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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

    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 .