Homological properties of quantum permutation algebras

Abstract

We show that As(n), the coordinate algebra of Wang's quantum permutation group, is Calabi-Yau of dimension 3 when n ≥ 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.

An In-Depth Look at Quotient Modules

“…This paper continues the study of the quotient module Q; among other things, extending the fundamental theorem of Hopf modules along the lines of Ulbrich in Theorem 2.7 below, answering a question, implied in the paper [4] and restricted to finite dimensions, in terms of a nonzero integral in Q and (ordinary nontwisted) Frobenius extensions (see Theorem 2.11), studying a Mackey theory of Q with labels allowing variation of Hopf-group subalgebra (Section 2.4), and the ascending chain of trace ideals of Q and its tensor powers in Section 5. In Section 4, we point out that the endomorphism algebra of Q is a generalized Hecke algebra, define and study from several points of view a tower of endomorphism algebras of increasing tensor powers of Q, which is an example of a topic of current study of endomorphism algebras of tensor powers of certain modules over various groups and quantum groups.…”

“…The theorem above does not deal with a general nonzero H/R-integral t where ε(t) = 0. The paper [4] suggests the next two examples.…”