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### Transcript of Finite groups acting on higher dimensional noncommutative ... operator_2014/slides/5_1_Jae_Hyup... n

• 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

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

. . . ...

 . 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 .