Does connected graded as an algebra imply connected as a coalgebra for Hopf algebras of finite Gel'fand–Kirillov dimension (Brown–Gilmartin–Zhang Question 0.4)?

The paper “Connected (graded) Hopf algebras” by Brown, Gilmartin and Zhang (Trans. Amer. Math. Soc. 372, 2019) poses as its central open question (Question 0.4) whether a Hopf algebra of finite Gel'fand–Kirillov dimension over an algebraically closed field of characteristic zero whose underlying algebra is connected graded must be connected as a coalgebra, i.e. whether its coradical equals the base field. The converse implication fails trivially (enveloping algebras of non-nilpotent Lie algebras), all known low-dimensional classifications and structure theorems are consistent with a positive answer, and the two classes share the strong homological properties of enveloping algebras. The question remains open: no proof or counterexample is known.

Background

A Hopf algebra over a field $k$ is a vector space carrying both an algebra structure and a compatible coalgebra structure — a comultiplication $\Delta:H\to H\otimes H$ and counit $\epsilon:H\to k$ that are algebra homomorphisms — together with an antipode $S$, the linear analogue of inversion in a group. Two unrelated “connectedness” conditions can be imposed on a Hopf algebra, and the relationship between them is the subject of this problem. On the coalgebra side, a nonzero element $g$ is group-like if $\Delta(g)=g\otimes g$; a simple subcoalgebra is a nonzero subcoalgebra with no proper nonzero subcoalgebra; the coradical $H_0$ is the sum of all simple subcoalgebras; and $H$ is connected as a coalgebra when $H_0=k$. The basic examples are the universal enveloping algebras $U(\mathfrak g)$ of Lie algebras $\mathfrak g$ with their standard cocommutative coalgebra, which are connected in this sense. On the algebra side, $H$ is connected graded as an algebra when $H=\bigoplus_{i\ge 0}H(i)$ with $H(0)=k$ and $H(i)H(j)\subseteq H(i+j)$ — the graded analogue of a polynomial ring with its usual grading. The Gel'fand–Kirillov dimension (GK dimension) measures the growth of a finitely generated algebra and serves as the notion of dimension for infinite-dimensional algebras; Hopf algebras of finite GK dimension form the natural infinite-dimensional analogue of finite-dimensional algebras.

Since Zhuang proved that connected Hopf algebras (connected as coalgebras) of finite GK dimension are noetherian domains with strong homological regularity (Auslander regularity, a chain-condition on injective dimension of modules, and GK-Cohen–Macaulayness, a dimension-equality condition between GK dimension and projective dimensions), Brown, Gilmartin and Zhang studied the two classes in parallel over an algebraically closed field $k$ of characteristic $0$. They proved that a Hopf algebra of finite GK dimension $n$ that is connected graded as an algebra is a noetherian domain that is Artin–Schelter regular (a homological regularity property modeled on polynomial rings: finite global dimension $n$ together with a Poincaré-duality condition on cohomology) and Calabi–Yau with $S^2=\mathrm{Id}$; and that for a connected Hopf algebra of finite GK dimension $n$ the underlying coalgebra is Artin–Schelter regular of global dimension $n$. The classes interlock tightly: by the classification of Wang, Zhang and Zhuang, a connected Hopf algebra of GK dimension at most $4$ is isomorphic as an algebra to $U(\mathfrak g)$, while Brown–Gilmartin–Zhang constructed one of GK dimension $5$ that is simultaneously connected graded as an algebra and connected as a coalgebra yet is not an enveloping algebra. The implication from coalgebra-connected to algebra-connected-graded fails trivially: $U(\mathfrak g)$ for any finite-dimensional non-nilpotent Lie algebra $\mathfrak g$ is connected as a coalgebra but admits no connected grading as an algebra. The converse direction is open and was called “a central underlying theme” of that paper. One elementary constraint shapes any search for a counterexample: in a connected graded domain the only units are the scalars, group-like elements are units, so all group-likes are trivial; hence a counterexample must be non-pointed (not spanned by group-likes in its coradical) — its coradical must contain a simple subcoalgebra of dimension greater than $1$. The question also draws motivation from the theorem of Etingof and Gelaki that connected Hopf algebras over $\mathbb C$ are exactly deformations of prounipotent proalgebraic groups (proalgebraic groups are inverse limits of algebraic groups, prounipotent meaning built from unipotent ones) equipped with Poisson data (a compatible Lie bracket on the coordinate ring), which classifies the coalgebra-connected class up to deformation data; whether the graded class sits inside it is a basic structural question about this classification.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $0$, and let $H$ be a Hopf $k$-algebra of finite Gel'fand–Kirillov dimension whose underlying $k$-algebra is connected graded, i.e. $H=\bigoplus_{i\ge 0}H(i)$ with $H(0)=k$ (hypothesis (CGA) of Brown–Gilmartin–Zhang). Let $H_0$ denote the coradical of $H$, the sum of all simple subcoalgebras of $H$. Must $H_0=k$, i.e. is $H$ necessarily connected as a coalgebra (hypothesis (CCA), making $H$ a connected Hopf algebra)? This is Question 0.4 of Brown–Gilmartin–Zhang: does (CGA) imply (CCA)? A complete answer is either a proof that the implication holds for every such Hopf algebra, or an explicit Hopf algebra of finite GK dimension that is connected graded as an algebra while its coradical strictly contains $k$.

References

1. K. A. Brown, P. Gilmartin and J. J. Zhang, “Connected (graded) Hopf algebras”, Transactions of the American Mathematical Society 372, no. 5 (2019), 3283–3317, DOI 10.1090/TRAN/7686; preprint arXiv:1601.06687 (2016), https://arxiv.org/abs/1601.06687
2. G.-S. Zhou, Y. Shen and D.-M. Lu, “The structure of connected (graded) Hopf algebras”, Advances in Mathematics 372 (2020), article 107292, DOI 10.1016/j.aim.2020.107292; preprint arXiv:1904.01918 (2019), https://arxiv.org/abs/1904.01918
3. C.-C. Li and G.-S. Zhou, “The structure of connected (graded) Hopf algebras revisited”, Journal of Algebra 610 (2022), 684–702, DOI 10.1016/j.jalgebra.2022.07.031; preprint arXiv:2109.01882 (2021), https://arxiv.org/abs/2109.01882
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5. K. A. Brown, “Report on AS-Gorenstein Hopf algebras”, arXiv:2605.00774 (2026), https://arxiv.org/abs/2605.00774
6. G. Zhuang, “Properties of pointed and connected Hopf algebras of finite Gelfand–Kirillov dimension”, Journal of the London Mathematical Society (2) 87, no. 3 (2013), 877–898, DOI 10.1112/jlms/jds079, https://doi.org/10.1112/jlms/jds079
7. D.-G. Wang, J. J. Zhang and G. Zhuang, “Connected Hopf algebras of Gelfand–Kirillov dimension four”, Transactions of the American Mathematical Society 367, no. 8 (2015), 5597–5632, DOI 10.1090/s0002-9947-2015-06219-9, https://doi.org/10.1090/s0002-9947-2015-06219-9
8. P. Etingof and S. Gelaki, “Quasisymmetric and unipotent tensor categories”, Mathematical Research Letters 15 (2008) 857–866, DOI 10.4310/mrl.2008.v15.n5.a3, https://doi.org/10.4310/mrl.2008.v15.n5.a3
9. B. Yu and G. Zhou, “The structure of iterated Hopf Ore extensions”, arXiv:2607.28234 (2026), https://arxiv.org/abs/2607.28234