Gluing Compact Matrix Quantum Groups

Gromada, Daniel

doi:10.1007/s10468-020-10010-0

Cited by 5 publications

(7 citation statements)

References 26 publications

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“…In [4, Theoreme 1(iv)] (see also [36,Remark 6.18]), Banica observed that 𝑈 + 𝑛 ≅ 𝑂 + 𝑛 * ℤ. In fact, Gromada [26,Theorem C/4.28] showed that one can replace ℤ by ℤ 2 , using the fact that the degree of reflection of 𝑂 + 𝑛 is 2. We will now generalize those observations to the matrix Hopf algebras (𝐸) and (𝐹).…”

Section: Matrix Hopf Algebras and Glued Productsmentioning

confidence: 99%

“…We can use the resolution for (𝐸) to first construct a projective resolution of (𝐸) ∶= (𝐸) * ℂℤ 2 and apply a result of Chirvasitu [16] to turn it into a free resolution for (𝐹). This part requires a generalization of the work of Gromada [26] from compact matrix quantum groups to matrix Hopf algebras.…”

Section: Introductionmentioning

confidence: 99%

“…We can use the resolution for

\mathcal {B}(E)

to first construct a projective resolution of

\mathcal {A}(E):=\mathcal {B}(E)\ast \mathbb {C}\mathbb {Z}_2

and apply a result of Chirvasitu [16] to turn it into a free resolution for

\mathcal {H}(F)

. This part requires a generalization of the work of Gromada [26] from compact matrix quantum groups to matrix Hopf algebras.…”

Section: Introductionmentioning

confidence: 99%

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Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

Baraquin,

Franz,

Gerhold

et al. 2024

Journal of London Math Soc

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We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

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Section: Matrix Hopf Algebras and Glued Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We can use the resolution for

\mathcal {B}(E)

to first construct a projective resolution of

\mathcal {A}(E):=\mathcal {B}(E)\ast \mathbb {C}\mathbb {Z}_2

and apply a result of Chirvasitu [16] to turn it into a free resolution for

\mathcal {H}(F)

. This part requires a generalization of the work of Gromada [26] from compact matrix quantum groups to matrix Hopf algebras.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

Baraquin,

Franz,

Gerhold

et al. 2024

Journal of London Math Soc

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“…When Γ (hence also Λ) is abelian, another description of the amalgamated free wreath product is possible, involving the notion of glued direct product from [20] in a specific case. Because Γ is assumed to be abelian, Λ is normal and we can consider the usual free wreath product H + N (Γ/Λ) = H + N (Γ/Λ, {e}) and build its direct product in the sense of compact quantum groups with Γ.…”

Section: 2mentioning

confidence: 99%

“…Remark 3.9. The previous construction is slightly outside the setting of [20]. However, it can easily be made to fit in if Γ is finitely generated.…”

Section: 2mentioning

confidence: 99%

Free wreath products with amalgamation

Freslon¹

2021

Preprint

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We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name "free wreath products of pairs" in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

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