Homological properties of quantum permutation algebras

Abstract

We show that $A_s(n)$, the coordinate algebra of Wang's quantum permutation group, is Calabi-Yau of dimension $3$ when $n\ge 4$, and compute its Hochschild cohomology with trivial coefficients. We also show that, for a larger class of quantum permutation algebras, including those representing quantum symmetry groups of finite graphs, the second Hochschild cohomology group with trivial coefficients vanishes, and hence these algebras have the AC property considered in quantum probability: all cocycles can be completed to a Schürmann triple.

Minor revision, 17 pages

References in corpus (2)

• Calabi-Yau algebras
• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups

Cited by in corpus (3)

• Second cohomology groups of the Hopf${}^{\ast }$-algebras associated to universal unitary quantum groups
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