A quantum homogeneous space is a right coideal subalgebra BB of a Hopf algebra HH over which HH is faithfully flat as a left and as a right module; the standard examples are coordinate rings of quantum spheres and Grassmannians and the coideal subalgebras of quantized enveloping algebras. Liu and Wu (J. Algebra 353 (2012) 121–141) proved that such BB inherit injective-dimension bounds, noetherianity, and the Auslander condition from a noetherian Hopf algebra HH with bijective antipode, constructed rigid dualizing complexes and Nakayama automorphisms for BB under pointed/liftability or normal-basis hypotheses, and classified all quantum homogeneous spaces of Uq(sl2)U_q(\mathfrak{sl}_2). They left open their Question 4.7 — whether BB must be Artin–Schelter Gorenstein whenever HH is — together with companion questions: whether l.gl.dim B=proj.dim⁡Bk=Hcdim B\mathrm{l.gl.dim}\,B=\operatorname{proj.dim}_B k=\mathrm{Hcdim}\,B unconditionally, and whether the group-like element controlling the Nakayama automorphism is always liftable.

Background

A Hopf algebra HH over a field kk is an algebra carrying a compatible comultiplication Δ:H→H⊗H\Delta:H\to H\otimes H, counit ε:H→k\varepsilon:H\to k, and antipode SS. A right coideal subalgebra is a subalgebra B⊆HB\subseteq H with Δ(B)⊆B⊗H\Delta(B)\subseteq B\otimes H. A quantum homogeneous space of HH is a right coideal subalgebra BB such that HH is faithfully flat as a left and as a right BB-module; this is the standard algebraic model of a quantum subgroup orbit, with guiding examples the coordinate rings of Podleś spheres and quantum Grassmannians inside the coordinate Hopf algebras Oq(G)O_q(G), and coideal subalgebras of quantized enveloping algebras Uq(g)U_q(\mathfrak g) such as their negative parts. A noetherian kk-algebra AA with augmentation A→kA\to k is Artin–Schelter Gorenstein (AS-Gorenstein) of dimension dd if AA has finite left and right injective dimension dd and Ext⁡Ai(k,A)\operatorname{Ext}^i_A(k,A) vanishes for i≠di\ne d and is one-dimensional for i=di=d, with the analogous right-module conditions; if in addition AA has finite global dimension it is AS-regular, and if it satisfies the Auslander condition — for every finitely generated module, each submodule appearing in the ii-th term of a minimal injective resolution has injective dimension at most ii — it is Auslander-Gorenstein. For noetherian Hopf algebras with bijective antipode, Brown–Zhang (2008) and Lu–Wu–Zhang (2009) developed the theory of rigid dualizing complexes — distinguished dualizing complexes, unique up to shift, that control duality for Hochschild (co)homology — and showed that AS-Gorenstein Hopf algebras are twisted Calabi–Yau: the top self-Ext bimodule of AA is a one-dimensional twist of AA, and the twisting automorphism (the Nakayama automorphism, measuring the failure of the duality to be symmetric) is S2S^2 composed with a winding automorphism Ξ[τ]\Xi[\tau], Ξ[τ](a)=∑τ(a(1))a(2)\Xi[\tau](a)=\sum\tau(a_{(1)})a_{(2)}, built from the character τ\tau of the one-dimensional homological integral Ext⁡Ad(k,A)\operatorname{Ext}^d_A(k,A) (here Δ(a)=∑a(1)⊗a(2)\Delta(a)=\sum a_{(1)}\otimes a_{(2)} is the Sweedler notation). Liu–Wu (2012) extended this program from Hopf algebras to quantum homogeneous spaces BB: they proved that injective-dimension bounds, noetherianity, and the Auslander-Gorenstein property descend from HH to any quantum homogeneous space BB; they established full AS-Gorenstein descent under the normal basis property (an isomorphism H≅B⊗H‾H\cong B\otimes\underline H as left BB-modules and right H‾\underline H-comodules, where H‾=H/B+H\underline H=H/B^+H and B+=ker⁡(ε∣B)B^+=\ker(\varepsilon|_B)) together with finite generation of HH as a BB-module, and for pointed Hopf algebras (those whose coradical, the sum of all simple subcoalgebras, is the group algebra spanned by the group-like elements); and they described the rigid dualizing complex and Nakayama automorphism ν=ad⁡(g)∘S2∘Ξ[τ]\nu=\operatorname{ad}(g)\circ S^2\circ\Xi[\tau] of BB, provided the group-like element g~∈H‾\tilde g\in\underline H governing the twist lifts to a group-like element gg of HH — automatic when HH is pointed, but not known otherwise. As a test of the theory they classified all quantum homogeneous spaces of Uq(sl2)U_q(\mathfrak{sl}_2) (qq not a root of unity) into six explicit families and verified each is AS-regular, and Krähmer (2012) supplied the Hochschild-theoretic lemmas the proofs rely on. The AS-Gorenstein descent question is answered affirmatively for quantized enveloping algebras (Liu–Wu 2014) and for connected Hopf algebras (Brown–Gilmartin 2016), while the general descent question, the unconditional dimension equality, and the liftability question remain open. The descent questions below ask how much of this survives with no hypothesis on HH beyond being AS-Gorenstein with bijective antipode.

Problem Statement

Let kk be a field, HH a noetherian Hopf kk-algebra with bijective antipode that is Artin–Schelter Gorenstein, and B⊆HB\subseteq H a quantum homogeneous space of HH, i.e. a right coideal subalgebra with Δ(B)⊆B⊗H\Delta(B)\subseteq B\otimes H such that HH is faithfully flat as a left and as a right BB-module. Determine, without any hypothesis on HH or BB beyond the above: (1) (Liu–Wu, Question 4.7, 2012) Must BB be AS-Gorenstein? Since noetherianity, the injective-dimension bound inj.dim⁡BB≤inj.dim⁡HH\operatorname{inj.dim}_B B\le\operatorname{inj.dim}_H H, and the Auslander condition are already known to descend in general, the open content is precisely the Artin–Schelter Ext condition: must Ext⁡Bi(k,B)\operatorname{Ext}^i_B(k,B) vanish for all i≠d′i\ne d' and be one-dimensional for i=d′i=d', where d′=inj.dim⁡BBd'=\operatorname{inj.dim}_B B and k=B/B+k=B/B^+ is the trivial module with B+=ker⁡(ε∣B)B^+=\ker(\varepsilon|_B), together with the analogous right-module statement? (2) Must the left global dimension of BB, the projective dimension of the trivial module kk, and the Hochschild cohomological dimension of BB (the largest degree ii in which the Hochschild cohomology HH⁡i(B)=Ext⁡B⊗Bopi(B,B)\operatorname{HH}^i(B)=\operatorname{Ext}^i_{B\otimes B^{op}}(B,B) is nonzero) coincide, l.gl.dim B=proj.dim⁡Bk=Hcdim B\mathrm{l.gl.dim}\,B=\operatorname{proj.dim}_B k=\mathrm{Hcdim}\,B? (Liu–Wu prove this when BB is a direct summand of HH as a BB-bimodule.) (3) Is the group-like element g~∈H‾=H/B+H\tilde g\in\underline H=H/B^+H governing the rigid dualizing complex always liftable to a group-like element of HH — or, failing that, does the Nakayama automorphism formula ν=ad⁡(g)∘S2∘Ξ[τ]\nu=\operatorname{ad}(g)\circ S^2\circ\Xi[\tau] for BB remain valid without the liftability hypothesis? (Liftability is automatic for pointed Hopf algebras and open otherwise.) The common objective of (1)–(3) is to make the full homological duality framework of HH (AS-Gorensteinness, dimension identities, rigid dualizing complexes with their Nakayama automorphisms) descend to every quantum homogeneous space. The related but separate task of classifying all quantum homogeneous spaces of Uq(sln+1)U_q(\mathfrak{sl}_{n+1}) for n≥2n\ge2 is recorded as an adjacent open direction and is not part of this problem's completion boundary.

References

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