Quantum isometry groups
Jyotishman Bhowmick
Indian Statistical Institute
07.11.2015
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
07.11.2015
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Outline
1
The set up
The set up
Classical
Compact Hausdorff space
Compact Group
Group Action
Riemannian manifold
Isometry group
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum
Unital C∗ algebra
Compact Quantum Group
Coaction
Spectral triple
Quantum Isometry Group
Quantum isometry groups
Background
Gelfand-Naimark
Woronowicz
Woronowicz
Connes
To be discussed
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Background
Alain Connes’ question ( 1995 )
What is the quantum symmetry group of a NONcommutative space?
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Background
Alain Connes’ question ( 1995 )
What is the quantum symmetry group of a NONcommutative space?
Answer
1
In 1998 Wang defined the notion of quantum symmetry groups for finite
NONcommutative spaces.
2
Banica and Bichon defined quantum symmetry groups for finite metric
spaces, finite graphs, etc.
3
Lots of examples computed leading to discovery of completely new
kinds of quantum groups
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
07.11.2015
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Background
Alain Connes’ question ( 1995 )
What is the quantum symmetry group of a NONcommutative space?
Answer
1
In 1998 Wang defined the notion of quantum symmetry groups for finite
NONcommutative spaces.
2
Banica and Bichon defined quantum symmetry groups for finite metric
spaces, finite graphs, etc.
3
Lots of examples computed leading to discovery of completely new
kinds of quantum groups
Remark
The above examples dealt with some finite or discrete structures.
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Quantum symmetry for ( NONCOMMUTATIVE ) manifolds
Quantum isometry groups ( Bhowmick + Goswami, J.F.A., 2009 )
Construction of the “biggest” Compact Quantum Group acting isometrically
on a noncommutative Riemannian manifold ( given by a spectral data ).
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Quantum symmetry for ( NONCOMMUTATIVE ) manifolds
Quantum isometry groups ( Bhowmick + Goswami, J.F.A., 2009 )
Construction of the “biggest” Compact Quantum Group acting isometrically
on a noncommutative Riemannian manifold ( given by a spectral data ).
Ingrediants
1
Geometry given by Spectral triples on noncommutative C∗ algebras.
2
Symmetry given by Compact Quantum Groups ( CQG ) and their
coactions.
3
Isometric coactions
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Quantum isometry groups
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Results for classical spaces
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Results for classical spaces
Theorem( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
QISO+ (C∞ (M), L2 (S), D) is classical for M = S1 , T2 , Sn .
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Quantum isometry groups
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Results for classical spaces
Theorem( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
QISO+ (C∞ (M), L2 (S), D) is classical for M = S1 , T2 , Sn .
Theorem ( Bhowmick + Banica, De Kommer, Ann. Math. Blaise
Pascal, 2012)
A classical Riemannian manifold cannot have genuine group dual isometries.
In particular, all the known examples of orthogonal easy quantum groups
cannot coact isometrically on a classical Riemannian manifold.
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Computation for group C∗ -algebras
For a group G, we compute QISO+ (Cr∗ (G)).
Proposition( Bhowmick-Skalski, Journal of Geometry and Physics,
2010 )
1
2
For G = Zn , n 6= 4, QISO+ is commutative,QISO+ = C∗ (Zn ) ⊕ C∗ (Zn ).
For G = Z4 , QISO+ = C∗ (D∞ × Z2 ). ( noncommutative and infinite
dimensional )
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Computation for Infinite cyclic groups
Proposition( Bhowmick-Skalski, Journal of Geometry and Physics,
2010 )
For G = Z, QISO+ (C∗ (Z)) = C(S1 o Z2 ).
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Quantum isometry groups
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Symmetric groups S3
Generating set: {(1, 2), (2, 3)}.
Proposition( Bhowmick-Skalski, Journal of Geometry and Physics,
2010 )
QISO+ (C∗ (S3 )) = C∗ (S3 ) ⊕ C∗ (S3 ).
Remark, Dalecki-Soltan
This was extended to all Sn by Dalecki-Soltan.
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Computation for Free group
Theorem ( Bhowmick-Skalski, Journal of Geometry and Physics, 2010
)
+
QISO+ (Cr∗ (F2 )) ∼
of Banica-Skalski.
= H2,0
New 2 parameter family of CQG ( Banica-Skalski)
+ is the universal C ∗ algebra generated by
Hp,q
1
(2p + q)2 partial isometries {Uz,y : z, y ∈ τp,q }
2
For iα, jβ ∈ τp,q , M, N ∈ {1, 2, ..., q},
3
{Uz,y : z, y ∈ τp,q } forms a unitary matrix.
4
∗
Uiα,N
= Uīα,N
5
∗
UM,jβ
= UM,j̄β
6
∗
UM,N
= UM,N
+
Moreover, QISO+ (Cr∗ (Fp )) ∼
.
= Hp,0
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Computation for other noncommutative C∗ algebras
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Computation for other noncommutative C∗ algebras
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
QISO+ for the Chakraborty-Pal triple on SUµ (2) is Uµ (2).
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Computation for other noncommutative C∗ algebras
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
QISO+ for the Chakraborty-Pal triple on SUµ (2) is Uµ (2).
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2010 )
2 ) ∼ SO (3).
QISO+ (Sµ,c
= µ
Jyotishman Bhowmick (Indian Statistical Institute)
Quantum isometry groups
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Computation for other noncommutative C∗ algebras
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
QISO+ for the Chakraborty-Pal triple on SUµ (2) is Uµ (2).
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2010 )
2 ) ∼ SO (3).
QISO+ (Sµ,c
= µ
Theorem (Bhowmick + D. Goswami, A. Skalski, Trans. Amer. Math.
Soc., 2011 )
Quantum isometry group commutes with inductive limits.
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Result for Rieffel deformations
Set up
1
2
3
4
A C∗ dynamical system (A, Tn , β)
A faithful representation π0 : A → B(H) and a spectral triple
(A∞ , H, D, R).
fn with a covering map γ : T
fn → Tn , Lie
A compact abelian group T
n
n
n
f and T identified with R .
algebras of both T
fn on H such that
A strongly continuous unitary representation V of T
−1
Vg̃ D = DVg̃ and Vg̃ π0 (a)Vg̃ = π0 (βγ(g̃) (a))
Theorem ( Bhowmick + Goswami, Journal of Functional Analysis,
2009 )
1
2
(A∞
J , H, D) is also a deformed spectral triple,
QISO+ (A∞ , H, D) ∼
= [QISO+ (A∞ , H, D)]J⊕−J .
J
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