The Brown–Goodearl questions for general noetherian weak Hopf algebras: finite injective dimension, homogeneous decomposition, and bijectivity of the antipode (Rogalski–Won–Zhang Questions 8.1–8.3)

Weak Hopf algebras generalize Hopf algebras the way groupoids generalize groups: the comultiplication and counit are allowed to be non-unital algebra maps, so that groupoid algebras and finite quantum groupoids are examples, and direct sums of weak Hopf algebras are again weak Hopf algebras. Rogalski, Won, and Zhang proved the Brown–Goodearl conjecture for weak Hopf algebras that are finitely generated modules over an affine center (with a mild monoidal hypothesis), and posed three questions for arbitrary noetherian weak Hopf algebras, each re-asking one conclusion of their theorem without the module-finiteness hypothesis: (8.1) must a noetherian weak Hopf algebra have finite injective dimension — more strongly, per their 2026 sequel, be a finite direct sum of Artin–Schelter Gorenstein weak Hopf algebras — even just when it is affine and satisfies a polynomial identity; (8.2) must a noetherian weak Hopf algebra of finite Gelfand–Kirillov dimension $d$ decompose as a direct sum of homogeneous weak Hopf subalgebras of Gelfand–Kirillov dimension $i$ for $0\le i\le d$; and (8.3) must its antipode be bijective. All three remain open: they contain, as the special case of ordinary Hopf algebras, the classical Brown–Goodearl conjecture and Skryabin's antipode-bijectivity conjecture, both still open after decades, and the only later work on the weak case (the authors' 2026 sequel on homological integrals) resolves these questions only in the same module-finite-over-affine-center regime as the original theorem.

Background

A Hopf algebra over a field $\Bbbk$ is an algebra equipped with a comultiplication $\Delta:H\to H\otimes H$ and a counit that are unital algebra maps, together with an antipode $S$ playing the role of inversion. A weak Hopf algebra relaxes this: $\Delta$ and the counit need not preserve the identity, but only up to idempotents; the precise axioms were introduced by Böhm, Nill, and Szlachányi in the late 1990s to axiomatize finite quantum groupoids. The group algebra of a finite groupoid is a weak Hopf algebra but not a Hopf algebra unless the groupoid is a group, and — crucially for what follows — a direct sum of weak Hopf algebras is again a weak Hopf algebra, whereas a nonzero Hopf algebra never decomposes as a direct sum of two nonzero weak Hopf algebras (the counit and antipode axioms force the counit to vanish on all but one summand). Every Hopf algebra is a weak Hopf algebra, so any question about weak Hopf algebras subsumes its Hopf counterpart. The weak structure singles out two counital subalgebras $H_s$ and $H_t$ (the images of the idempotents $(\operatorname{id}\otimes\varepsilon)\Delta$ and $(\varepsilon\otimes\operatorname{id})\Delta$), which replace $\Bbbk\cdot 1$ as the 'base' of the theory.

The homological questions studied here come from noncommutative ring theory. An algebra is noetherian if it is left and right noetherian as a ring, affine if finitely generated as an algebra, and PI if it satisfies a polynomial identity (a noncommutative analog of commutativity). The Gelfand–Kirillov dimension $\operatorname{GKdim}$ is a noncommutative measure of growth rate generalizing Krull dimension. The injective dimension of $H$ as a left module over itself, written $\operatorname{injdim}{}_{H}H$, is the homological quantity central to Gorenstein theory. In the weak-world sense used here (introduction of arXiv:2504.03057; see also Sections 1 and 5 of arXiv:1912.11922), $H$ is Artin–Schelter (AS) Gorenstein of injective dimension $d$ if $\operatorname{injdim}{}_{H}H=d<\infty$, if $\operatorname{Ext}^i_H(V,H)=0$ for every finite-dimensional left $H$-module $V$ and every $i\ne d$ while $\operatorname{Ext}^d_H(V,H)$ is finite-dimensional over $\Bbbk$, and if the analogous right-sided conditions hold; when $H$ is an ordinary Hopf algebra this is equivalent to the classical requirement that $\operatorname{Ext}^i_H(\Bbbk,H)$ vanish for $i\ne d$ and be one-dimensional for $i=d$. The corresponding weak homological integral is the right $H$-module $\operatorname{Ext}^d_H(H_t,H)$ (Definition 3.1 of arXiv:2504.03057), which for a noetherian AS Gorenstein weak Hopf algebra is an invertible object of the module category rather than a one-dimensional space. Stronger still, Auslander–Gorenstein and Cohen–Macaulay properties encode duality between module categories; the Van den Bergh condition ensures existence of a well-behaved dualizing complex.

In the 1990s, Brown and Goodearl conjectured that every noetherian Hopf algebra is AS Gorenstein — in particular, has finite injective dimension. This has been proved for major classes: Wu and Zhang (2003) settled the affine PI case, using which Skryabin (2006) deduced that noetherian affine PI Hopf algebras have bijective antipodes; Lü, Oh, Wang, and Yu (2018) proved that any noetherian AS Gorenstein Hopf algebra has a bijective antipode. Skryabin conjectured more generally that the antipode of every noetherian Hopf algebra is bijective (known for semiprime right noetherian Hopf algebras). Both the Brown–Goodearl conjecture and Skryabin's conjecture remain open in general, as recorded in Brown's 2026 survey, which reports that the injective-dimension question 'remains open after 30 years'.

Rogalski, Won, and Zhang (2021) extended this program to weak Hopf algebras that are finitely generated modules over an affine center (plus a mild monoidal hypothesis), proving (their Theorem 0.3) that such $H$ has finite injective dimension and decomposes as a finite direct sum of AS Gorenstein, Auslander–Gorenstein, Cohen–Macaulay, GK-homogeneous summands. Their Section 8 then poses, for the general noetherian weak Hopf algebra, the three questions studied here — the finite-injectivity question (their Question 8.1), the decomposition question (Question 8.2), and the antipode-bijectivity question (Question 8.3) — each asking whether one conclusion of the theorem survives without module-finiteness over the center. Their 2026 sequel on homological integrals for weak Hopf algebras strengthens the module-finite theory (antipode bijective; decomposition into AS Gorenstein summands satisfying the Van den Bergh condition; AS property equivalent to the Van den Bergh condition for noetherian weak Hopf algebras) and reformulates Question 8.1 in its strongest weak-world form — whether every noetherian weak Hopf algebra is a finite direct sum of AS Gorenstein weak Hopf algebras, the direct sum being unavoidable since direct sums of weak Hopf algebras with different injective dimensions cannot themselves be AS Gorenstein.

Problem Statement

For a weak Hopf algebra $H$ over a field that is noetherian, but with no module-finiteness-over-center hypothesis, three linked structural questions (Rogalski–Won–Zhang, Questions 8.1–8.3), each asking whether one conclusion of the module-finite theory extends to the general noetherian weak Hopf algebra — in particular already under the additional hypotheses that $H$ is affine and satisfies a polynomial identity:

1. (Brown–Goodearl question, Question 8.1, in the strengthened form of Question 5.6 of the 2026 sequel) Must $H$ have finite injective dimension as a module over itself; more strongly, must $H$ be isomorphic to a finite direct sum of Artin–Schelter Gorenstein weak Hopf algebras? (The direct-sum form is the strongest weak-world analog: direct sums of weak Hopf algebras are weak Hopf algebras, and a direct sum of summands of different injective dimensions cannot itself be AS Gorenstein; an affirmative answer implies the literal finite-injectivity statement.)

2. (Decomposition question, Question 8.2) If $H$ has finite Gelfand–Kirillov dimension $d$, must $H$ decompose as a direct sum of homogeneous weak Hopf subalgebras $H_i$ with $\operatorname{GKdim} H_i=i$ for integers $0\le i\le d$?

3. (Skryabin-type question, Question 8.3) Must the antipode $S$ of $H$ be bijective?

The common objective is to extend the Brown–Goodearl structural picture — Gorenstein homological properties, homogeneous direct-sum decomposition into weak Hopf summands, and invertibility of the antipode — from the module-finite-over-affine-center regime, where it is fully established, to all noetherian weak Hopf algebras (or to exhibit a noetherian, e.g. affine PI, weak hopf algebra violating one of the three properties). Note that affirmative answers in the special case of ordinary Hopf algebras would resolve the long-open classical Brown–Goodearl conjecture and Skryabin's antipode-bijectivity conjecture.

References

1. Daniel Rogalski, Robert Won, James J. Zhang, A proof of the Brown–Goodearl conjecture for module-finite weak Hopf algebras, Algebra & Number Theory 15 (2021), no. 4, 971–997. DOI: 10.2140/ant.2021.15.971; arXiv:1912.11922 (the arXiv preprint is titled 'A proof of the Brown–Goodearl Conjecture for weak Hopf algebras'); https://arxiv.org/abs/1912.11922
2. Daniel Rogalski, Robert Won, James J. Zhang, Homological integrals for weak Hopf algebras, Journal of Algebra 692 (2026), 173–204. DOI: 10.1016/j.jalgebra.2025.12.009; arXiv:2504.03057; https://arxiv.org/abs/2504.03057
3. Ken A. Brown, Report on $AS$-Gorenstein Hopf algebras, survey preprint, 2026. arXiv:2605.00774; https://arxiv.org/abs/2605.00774
4. Serge Skryabin, New results on the bijectivity of antipode of a Hopf algebra, Journal of Algebra 306 (2006), no. 2, 622–633. DOI: 10.1016/j.jalgebra.2006.04.017; https://doi.org/10.1016/j.jalgebra.2006.04.017
5. Q.-S. Wu, J. J. Zhang, Noetherian PI Hopf algebras are Gorenstein, Transactions of the American Mathematical Society 355 (2003), no. 3, 1043–1066. DOI: 10.1090/s0002-9947-02-03106-9; https://doi.org/10.1090/s0002-9947-02-03106-9
6. Jiafeng Lü, Sei-Qwon Oh, Xingting Wang, Xiaolan Yu, A note on the bijectivity of the antipode of a Hopf algebra and its applications, Proceedings of the American Mathematical Society 146 (2018), no. 11, 4619–4631. DOI: 10.1090/proc/14140; https://doi.org/10.1090/proc/14140