Actions of Compact Quantum Groups VI Free and homogeneous actions II Kenny De Commer (VUB, Brussels, Belgium)
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Outline Homogeneous and free: Galois objects From homogeneous to free and back Homogeneous actions of SU q (2)
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Galois objects Definition α G Galois object (or quantum torsor) if X � 1. α free, 2. α homogeneous, 3. C ( X ) � = { 0 } . Lemma α G Galois object, then X ∼ If X � = G equivariantly. No longer true in quantum case!
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Quantum torus Example (Quantum torus) Let θ ∈ [0 , 2 π ] . Put C ( T 2 θ ) = C ∗ ( U, V | U, V unitary , UV = e iθ V U. } . θ � T 2 by Then free and homogeneous T 2 α ( w,z ) ( U ) = wU, α ( w,z ) V = zV. Remarks: ◮ Check that C ( T 2 q ) not trivial. ◮ Instance of general construction: 2-cocycles on discrete quantum groups.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Twisting procedure Theorem (Bichon-De Rijdt-Vaes) 1. There is a one-to-one-correspondence between (classes of) ◮ Galois objects X for G , ◮ Fiber functors F on Rep( G ) (into Hilbert spaces). 2. Let X � α G Galois object. Then ∃ ! H such that ◮ H � β X is (left) Galois object, ◮ α and β commute. Remark: ◮ Abstractly: H from Tannaka-Krein on F . ◮ Concretely: C ( H red ) ⊆ C ( X red ) ⊗ C ( X red ) op . ◮ One says G and H monoidally equivalent.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Another look at quantum SU (2) and O + ( n ) Definition Take F ∈ GL n ( C ) with F ¯ F ∈ R . Then u ( F )) = C ∗ ( u ij | 1 ≤ i, j ≤ n, U unitary , FUF − 1 = U ) C ( O + becomes compact quantum group for � ∆( u ij ) = u ik ⊗ u kj . k Example � � 0 1 , C ( O + 1. For F = u ( F )) = C ( SU q (2)) . − q − 1 0 2. For F = I n , C ( O + u ( I n )) = C ( O + n ) .
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Classification of all Galois objects of SU q (2) Notation For F ∈ GL n ( C ) with F ¯ F ∈ R , write c F = − sign( F ¯ F ) Tr ( F ∗ F ) . Remark: Always | c F | ≥ 2 . Theorem (Bichon-De Rijdt-Vaes) ◮ { O + ( F ) } is complete w.r.t. monoidal equivalence. ∼ ◮ O + ( F 1 ) mon. eq. O + ( F 2 ) iff c F 1 = c F 2 . = mon. eq. SU q (2) for q + q − 1 = c F . ∼ ◮ O + ( F ) = In fact, Galois object between O + ( F 1 ) and O + ( F 2 ) 1 ≤ i ≤ dim ( F 1 ) , 1 ≤ j ≤ dim ( F 2 ) u ( F 1 , F 2 )) = C ∗ � � C ( O + u ij | . U unitary , F 1 UF − 1 = U 2
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Morita base change Lemma Let X � G free, Y = X / G . Assume p ∈ M ( C 0 ( Y )) full projection: [ C 0 ( Y ) pC 0 ( Y )] = C 0 ( Y ) . Then, with C 0 ( X p ) = pC 0 ( X ) p , free action X p � G by α p : C 0 ( X p ) → C 0 ( X p ) ⊗ C ( G ) , a �→ α ( a ) . Moreover, with C 0 ( Y p ) = pC 0 ( Y ) p , X p / G = Y p . Remarks: ◮ C 0 ( Y p ) and C 0 ( Y ) (strongly) Morita equivalent (‘non-commutative isomorphism Y p ∼ = Y ’ ). ◮ Then also C 0 ( X p ) and C 0 ( X ) Morita equivalent.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Proof ◮ Well-defined coaction: clear. ◮ Free: using C 0 ( X ) = [ C 0 ( Y ) C 0 ( X ) C 0 ( Y )] , [ α p ( C 0 ( X p ))( C 0 ( X p ) ⊗ 1)] = [( p ⊗ 1) α ( C 0 ( X ))( pC 0 ( X ) p ⊗ 1)] = [( p ⊗ 1) α ( C 0 ( X ))( C 0 ( Y ) pC 0 ( Y ) C 0 ( X ) p ⊗ 1)] = [( p ⊗ 1) α ( C 0 ( X ))( C 0 ( Y ) C 0 ( X ) p ⊗ 1)] = [( p ⊗ 1) α ( C 0 ( X ))( C 0 ( X ) p ⊗ 1)] = [ pC 0 ( X ) p ⊗ C ( G )] = C 0 ( X p ) ⊗ C ( G ) . ◮ X p / G = Y p as exercise.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Reduction to free actions Recall: X � G homogeneous, then C 0 ( X ⋊ G ) ∼ = ⊕ i ∈ I B 0 ( H i ) . Corollary (Wassermann construction) Consider e ( i ) 00 fixed matrix unit in B 0 ( H i ) , and put p = ⊕ e ( i ) 00 ∈ M ( ⊕ i ∈ I B 0 ( H i )) . ◮ p full projection for ⊕ i ∈ I B 0 ( H i ) . ◮ X free � G free with C 0 ( X free ) = pC 0 ( X ⋊ G ⋊ � G ) p . ◮ C 0 ( X free / G ) = c 0 ( I ) . Proof. Use ( X ⋊ G ⋊ � G ) / G = X ⋊ G .
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) From free to homogeneous and back Lemma X � G with C 0 ( X / G ) = c 0 ( I ) , and C ( X i ) = δ i C 0 ( X ) δ i . Then X i � G homogeneous. Lemma Let X � G homogeneous. ◮ C 0 ( X ⋊ G ) ։ B ( L 2 Y ( X )) ⇒ distinguished block B 0 ( H i 0 ) ⊆ C 0 ( X ⋊ G ) . ◮ Associated projection δ i 0 ∈ c 0 ( I ) ⊆ C 0 ( X free ) is full. ∼ ⇒ C 0 ( X free ) Morita C 0 ( X ) . = Theorem G CQG. The above gives one-to-one correspondence between ◮ (Irreducible) free actions X ′ � G with X ′ / G classical discrete set (up to iso) ◮ Homogeneous actions X � G (up to ‘equivariant Morita equivalence ’ ). ⇒ classifying homogeneous actions = classifying certain free actions.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Free actions and fiber functors Definition I a set. Monoidal category ( I Hilb I , ⊠ ) : ◮ Objects: I -bigraded Hilbert spaces, H = ⊕ k H l , k,l ◮ Tensor product: k ( H ⊠ G ) l = ⊕ m k H m ⊗ m G l . Theorem (DC-Yamashita) There is a one-to-one correspondence between 1. Free actions X � G with X / G classical discrete set I (up to isomorphism) 2. Tensor C ∗ -functors Rep fd ( G ) → I Hilb I (up to ‘equivalence’). � Concrete Tannaka-Krein reconstruction process.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Reduction scheme Classifying homogeneous actions of SU q (2) . � Classifying free actions of SU q (2) with discrete quotient space � Classifying Monoidal C ∗ -functors Rep fd ( SU q (2)) → I Hilb I . But... Rep fd ( SU q (2)) easy generators and relations... Classifying Monoidal C ∗ -functors Rep fd ( SU q (2)) → I Hilb I . � Combinatorial data. Remark: ∃ q, Rep( O + ( F )) = Rep( SU q (2)) , so: classification homogeneous X � O + ( F ) .
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Representation category of SU q (2) Lemma Rep fd ( SU q (2)) is ‘completion’ of tensor C ∗ -category with ◮ Objects: finite ordinals ◮ Basis for morphisms: non-crossing 2-partitions ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ and ◮ Tensor product: horizontal juxtaposition ◮ Composition: vertical stacking with rule ★ = − q − q − 1 . ◮ ∗ -structure: ∩ ∗ = − sgn( q ) ∪ .
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Reciprocal random walks Definition (DC-Yamashita) Let δ ∈ R 0 . A δ -reciprocal random walk consists of a quadruple (Γ , w, sgn , i ) where ◮ Γ = (Γ (0) , Γ (1) , s, t ) is a graph with source and target maps s and t , ◮ w is a weight function w : Γ (1) → R + 0 , ◮ sgn a sign function sgn: Γ (1) → {± 1 } , ◮ i is an involution e �→ e on Γ (1) interchanging source and target, s.t. ◮ for all e , w ( e ) w (¯ e ) = 1 , ◮ for all e , sgn( e )sgn(¯ e ) = sgn( δ ) , 1 ◮ for all v , � | δ | w ( e ) = 1 . s ( e )= v
� � � � � � Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Examples ◮ Action SU q (2) on non-standard Podle´ s sphere S 2 q,x qx + q − x qx +1+ q − x − 1 qx − 1+ q − x +1 qx + q − x � • · · · � • · · · • qx − 1+ q − x +1 qx + q − x qx + q − x qx +1+ q − x − 1 Figure: δ = − ( q + q − 1 ) ( q > 0 , x ∈ R ) ◮ Action O + n on S N − 1 + 1 1 N ( N − 2) N − 1 N − 1 � • · · · � • • 1 N − 1 N − 1 N ( N − 2) Figure: δ = N
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) Abundance of reciprocal random walks Lemma (DC-Yamashita) Let (Γ , w, sgn , i ) δ -reciprocal random walk. Then Γ bounded degree: v ∈ Γ (0) # { e ∈ Γ (1) | s ( e ) = v } < ∞ . sup Theorem (DC-Yamashita) Γ bounded degree ⇒ ∃ δ and δ -reciprocal random walk on Γ . ‘Proof’. By Frobenius-Perron theory. Theorem (Kronecker) ADE-classification for 2 -reciprocal random walks.
Homogeneous and free: Galois objects Homogeneity versus freeness Homogeneous actions of SU q (2) A one-to-one correspondence Theorem (DC-Yamashita) Fix q � = 0 , put δ q = − q − q − 1 . There is (up to appropriate equivalence) a one-to-one correspondence between ◮ Tensor C ∗ -functors F : Rep → I Hilb I , and ◮ δ q -reciprocal random walks Γ = (Γ (0) , Γ (1) , s, t ) with Γ (0) = I . Construction of F from Γ : ◮ F ( ) = l 2 (Γ (0) ) , ◮ F ( • ) = l 2 (Γ (1) ) , ◮ F ( ∩ )( δ v ) = sgn( e ) w ( e ) 1 / 2 δ e ⊗ δ ¯ � e . s ( e )= v
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