Relative Fourier transforms and expectations on coideal subalgebras

Abstract

For an algebraic compact quantum group H we establish a bijection between the set of right coideal * -subalgebras A → H and that of left module quotient * -coalgebras H → C. It turns out that the inclusion A → H always splits as a map of right A-modules and right H-comodules, and the resulting expectation E : H → A is positive (and lifts to a positive map on the full C * completion on H) if and only if A is invariant under the squared antipode of H.The proof proceeds by Tannaka-reconstructing the coalgebra C corresponding to A → H by means of a fiber functor from H-equivariant A-modules to Hilbert spaces, while the characterization of those A → H which admit positive expectations makes use of a Fourier transform turning elements of H into functions on C.

On the monoidal invariance of the cohomological dimension of Hopf algebras

“…Similarly the natural map A * n → A * w A s (n) has a retraction, and hence A * n stands as left coideal * -subalgebra of A * w A s (n). By the results in [13], A * w A s (n) is thus faithfully flat as A * n -module, hence projective [28]. We then have, using [7,Corollary 5.3 The converse inequality obviously holds is cd(A) is infinite, hence we can assume that cd(A) is finite, and hence, in view of our assumption, that cd GS (A) is finite.…”

Faithful flatness of Hopf algebras over coideal subalgebras with a conditional expectation

“…Theorem 4.3 therefore ensures that if A has bijective antipode, then A * w A s (n) is projective as a left and right A * n -module. This was shown and used in the proof of [4,Theorem 8.4], using [7], under the additional assumption that k = C and that A is a compact Hopf algebra.…”

“…(2) Skryabin [17] has shown that finite-dimensional Hopf algebras are free over their coideal subalgebras. (3) Compact Hopf algebras are faithfully flat over their coideal * -subalgebras: this was shown by Chirvasitu [7]. The primary motivation for this work was the idea that it would be useful to write a self-contained proof of the above mentioned result [6, Theorem 2.1] of Chirvasitu on the faithful flatness of a cosemisimple Hopf algebra over its Hopf subalgebras.…”

Dolbeault-Dirac Fredholm Operators for Quantum Homogeneous Spaces

“…In Takeuchi's original formulation [79], the equivalence is stated for a coideal C of a Hopf algebra A, such that the functor A ⊗ C −, from left C-modules to vector spaces, is faithfully flat. As shown in [11,Corollary 3.4.5], for any coideal * -subalgebra of a CQGA, faithful flatness is automatic. In particular, it is automatic for any CQGA-homogeneous space.…”

“…∂ is Fredholm, then it is closed by definition, and hence by (10) and (11), it must have finite-dimensional anti-holomorphic cohomology groups. Conversely, if D + ∂ is closed and has finite-dimensional anti-holomorphic cohomology groups, then its kernel and cokernel must be finite dimensional.…”