Twisted Calabi–Yau property of right coideal subalgebras of quantized enveloping algebras

Quantum homogeneous spaces of connected Hopf algebras

“…while easily seen to be an algebra homomorphism, might not take values in T . But this is in fact so for all such quantum homogeneous spaces T in H, by [23,Lemma 3.9], building on work of [18]. Consequently, we obtain:…”

“…1.3. Following [23], we call a left or right coideal subalgebra T of a Hopf algebra H a quantum homogeneous space if H is faithfully flat as a left and as a right T -module. Thus (1) of the theorem ensures that all coideal subalgebras of H as in the theorem are quantum homogeneous spaces.…”

“…For the unexplained terminology in (3), see §4.3. The determination of the Nakayama automorphism in (3) depends crucially on earlier work of Kraehmer [18] and of Liu and Wu [23], [24].…”

“…For example, Kraehmer in [18] defines a left quantum homogeneous space to be a left coideal subalgebra C of a Hopf algebra H such that H is a faithfully flat left C-module. We adopt in this paper the formally more restrictive definition used in [23]: a left quantum homogeneous space of the Hopf algebra H with a bijective antipode S is a left coideal subalgebra T of H such that H is a faithfully flat left and right T -module. In fact, for the connected Hopf algebras which are the object of study in this paper, the distinction is irrelevant, as shown by the result below.…”

The Nodal Cubic is a Quantum Homogeneous Space

“…in [2,7,11,[14][15][16][17][18][19]21] generalise affine varieties with a transitive action of an algebraic group: Definition A quantum homogeneous space is a right coideal subalgebra B of a Hopf algebra A which is faithfully flat as a left B-module.…”