 + q|−2.

Corollary

The End

Thank you for your attentions.

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

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