Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Homomorphisms of quantum groups Sutanu Roy (joint work with R. Meyer and S.L.Woronowicz) Mathematics Institute Georg-August-University G¨ ottingen 29 June 2011 XXX Workshop on Geometric Methods in Physics, Bia� lowie˙ za, Poland Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary I bought a new car Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5 Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5 Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5 Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5 Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5 Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Multiplicative unitary Definition An operator W ∈ U ( H ⊗ H ) is said to be multiplicative unitary on the Hilbert space H if it satisfies the pentagon equation W 23 W 12 = W 12 W 13 W 23 . Examples Consider H G = L 2 ( G , λ ) for a locally compact group G with a right Haar measure λ . Then, W G ∈ U ( L 2 ( G × G , λ × λ )) defined by W G T ( x , y ) = T ( xy , y ) is a multiplicative unitary on H G . Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Multiplicative unitary Definition An operator W ∈ U ( H ⊗ H ) is said to be multiplicative unitary on the Hilbert space H if it satisfies the pentagon equation W 23 W 12 = W 12 W 13 W 23 . Examples Consider H G = L 2 ( G , λ ) for a locally compact group G with a right Haar measure λ . Then, W G ∈ U ( L 2 ( G × G , λ × λ )) defined by W G T ( x , y ) = T ( xy , y ) is a multiplicative unitary on H G . Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Observations One can define two non-degenerate, normal, coassociative ∗ -homomorphisms from B ( H ) to B ( H ⊗ H ): ∆( x ) = W ( x ⊗ I ) W ∗ � ∆( y ) = Ad(Σ) ◦ ( W ∗ ( I ⊗ y ) W ) . for all x , y ∈ B ( H ) and Σ is the flip operator acting on H ⊗ H . Consider the slices/legs of the multiplicative unitaries: � . � C = { ( ω ⊗ id) W : ω ∈ B ( H ) ∗ } � . � . � C = { (id ⊗ ω ) W : ω ∈ B ( H ) ∗ } Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Observations One can define two non-degenerate, normal, coassociative ∗ -homomorphisms from B ( H ) to B ( H ⊗ H ): ∆( x ) = W ( x ⊗ I ) W ∗ � ∆( y ) = Ad(Σ) ◦ ( W ∗ ( I ⊗ y ) W ) . for all x , y ∈ B ( H ) and Σ is the flip operator acting on H ⊗ H . Consider the slices/legs of the multiplicative unitaries: � . � C = { ( ω ⊗ id) W : ω ∈ B ( H ) ∗ } � . � . � C = { (id ⊗ ω ) W : ω ∈ B ( H ) ∗ } Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Special class of multiplicative unitaries Manageability and modularity Manageable multiplicative unitary. [Woronowicz, 1997] Modular multiplicative unitary. [So� ltan-Woronowicz, 2001] Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Definition Hopf *-homomorphisms Legs of a multiplicative unitary Equivalent pictures of homomorphisms of quantum groups Summary Nice legs of modular multiplicative unitaries Theorem (So� ltan, Woronowicz, 2001) Let, W ∈ U ( H ⊗ H ) be a modular multiplicative unitary. Then, C and � C are C ∗ -sub algebras in B ( H ) and W ∈ UM ( � C ⊗ C ) . there exists a unique ∆ C ∈ Mor( C , C ⊗ C ) such that (id � C ⊗ ∆)W = W 12 W 13 . ∆ C is coassociative: (∆ C ⊗ id C ) ◦ ∆ C = (id C ⊗ ∆ C ) ◦ ∆ C . ∆( C )(1 ⊗ C ) and ( C ⊗ 1)∆( C ) are linearly dense in C ⊗ C. There exists an involutive normal antiautomorphism R C of C. Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Outline Multiplicative unitaries 1 Locally compact quantum groups 2 Hopf *-homomorphisms 3 Equivalent pictures of homomorphisms of quantum groups 4 Bicharacters Universal bicharacter Right or left coactions as homomorphisms Morphism as a functor between coaction categories Summary 5
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Locally compact quantum groups Definition [So� ltan-Woronowicz, 2001] The pair G = ( C , ∆ C ) is said to be a locally compact quantum group if the C ∗ -algebra C and ∆ C ∈ Mor( C , C ⊗ C ) comes from a modular multiplicative unitary W . We say W giving rise to the quantum group G = ( C , ∆ C ). Observation The unitary operator � W = Ad(Σ)( W ∗ ) gives rise to the quantum group � G = ( � C , ∆ ˆ C ) which is dual to G = ( C , ∆ C ). Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary Locally compact quantum groups Definition [So� ltan-Woronowicz, 2001] The pair G = ( C , ∆ C ) is said to be a locally compact quantum group if the C ∗ -algebra C and ∆ C ∈ Mor( C , C ⊗ C ) comes from a modular multiplicative unitary W . We say W giving rise to the quantum group G = ( C , ∆ C ). Observation The unitary operator � W = Ad(Σ)( W ∗ ) gives rise to the quantum group � G = ( � C , ∆ ˆ C ) which is dual to G = ( C , ∆ C ). Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
Multiplicative unitaries Locally compact quantum groups Hopf *-homomorphisms Equivalent pictures of homomorphisms of quantum groups Summary From groups to quantum groups Given a locally compact group G : G = ( C 0 ( G ) , ∆) is a locally compact quantum group with ∆ f ( x , y ) = f ( xy ). � r ( G ) , ˆ G = (C ∗ ∆) is the dual quantum group of G with ∆( λ g ) = λ g ⊗ λ g for all g ∈ G . G u = ( C ∗ ( G ) , ˆ � ∆ u ) is a C ∗ -bialgebra which is known as quantum group C ∗ -algebra of G . Sutanu Roy (G¨ ottingen) Homomorphisms of quantum groups
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