Homology of free quantum groups

Abstract

We compute the Hochschild homology of the free orthogonal quantum group $A_o(n)$. We show that it satisfies Poincaré duality and should be considered to be a 3-dimensional object. We then use recent results of R. Vergnioux to derive results about the ${\ell }^2$-homology of $A_o(n)$ and estimates on the free entropy dimension of its set of generators. In particular, we show that the ${\ell }^2$ Betti-numbers of $A_o(n)$ all vanish and that the free entropy dimension is less than 1.

8 pages

References in corpus (1)

• L^2-Homology for von Neumann Algebras

Cited by in corpus (14)

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• Homological properties of quantum permutation algebras
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• Lévy-Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups
• Higher l^2-Betti numbers of universal quantum groups
• Cohomological dimensions of universal cosovereign Hopf algebras
• Second cohomology groups of the Hopf${}^{\ast }$-algebras associated to universal unitary quantum groups
• Cyclic Homology and Quantum Orbits
• Cohomology Rings of Plactic Monoid Algebra via Groubner-Shirshov Basis
• Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras
• Reduktionssysteme zur Berechnung einer Auflösung der orthogonalen freien Quantengruppen $A_o(n)$