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Finite groups acting on higher dimensional noncommutative tori

Jae Hyup Lee Seoul 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 ∗-algebra generated by two unitaries u1, u2 satisfying

u2u1 = e 2πiθu1u2.

If θ ∈ Z, then Aθ ∼= C (T× T). So rotation algebras are also called 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 if θ ≡ ±θ′ mod Z.

2 / 23 Finite groups acting on higher dimensional noncommutative tori

Actions of SL2(Z) on rotation algebras

Given A =

( a b c d

) ∈ SL2(Z), the map

u1 7→ eπiacθua1uc2 ,

u2 7→ eπibdθub1 ud2 determines an automorphism αA of Aθ, and the choice of the scalars 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 to conjugate which are necessarily

Z2 = 〈 ( −1 0 0 −1

) 〉, Z3 = 〈

( −1 −1 1 0

) 〉,

Z4 = 〈 (

0 −1 1 0

) 〉, Z6 = 〈

( 0 −1 1 1

) 〉.

Theorem (2010, Echterhoff-Lück-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 d dimensional noncommutative torus AΘ associated with Θ is defined by the universal C ∗-algebra generated by d unitaries u1, · · · , ud subject to the relations

uj ui = e 2πiθij 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 group actions by some consistent matrix group which acts on every torus of 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 crossed products 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) = e πi〈Θx ,y〉

for x , y ∈ Zd .

The regular ωΘ-representation `Θ : Zd → U(`2(Zd )) is defined 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 defined by 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 of B(`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 group homomorphism 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 be called 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 as the SL2(Z) action on the rotation algebra since GΘ = SL2(Z) and

Ad UA(u1) = e πiθacua1u

c 2 ,

Ad UA(u2) = e πiθbd ub1 u

d 2 ,

where A =

( a b c 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 be degenerate 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 cyclic groups

Theorem (Jeong-L.)

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

(Proof) We use the complete list of elements of non-trivial finite orders in GL3(Z) up to conjugate which consists of only 15 elements due to [1971, K. Tahara]. For AΘ on where elements in that 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 on a simple AΘ. Then

α has the tracial Rokhlin property [ELPW]. So AΘ oα G is simple 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 unital separable nuclear C ∗-algebras with tracial rank zero which satisfy the 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 nuclear C ∗-algebra with tracial rank zero and which satisfies the UCT. Then A is a simple AH-algebra with real rank zero and no dimension growth. If K∗(A) is torsion free, then A is an AT-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, then Ki (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 , ω̃Θ) which has the 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, Lück) Let n, d ∈ N. Consider the extension of groups 1→ 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((Λ 2lZd )Zn ) +

∑ (M)∈M

(|M| − 1),

s1 = ∑ l≥0

rkZ((Λ 2l+1Zd )Zn ).

If n is even, then s1 = 0.

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

2n .

(Remark) If n is a prime number and d = n − 1, then s1 = 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 the polynomial given by

Φn(x) = ∏

1≤k≤n, g .c.d(k,n)=1

(x − e2πi k n ).

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-dimensional tori AΘ on which the group Zn = 〈Cn〉 acts canonically.

(Proof) Choose any irrational number θ and put Θ := θ

∑n−1 k=0(C

k n )

t(C tn − Cn)C kn .

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