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

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Interpolating products of quantum groups Interpolating products of quantum groups Interpolating products of quantum groups

Daniel Gromada, Universität des Saarlandes, Saarbrücken Joint with Moritz Weber arXiv:1907.08462

Supported by SFB-TRR 195

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Compact matrix quantum groups

. A compact matrix quantum group is a compact quantum group with

a distinguished fundamental representation.

. That is, a pair G = (A, u), where A is a C*-algebra and u ∈ MN(A) such

that

• 1. the elements uij i, j = 1, . . . , N generate A,
• 2. the matrices u and ut = (uji) are invertible,
• 3. the map ∆: A → A ⊗min A defined as ∆(uij) := N

k=1 uik ⊗ ukj extends

to a ∗-homomorphism.

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3+.

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Products of quantum groups

For classical matrix groups G, H, we have G × H =

• g

h

• | g ∈ G, h ∈ H

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3+.

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Products of quantum groups

For classical matrix groups G, H, we have G × H =

• g

h

• | g ∈ G, h ∈ H
• For G = (A, u), H = (B, v) CMQGs:

. The tensor product G × H = (A × B, u ⊕ v)

[Wang 1995]

. The free product G ∗ H = (A ∗ B, u ⊕ v)

[Wang 1995]

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Products of quantum groups

For classical matrix groups G, H, we have G × H =

• g

h

• | g ∈ G, h ∈ H
• For G = (A, u), H = (B, v) CMQGs:

. The tensor product G × H = (A × B, u ⊕ v)

[Wang 1995]

. The free product G ∗ H = (A ∗ B, u ⊕ v)

[Wang 1995] Question: Are there some quantum groups between G × H and G ∗ H?

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Interpolating products

We define quantum subgroups of G ∗ H by imposing new relations:

. G ×

× H: ab∗x = xab∗, a∗bx = xa∗b, , ,

. G ×

× H: ax∗y = x∗ya, axy ∗ = xy ∗a, , ,

. G ×0 H := G ×

× H ∩ G × × H (i.e. by all four together)

. G ×2k H:

a1x1 · · · akxk = x1a1 · · · xkak, ( )⊗k, where a, b, a1, . . . , ak ∈ {uij} and x, y, x1, . . . , xk ∈ {vij}. Theorem [G., Weber]: For l a divisor of k, we have G ∗ H ⊃ G × × H ⊃ ⊃ G × × H ⊃ G ×0 H ⊃ G ×2k H ⊃ G ×2l H ⊃ G ×2 H = G × H, The last three inclusions are strict if and only if the degree of reflection

• f both G and H is different from one.

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Partitions

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

Examples:

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Partitions with extra singletons

. can be used to describe quantum subgroups G ⊂ O+

N ∗ ˆ

Z2 O+

N = (C(O+ N), v),

ˆ Z2 = (C ∗(Z2), r) −→ G = (C(G), v ⊕ r) ⊂ O+

N ∗ ˆ

Z2 = (C(O+

N) ∗ C ∗(Z2), v ⊕ r)

↔ v, ↔ r ↔ vijr = rvij −→ O+

N × ˆ

Z2 ↔ vijvklr = rvijvkl −→ O+

N ×

× ˆ Z2 ( )⊗k ↔ vi1j1 · · · vikjkr = rvi1j1 · · · vikjk −→ O+

N ×2k ˆ

Z2

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Classification of partitions with extra singletons

. We have a correspondence

partitions with extra singletons ↔ two-colored partitions ↔ G ⊂ O+

N ∗ ˆ

Z2 ↔ ˜ G ⊂ U+

N

v ⊕ r ↔ ˜ v = vr

. There are some classification results for two-colored categories . Non-crossing

[Tarrago–Weber, 2018]

. Globally colorized

[D. G., 2018]

. Pairs with neutral color sum

[Mang–Weber, 2019]

. Non-hyperoctahedral categories

[Mang–Weber, yesterday]

. Ongoing research by Maaßen, Mang, Weber. . .

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Summary

. We introduced new products of compact matrix quantum groups

interpolating the free and the tensor product G ∗ H ⊃ G × × H ⊃ ⊃ G × × H ⊃ G ×0 H ⊃ G ×2k H ⊃ G ×2l H ⊃ G ×2 H = G × H,

. We adapted the framework of partition categories to describe the

quantum subgroups of O+

N ∗ ˆ

Z2 by introducing partitions with extra singletons

Thank you for your attention!