The Nodal Cubic is a Quantum Homogeneous Space

Abstract

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.Keywords Hopf algebra · Quantum homogeneous space · Singular curve · Noncommutative Galois extension Mathematics Subject Classification (2010) 16T05 · 16T20 · 14H50 Just as quantum groups (Hopf algebras) generalise affine algebraic groups, quantum homogeneous spaces as studied e.g. 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.There is also an analytic theory of transitive or more generally ergodic actions of compact or locally compact quantum groups, see e.g. [8,13] and the references therein.

Plane Curves Which are Quantum Homogeneous Spaces

“…Goodearl and Zhang showed [11,Construction 1.2] that every cusp y m = x n , where m and n are coprime integers with m > n ≥ 2, is a quantum homogeneous space in a noetherian Hopf domain of Gel'fand-Kirillov dimension 2. In [17], Kraehmer and the second author showed that the nodal cubic y 2 = x 3 + x 2 is a quantum homogeneous space, and Kraehmer and Martins refined this result in [16] to show that the containing Hopf algebra H can in this case be taken to be affine noetherian of Gel'fand-Kirillov dimension 1. From a slightly different perspective, some further embeddings of this sort appear in Liu's endeavour [21] aimed at extending the classification of prime affine Hopf k-algebras of Gel'fand-Kirillov dimension 1 to the non-regular case.…”

“…The study of quantum homogeneous spaces has been a pervasive theme of research in Hopf algebras over the past 25 years, from both the algebraic and analytic perspectives. For the algebraic point of view which concerns us here see, for example, [5,16,17,23,24,28].…”

“…(v) It is clear from the defining relations and the PBW theorem that A(g, f ) is not a domain, since a = ±b, but (a − b)(a + b) = 0. The easiest way to see that A(g, f ) is prime is to use the crossed product description of A(g, f ) found in the proof of This is precisely the Hopf algebra presented in [17], for the parameters p = q = 0. As noted in [16, §2.3], the other values of p and q yield isomorphic Hopf algebras.…”

“…The first three questions address specific issues concerning the algebras A(x, a, g); then, in the last two questions, we revisit the fundamental issue which motivated this paper. The stimulus for this paper was [17], but the following basic question, which was raised there, remains wide open. Question 8.4 Which affine algebraic varieties V can be realised as a quantum homogeneous space of a pointed affine Hopf k-algebra?…”

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

“…Indeed, several such examples are known by now, which include: (i) (co)action of S + n on the connected compact space formed by topologically gluing n copies of a given compact connected space [18]; (ii) Co-action of the group C * algebra C * (S 3 ) of the group of permutations of 3 objects on the coordinate ring of the variety {xy = 0} as in [11] (iii) algebraic co-action of Hopf-algebras corresponding to genuine non-compact quantum groups on commutative domains associated with affine varieties as in [35](Example 2.20). (iv) Algebraic co-action of (non-commutative) Hopf algebras on the coordinate ring of singular curves like the cusp and the nodal cubic, given by Krahmer and his collaborators (see [23], [24]).…”