Finite groups acting on higher dimensionalnoncommutative tori

Jae Hyup LeeSeoul National University

(Joint work with Ja A Jeong)

Aug 10, 2014

1 / 23 Finite groups acting on higher dimensional noncommutative tori

Rotation algebra

Definition

The Rotation algebra Aθ (θ ∈ R) is the universal C ∗-algebragenerated by two unitaries u1, u2 satisfying

u2u1 = e2πiθu1u2.

If θ ∈ Z, then Aθ ∼= C (T× T). So rotation algebras are alsocalled 2-dimensional noncommutative tori.

Aθ is simple if and only if θ is irrational.

For irrational θ and θ′, we have Aθ ∼= Aθ′ if and only ifθ ≡ ±θ′ mod Z.

2 / 23 Finite groups acting on higher dimensional noncommutative tori

Actions of SL2(Z) on rotation algebras

Given A =

(a bc d

)∈ SL2(Z), the map

u1 7→ eπiacθua1uc

2 ,

u2 7→ eπibdθub1 ud

2

determines an automorphism αA of Aθ, and the choice of thescalars enables

α : SL2(Z)→ Aut(Aθ)

A 7→ αA

to be a group action.

3 / 23 Finite groups acting on higher dimensional noncommutative tori

Aθ o Zn for simple Aθ and Zn ≤ SL2(Z)

There are only 4 nontrivial finite subgroups of SL2(Z) up toconjugate which are necessarily

Z2 = 〈(−1 00 −1

)〉, Z3 = 〈

(−1 −11 0

)〉,

Z4 = 〈(

0 −11 0

)〉, Z6 = 〈

(0 −11 1

)〉.

Theorem (2010, Echterhoff-Luck-Phillips-Walters)Aθ o Zn are all AF-aglebras if θ is irrational. If θ, θ′ are irrational,then Aθ o Zn

∼= Aθ′ o Zm if and only if n = m and θ ≡ ±θ′mod Z.

4 / 23 Finite groups acting on higher dimensional noncommutative tori

Higher dimensional noncommutative torus

Definition

Given d × d (d ≥ 2) real skew symmetric matrix Θ = (θij ), the ddimensional noncommutative torus AΘ associated with Θ isdefined by the universal C ∗-algebra generated by d unitariesu1, · · · , ud subject to the relations

uj ui = e2πiθij ui uj , (i , j = 1, · · · , d).

Rotation algebra Aθ is a NC 2-torus AΘ associated with

Θ =

(0 θ−θ 0

).

5 / 23 Finite groups acting on higher dimensional noncommutative tori

Action of Z2 on AΘ

For every AΘ, the flip automorphism

ui 7→ u∗i

determines the flip action : Z2 → Aut(AΘ).

Theorem (ELPW)If AΘ is simple, then AΘ oflip Z2 is an AF-algebra.

Note that flip action is coming from the matrix −Id .

6 / 23 Finite groups acting on higher dimensional noncommutative tori

Problem

Except the flip action, it is considerably difficult to find groupactions by some consistent matrix group which acts on every torusof given dimension d > 2. Instead, we want

to find an action of GΘ on AΘ which is the action of SL2(Z)on Aθ in case of d = 2,

to find finite subgroups of GΘ and to study the crossedproducts of simple AΘ by finite group actions.

7 / 23 Finite groups acting on higher dimensional noncommutative tori

AΘ∼= C ∗r (Zd , ωΘ)

Let Θ be a d × d real skew symmetrix matrix.

Let ωΘ : Zd × Zd → T be the 2 cocycle on Zd defined by

ωΘ(x , y) = eπi〈Θx ,y〉

for x , y ∈ Zd .

The regular ωΘ-representation `Θ : Zd → U(`2(Zd )) isdefined by

(`Θ(y)ξ)(x) = ωΘ(y , x − y)ξ(x − y)

for ξ ∈ `2(Zd ) and x , y ∈ Zd .

The reduced twisted group C ∗-algebra C ∗r (Zd , ωΘ) is definedby the C ∗-subalgebra generated by `Θ(Zd ) in B(`2(Zd )).

C ∗r (Zd , ωΘ) := C ∗{`Θ(Zd )}.

8 / 23 Finite groups acting on higher dimensional noncommutative tori

(Continued)

We have the following isomorphism

AΘ∼=−→ C ∗r (Zd , ωΘ) ⊂ B(`2(Zd ))

ui 7−→ `Θ(ei ),

So we consider every d dimensional torus as a C ∗-subalgebra ofB(`2(Zd )).

9 / 23 Finite groups acting on higher dimensional noncommutative tori

AdUA : AΘ∼= A(A−1)tΘA−1

Let A 7→ UA : GLd (Z)→ U(`2(Zd )) be a grouphomomorphism where

(UAξ)(x) = ξ(A−1x)

for ξ ∈ `2(Zd ) and x ∈ Zd .

We have an isomorphism

Ad UA : AΘ∼=−→ A(A−1)t ΘA−1

`Θ(x) 7−→ `(A−1)t ΘA−1(Ax),

since

Ad UA(`Θ(x)) = UA ◦ `Θ(x) ◦ UA−1

= `(A−1)t ΘA−1(Ax).

10 / 23 Finite groups acting on higher dimensional noncommutative tori

Canonical action

If (A−1)tΘA−1 = Θ, then Ad UA is an automorphism of AΘ

such that Ad UA(`Θ(x)) = `Θ(Ax). Moreover,

GΘ := {A ∈ GLd (Z) : Θ = (A−1)tΘA−1}

forms a subgroup of GLd (Z).

So we have a group action

GΘ → Aut(AΘ)

A 7→ Ad UA.

The restriction of this action to a subgroup G of GΘ will becalled the canonical action of G on AΘ.

11 / 23 Finite groups acting on higher dimensional noncommutative tori

Canonical action=SL2(Z) action if d = 2

If d = 2 and Θ 6= 0, then the canonical action of GΘ is same asthe SL2(Z) action on the rotation algebra since GΘ = SL2(Z) and

Ad UA(u1) = eπiθacua1uc

2 ,

Ad UA(u2) = eπiθbd ub1 ud

2 ,

where A =

(a bc d

)∈ SL2(Z).

12 / 23 Finite groups acting on higher dimensional noncommutative tori

Simple torus

A d × d real skew symmetric matrix Θ is said to bedegenerate if there exists nonzero x ∈ Zd \ {0} satisfying

e2πi〈x ,Θy〉 = 1

for all y ∈ Zd . Otherwise, Θ is said to be nondegenerate.

AΘ is simple if and only if Θ is nondegenerate.

13 / 23 Finite groups acting on higher dimensional noncommutative tori

Canonical actions on simple NC 3-torus by finite cyclicgroups

Theorem (Jeong-L.)

There are no canonical actions on 3 dimensional simple tori byfinite cyclic groups except the flip action.

(Proof) We use the complete list of elements of non-trivial finiteorders in GL3(Z) up to conjugate which consists of only 15elements due to [1971, K. Tahara]. For AΘ on where elements inthat list can act canonically, we get only degenerate Θ except −I3.

14 / 23 Finite groups acting on higher dimensional noncommutative tori

AΘ o G for simple AΘ and finite G ≤ GΘ

Let α : G → Aut(AΘ) be a canonical action of a finite group G ona simple AΘ. Then

α has the tracial Rokhlin property [ELPW]. So AΘ oα G issimple and has tracial rank zero [N. Phillips].

AΘ oα G satisfies the UCT [ELPW].

So in this case, AΘ o G belongs to the class of simple unitalseparable nuclear C ∗-algebras with tracial rank zero which satisfythe UCT.

Theorem (H. Lin)Algebras in this class can be classified by K-groups.

15 / 23 Finite groups acting on higher dimensional noncommutative tori

(Continued)

Theorem (N. Phillips)Let A be a simple infinite dimensional separable unital nuclearC ∗-algebra with tracial rank zero and which satisfies the UCT.Then A is a simple AH-algebra with real rank zero and nodimension growth. If K∗(A) is torsion free, then A is anAT-algebra. If, in addition, K1(A) = 0, then A is an AF-algebra.

Theorem (ELPW)If AΘ is a NC d-torus and G is a finite subgroup of GΘ, thenKi (AΘ o G ) = Ki (C ∗r (Zd o G )) (i = 0, 1).

(Proof) AΘ o G ∼= C ∗r (Zd , ωΘ) o G ∼= C ∗r (Zd o G , ωΘ) which hasthe same K -groups with C ∗r (Zd o G ).

16 / 23 Finite groups acting on higher dimensional noncommutative tori

Ki (C∗r (Zd o Zn))

Theorem (2012, Langer, Luck)Let n, d ∈ N. Consider the extension of groups1→ Zd → Zd o Zn → Zn → 1 such that conjugation action of Zn

on Zd is free outside the origin. Then Ki (C ∗r (Zd o Zn)) = Zsi

where

s0 =∑l≥0

rkZ((Λ2lZd )Zn ) +∑

(M)∈M

(|M| − 1),

s1 =∑l≥0

rkZ((Λ2l+1Zd )Zn ).

If n is even, then s1 = 0.

If n is a odd prime number and d = n − 1, then s1 = 2n−1−(n−1)2

2n .

(Remark) If n is a prime number and d = n − 1, thens1 = 0 if and only if n = 2, 3, 5.

17 / 23 Finite groups acting on higher dimensional noncommutative tori

〈Cn〉 ∼= Zn ≤ GLφ(n)(Z)

For a monic polynomial p(x) = a0 + a1x + · · ·+ ad−1xd−1 + xd ,the companion matrix C (p) is the d × d matrix given by

C (p) :=

−a0

1 −a1

1 −a2

. . ....

1 −ad−1

.

Given n ∈ N, the nth cyclotomic polynomial Φn(x) is thepolynomial given by

Φn(x) =∏

1≤k≤n, g .c.d(k,n)=1

(x − e2πi kn ).

Then Cn := C (Φn) is a φ(n)× φ(n) integral matrix of order n.(Here, φ denotes the Euler phi-function.)

〈Cn〉 ∼= Zn ≤ GLφ(n)(Z).

18 / 23 Finite groups acting on higher dimensional noncommutative tori

Simple AΘ on which 〈Cn〉 ∼= Zn acts canonically

Theorem (Jeong-L.)

Let n ≥ 3 and d := φ(n). Then there exists simple d-dimensionaltori AΘ on which the group Zn = 〈Cn〉 acts canonically.

(Proof) Choose any irrational number θ and putΘ := θ

∑n−1k=0(C k

n )t(C tn − Cn)C k

n . Then it is easy to see that Θ isnondegenerate and Cn ∈ GΘ.

19 / 23 Finite groups acting on higher dimensional noncommutative tori

AΘ o Zn for simple AΘ

Let d = φ(n). Then the conjugation action of Zn∼= 〈Cn〉 on Zd is

free outside the origin.

Theorem (Jeong-L.)

Let α : Zn → Aut(AΘ) be a canonical action of Zn∼= 〈Cn〉 on

simple AΘ. If n is even then AΘ oα Zn ia an AF-algebra. If n is aprime number, then AΘ oα Zn is an AF-algebra if and only ifn = 2, 3, 5.

20 / 23 Finite groups acting on higher dimensional noncommutative tori

Examples

Example (Jeong-L.)

The map given by

u1 7→ u2u∗4 , u2 7→ eπiθu∗1u∗2 ,u3 7→ u4, u4 7→ eπiθu∗1u∗2u∗3

determines an action of Z5 on Aθ ⊗Aθ.

The map given by

u1 7→ u2, u2 7→ u∗1u3, u3 7→ u2u4,u4 7→ u∗3u5, u5 7→ u2u4u6, u6 7→ eπiθu∗2u∗4u∗5u∗6

determines an action of Z7 on Aθ ⊗Aθ ⊗Aθ.

These actions are conjugate to the canonical actions of 〈C5〉 and〈C7〉 on some 4 and 6 dimensional tori, resp. If θ is irrational, then(Aθ ⊗Aθ) o Z5 is AF, (Aθ ⊗Aθ ⊗Aθ) o Z7 is not AF.

21 / 23 Finite groups acting on higher dimensional noncommutative tori

AΘ o Zn for simple AΘ and odd n

Theorem (Jeong-L.)

Let n ≥ 3 be an odd integer with 2φ(n) ≥ n + 5. Ifα : Zn → Aut(AΘ) is a canonical action of Zn

∼= 〈Cn〉 on simpleAΘ, then AΘ oα Zn is not an AF-algebra.

(Remark) The odd n’s for which 2φ(n) ≥ n + 5 do not hold are9, 15, 21, 45, . . . etc. Davis and Luck also provided a simpleformula for a prime power n. According to them, K1-group istrivial for n = 9. We don’t know yet for the rest.

22 / 23 Finite groups acting on higher dimensional noncommutative tori

Thank you for listening.

23 / 23 Finite groups acting on higher dimensional noncommutative tori