The antipode of a Hopf algebra is required by the axioms to invert the identity only for the convolution product, and bijectivity can genuinely fail for arbitrary Hopf algebras. Skryabin (2006) proved that the antipode of any one-sided noetherian Hopf algebra is injective and conjectured that it is always bijective. The conjecture remains open as of May 2026; bijectivity is known for large classes (semiprime noetherian, affine PI, AS-Gorenstein, residually finite-dimensional, co-Frobenius, weak Hopf analogues), and it would follow from the Brown–Goodearl conjecture that every noetherian Hopf algebra is AS-Gorenstein, itself open after 30 years. Because the homological theory of noetherian Hopf algebras routinely assumes a bijective antipode, resolution in either direction directly changes the core structural knowledge of the field.

Background

A Hopf algebra over a field kk is an associative unital algebra HH that is simultaneously a coalgebra: it carries algebra homomorphisms Δ:H→H⊗kH\Delta: H \to H \otimes_k H (comultiplication) and ε:H→k\varepsilon: H \to k (counit) satisfying coassociativity and the counit axiom. A bialgebra becomes a Hopf algebra when it admits an antipode, a kk-linear map S:H→HS: H \to H satisfying m∘(S⊗id)∘Δ  =  m∘(id⊗S)∘Δ  =  u∘ε,m \circ (S \otimes \mathrm{id}) \circ \Delta \;=\; m \circ (\mathrm{id} \otimes S) \circ \Delta \;=\; u \circ \varepsilon, where mm and uu denote multiplication and unit. These axioms force SS to be an algebra anti-homomorphism and a coalgebra anti-homomorphism, but they do not force SS to be a bijection. Bijectivity genuinely fails for unrestricted Hopf algebras: Takeuchi's free Hopf algebras (1971) have antipodes that are injective but not surjective, and examples with non-injective antipode are also known (both reported in the surveys of Goodearl, 2012, and Andruskiewitsch, 2023, cited below). In contrast, the antipode of any finite-dimensional Hopf algebra is bijective and has finite order (Larson–Sweedler, 1969; Radford, 1976); S2=idS^2 = \mathrm{id} holds in any commutative or cocommutative Hopf algebra; and Hopf algebras admitting a nonzero integral (a linear functional t∈H∗t \in H^* with (t⊗f)∘Δ=f(1)t(t \otimes f) \circ \Delta = f(1)t for all f∈H∗f \in H^* — the co-Frobenius property) have bijective antipode.

The natural finiteness condition under which bijectivity is expected is noetherianity of the underlying algebra (the ascending chain condition on left, or right, ideals). Skryabin (2006) proved that the antipode of any one-sided noetherian Hopf algebra is injective, and that it is bijective when HH is semiprime noetherian (no nonzero nilpotent two-sided ideals) or affine (finitely generated as an algebra) and PI (satisfying a polynomial identity — a noncommutative analogue of commutativity); he then conjectured bijectivity in general. The question is load-bearing for the field: the homological theory of noetherian Hopf algebras — dualising complexes and twisted Hochschild cohomology (Brown–Zhang, 2008) and homological integrals, i.e. functionals on HH obtained as generators of the one-dimensional top space ExtHd(k,H)\mathrm{Ext}^d_H(k, H) when the AS-Gorenstein condition defined below holds (Lu–Wu–Zhang, 2007) — adopts a bijective antipode as a standing hypothesis, and Lü–Oh–Wang–Yu (2018) proved that every noetherian AS-Gorenstein Hopf algebra has bijective antipode, where AS-Gorenstein (Artin–Schelter Gorenstein) means that HH has finite injective dimension dd, that ExtHi(k,H)=0\mathrm{Ext}^i_H(k, H) = 0 for i≠di \neq d while ExtHd(k,H)≅k\mathrm{Ext}^d_H(k, H) \cong k as left modules, and that the mirrored right-handed conditions hold (kk being the trivial module via the counit). Consequently, the conjecture of Brown and Goodearl that every noetherian Hopf algebra is AS-Gorenstein — still, in Brown's words (2026), open after 30 years — would imply Skryabin's conjecture. Skryabin (2021) also proved bijectivity for noetherian Hopf algebras that are residually finite-dimensional (separated by their finite-dimensional representations), and the antipode of any Hopf domain (a Hopf algebra without zero divisors) of finite Gelfand–Kirillov dimension (a polynomial-growth invariant measuring the growth of finitely generated subalgebras) is bijective. Rogalski–Won–Zhang (2021, 2026) established the corresponding statements for weak Hopf algebras (a generalisation in which the comultiplication of the unit and the counit of the identity need not be trivial), proving the antipode bijective for noetherian weak Hopf algebras that are finite modules over their affine centers, and more generally finite direct sums of AS-Gorenstein algebras. Every noetherian Hopf algebra known at present has a bijective antipode; the general case is the open conjecture recorded here.

Problem Statement

Let HH be a Hopf algebra over an arbitrary field — a bialgebra equipped with an antipode S:H→HS: H \to H, the linear map inverting the identity for the convolution product — and assume the underlying algebra of HH is left noetherian or right noetherian (it satisfies the ascending chain condition on left, respectively right, ideals). Must SS be a bijective linear map? Skryabin's injectivity theorem (2006) already gives that SS is injective for every such HH, so the conjecture is equivalent to: prove that the antipode of every one-sided noetherian Hopf algebra is surjective, or exhibit a one-sided noetherian Hopf algebra whose antipode is not surjective.

References

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