1 . . . . . . . . Interpolating products of Interpolating - PowerPoint PPT Presentation

1 . . . . . . . . Interpolating products of Interpolating products of Interpolating products of quantum groups quantum groups quantum groups Daniel Gromada , Universität des Saarlandes, Saarbrücken Joint with Moritz Weber arXiv:1907.08462 Supported by SFB-TRR 195

. 2 . . . Compact matrix quantum groups . . A compact matrix quantum group is a compact quantum group with . . . That is, a pair G = ( A , u ) , where A is a C*-algebra and u ∈ M N ( A ) such a distinguished fundamental representation . . that 1. the elements u ij i , j = 1 , . . . , N generate A , 2. the matrices u and u t = ( u ji ) are invertible, 3. the map ∆: A → A ⊗ min A defined as ∆( u ij ) := � N k = 1 u ik ⊗ u kj extends to a ∗ -homomorphism.

. . 3+ . . Products of quantum groups . . For classical matrix groups G , H , we have . . �� � � g 0 G × H = | g ∈ G , h ∈ H 0 h

. . 3+ . . Products of quantum groups . . For classical matrix groups G , H , we have . . �� � � g 0 G × H = | g ∈ G , h ∈ H 0 h For G = ( A , u ) , H = ( B , v ) CMQGs: . The tensor product G × H = ( A × B , u ⊕ v ) . The free product G ∗ H = ( A ∗ B , u ⊕ v ) [Wang 1995] [Wang 1995]

. . 3 . . Products of quantum groups . . For classical matrix groups G , H , we have . . �� � � g 0 G × H = | g ∈ G , h ∈ H 0 h For G = ( A , u ) , H = ( B , v ) CMQGs: . The tensor product G × H = ( A × B , u ⊕ v ) . The free product G ∗ H = ( A ∗ B , u ⊕ v ) [Wang 1995] [Wang 1995] Question: Are there some quantum groups between G × H and G ∗ H ?

. . . 4 . Interpolating products . . . G × We define quantum subgroups of G ∗ H by imposing new relations: . . ab ∗ x = xab ∗ , a ∗ bx = xa ∗ b , × H : . G × , , axy ∗ = xy ∗ a , ax ∗ y = x ∗ ya , × H : . G × 0 H := G × , , . G × 2 k H : × H ∩ G × × H (i.e. by all four together) ) ⊗ k , a 1 x 1 · · · a k x k = x 1 a 1 · · · x k a k , ( where a , b , a 1 , . . . , a k ∈ { u ij } and x , y , x 1 , . . . , x k ∈ { v ij } . Theorem [G., Weber] : For l a divisor of k , we have × H G ∗ H ⊃ G × ⊃ ⊃ G × 0 H ⊃ G × 2 k H ⊃ G × 2 l H ⊃ G × 2 H = G × H , × H ⊃ G × The last three inclusions are strict if and only if the degree of reflection of both G and H is different from one.

. . . . Partitions 5 . . Partitions stand for quantum group relations. . . Categories of partitions define quantum groups. . . . Nice combinatorial way how to look for new quantum groups. Examples:

. . . . Partitions with extra singletons . . can be used to describe quantum subgroups G ⊂ O + 6 . N ∗ ˆ Z 2 . . ˆ Z 2 = ( C ∗ ( Z 2 ) , r ) O + N = ( C ( O + N ) , v ) , N ∗ ˆ N ) ∗ C ∗ ( Z 2 ) , v ⊕ r ) G = ( C ( G ) , v ⊕ r ) ⊂ O + Z 2 = ( C ( O + −→ ↔ v , ↔ r N × ˆ O + ↔ v ij r = rv ij −→ Z 2 × ˆ O + ↔ v ij v kl r = rv ij v kl −→ N × Z 2 ) ⊗ k ↔ v i 1 j 1 · · · v i k j k r = rv i 1 j 1 · · · v i k j k N × 2 k ˆ O + ( −→ Z 2

. . . . Classification of partitions with extra singletons . . We have a correspondence . 7 . . partitions with extra singletons ↔ two-colored partitions ↔ N ∗ ˆ ˜ G ⊂ O + G ⊂ U + ↔ Z 2 N v ⊕ r ↔ v = vr ˜ . There are some classification results for two-colored categories . Non-crossing . Globally colorized [Tarrago–Weber, 2018] . Pairs with neutral color sum [D. G., 2018] . Non-hyperoctahedral categories [Mang–Weber, 2019] . Ongoing research by Maaßen, Mang, Weber . . . [Mang–Weber, yesterday]

. . . . Summary . . We introduced new products of compact matrix quantum groups . . interpolating the free and the tensor product 8 . × H G ∗ H ⊃ G × ⊃ ⊃ G × 0 H ⊃ G × 2 k H ⊃ G × 2 l H ⊃ G × 2 H = G × H , × H ⊃ G × . We adapted the framework of partition categories to describe the N ∗ ˆ quantum subgroups of O + Z 2 by introducing partitions with extra singletons Thank you for your attention!