Does connected-graded-as-an-algebra imply connected-as-a-coalgebra for Hopf algebras, and the associated Nakayama, Ore, and Weyl-skew-field questions of Brown–Gilmartin–Zhang

A bundle of four structural open questions posed by Brown, Gilmartin and Zhang in 'Connected (graded) Hopf algebras' (2016; Transactions of the AMS 2019) about Hopf algebras over an algebraically closed field of characteristic 0, primarily of finite Gel'fand–Kirillov dimension: (i) whether a Hopf algebra that is connected graded as an algebra must be connected as a coalgebra (their Question 0.4); (ii) whether for a connected Hopf algebra of finite Gel'fand–Kirillov dimension the Nakayama automorphism of the coalgebra is the square of the antipode (their Conjecture 4.2); (iii) under what circumstances a connected Hopf algebra is an Ore domain (their Question 4.7); and (iv) what conditions ensure that its quotient division ring is a Weyl skew field (their Question 4.8). These remain open questions of Brown–Gilmartin–Zhang.

Background

A Hopf algebra over a field $k$ is at once an associative algebra and a coassociative coalgebra, with a comultiplication $\Delta$, a counit, and an antipode $S$ (a linear map playing the role of an inverse) compatible with the multiplication. The coradical of a coalgebra is the sum of its simple subcoalgebras, and a Hopf algebra is called connected (as a coalgebra) when its coradical is the one-dimensional field $k$ — informally, the object has no nontrivial group-like elements, the quantum analogue of a unipotent group. Two classical structure theorems anchor the subject: over a perfect field, a commutative connected Hopf algebra is a polynomial algebra, and in characteristic 0 a cocommutative connected Hopf algebra is the universal enveloping algebra $U(\mathfrak g)$ of a Lie algebra $\mathfrak g$. On the other side, an $\mathbb N$-graded algebra $H=\bigoplus_{i\ge 0}H_i$ is connected graded (as an algebra) when $H_0=k$; a graded Hopf algebra is required to be graded as both an algebra and a coalgebra, with $\Delta$ respecting the grading. The Gel'fand–Kirillov dimension (GK dimension) of an algebra measures the polynomial growth rate of its finitely generated subspaces, and is the standard finiteness invariant for infinite-dimensional algebras. Brown, Gilmartin and Zhang (BGZ) studied, over an algebraically closed field of characteristic 0, the two hypotheses (CGA): a Hopf algebra is connected graded as an algebra, with no grading condition on $\Delta$; and (CCA): it is connected as a coalgebra. The implication (CCA)$\Rightarrow$(CGA) fails — the enveloping algebra of any finite-dimensional non-nilpotent Lie algebra, with its standard cocommutative coproduct, is connected as a coalgebra but admits no connected grading as an algebra. BGZ proved that a (CGA) Hopf algebra of finite GK dimension $n$ is a noetherian domain which is Artin–Schelter regular (homologically regular in the sense of Artin–Schelter: finite global dimension $n$, finite injective dimension, and one-dimensional trivial Ext spaces) and Calabi–Yau with $S^2=\mathrm{id}$, and that for a (CCA) Hopf algebra of finite GK dimension $n$ the underlying coalgebra is Artin–Schelter regular. Both classes behave much like enveloping algebras, yet BGZ constructed a connected graded Hopf algebra of GK dimension 5 that is not isomorphic as an algebra to any $U(\mathfrak g)$. A key subsequent structure concept is the iterated Hopf Ore extension (IHOE): a Hopf algebra built from $k$ by a chain of Hopf subalgebras $k=H_0\subset H_1\subset\cdots\subset H_n=H$ in which each step adjoins a variable by an Ore extension (skew polynomial) construction; connected Hopf algebras of GK dimension at most 4 over algebraically closed fields of characteristic 0 have been completely classified (as recounted in the 2021 Brown–Zhang survey) and are IHOEs, while Zhou, Shen and Lu proved that every Hopf algebra of finite GK dimension that is connected graded both as an algebra and as a coalgebra is an IHOE of derivation type. For the homological component: a Frobenius algebra has a Nakayama automorphism measuring the asymmetry of its Frobenius pairing, and a dual theory defines the Nakayama automorphism of suitable coalgebras via Ext groups of comodules; BGZ's Conjecture 4.2 asserts, by analogy with the known computation of the Nakayama automorphism for connected graded Artin–Schelter regular Hopf algebras, that for a connected Hopf algebra of finite GK dimension this coalgebra invariant is exactly $S^2$. Finally, a noetherian domain satisfies the Ore condition precisely when its classical quotient division ring exists, and the Weyl skew field is the skew field of fractions of a Weyl algebra (the algebra of polynomial differential operators on affine space); the classical Gelfand–Kirillov conjecture predicted that enveloping algebras and related algebras have quotient division rings isomorphic to Weyl skew fields, so BGZ's Questions 4.7–4.8 ask for the Ore property and the Gelfand–Kirillov phenomenon for connected Hopf algebras: by a classical theorem quoted in BGZ, every domain of finite GK dimension is an Ore domain, so Question 4.7 genuinely concerns connected Hopf algebras without the finite GK hypothesis, and BGZ's own partial result is the dichotomy that a connected Hopf algebra of finite GK dimension is either commutative or has a quotient division ring containing a copy of the first Weyl algebra.

Problem Statement

Over an algebraically closed field $k$ of characteristic 0, resolve the following four questions of Brown–Gilmartin–Zhang, whose common objective is to determine how much coalgebra-side and homological structure is forced by the algebra-side hypothesis of connected grading, and to describe the localization and duality invariants of connected Hopf algebras:

(Q1) (BGZ Question 0.4.) If a Hopf $k$-algebra $H$ is connected graded as an algebra (hypothesis CGA, with no grading assumed for the comultiplication), must $H$ be connected as a coalgebra, i.e. must its coradical be $k$ (hypothesis CCA)? Equivalently, does (CGA) imply (CCA)? Note that the converse implication is false, since the enveloping algebra of any finite-dimensional non-nilpotent Lie algebra is connected as a coalgebra but not connected graded as an algebra.

(Q2) (BGZ Conjecture 4.2.) Let $H$ be a connected Hopf algebra of finite Gel'fand–Kirillov dimension. Is the Nakayama automorphism of the coalgebra $H$ (the comodule-Ext-theoretic invariant dual to the classical Nakayama automorphism of a Frobenius algebra) equal to the square $S^2$ of the antipode?

(Q3) (BGZ Question 4.7.) Let $H$ be a connected Hopf $k$-algebra, with no finite Gel'fand–Kirillov dimension or affineness assumed. Under what circumstances is $H$ an Ore domain, i.e. when do its nonzero elements form a right (equivalently left) Ore set so that the classical quotient division ring exists? (The finite GK-dimensional case is known to be affirmative, so the question concerns the general case.)

(Q4) (BGZ Question 4.8.) Let $H$ be a connected Hopf algebra with finite Gel'fand–Kirillov dimension. What conditions on $H$ ensure that its quotient division ring is isomorphic to a Weyl skew field (the skew field of fractions of a Weyl algebra)?

References

1. Ken Brown, Paul Gilmartin, James J. Zhang, "Connected (graded) Hopf algebras", arXiv:1601.06687 (2016); Transactions of the American Mathematical Society 372 (2019), no. 5, 3283–3317, DOI 10.1090/tran/7686, https://arxiv.org/abs/1601.06687
2. G.-S. Zhou, Y. Shen, D.-M. Lu, "The structure of connected (graded) Hopf algebras", arXiv:1904.01918 (2019); Advances in Mathematics 372 (2020), article 107292, DOI 10.1016/j.aim.2020.107292, https://arxiv.org/abs/1904.01918
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