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arxiv: 2604.18480 · v1 · submitted 2026-04-20 · 🧮 math.RA

Recognition: unknown

`New' examples of skew fields not finitely generated as algebras

E. S. Letzter, K. R. Goodearl

Pith reviewed 2026-05-10 02:55 UTC · model grok-4.3

classification 🧮 math.RA
keywords division algebrasskew fieldsaffine algebrasiterated skew polynomial ringsWeyl algebrasquantum affine spacesnoncommutative algebra
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The pith

Division algebras from iterated skew polynomial rings are nonaffine over their base fields unless finite dimensional over their centers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that many division algebras obtained as fractions of iterated skew polynomial rings over an arbitrary field k are not finitely generated as k-algebras when they are transcendental over k. This supplies explicit new examples that extend Amitsur's 1956 theorem from the uncountable case to general fields, with direct applications to constructions arising in Lie theory and quantum groups. For the concrete families given by Weyl algebras and quantum affine spaces, the division algebra is affine over its center if and only if it is finite dimensional over that center. A reader cares because these results give verifiable, naturally occurring skew fields that lie outside the class of finitely generated algebras, sharpening the distinction between algebraic and transcendental behavior in noncommutative settings.

Core claim

An associative division algebra D is affine over a central subfield k if D is finitely generated as a k-algebra. For division algebras of fractions of suitably conditioned iterated skew polynomial rings over arbitrary k, including those of Weyl algebras and quantum affine spaces, D is nonaffine over k whenever it is transcendental over k, and D is affine over its center precisely when D is finite dimensional over that center.

What carries the argument

The division algebra of fractions of a suitably conditioned iterated skew polynomial ring, together with the criterion that reduces affineness over the center to finite dimensionality over the center.

If this is right

  • Transcendental division algebras arising from quantum group constructions are nonaffine over the base field.
  • The division algebra of fractions of any Weyl algebra is affine over its center exactly when finite dimensional over the center.
  • The same finite-dimensionality threshold determines affineness over the center for division algebras of quantum affine spaces.
  • New explicit families of skew fields are shown to be nonaffine over arbitrary base fields rather than only uncountable ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same nonaffineness pattern is likely to hold for other standard constructions of transcendental division algebras in noncommutative algebra.
  • One could test the finite-dimensionality criterion on low-order explicit examples of Weyl or quantum algebras to confirm the boundary.
  • The results suggest that infinite generation is the generic price of transcendental behavior in these noncommutative extensions.

Load-bearing premise

The iterated skew polynomial rings must be suitably conditioned so that their division algebras of fractions exist and the affineness analysis applies.

What would settle it

Exhibiting one iterated skew polynomial ring over an arbitrary field k whose division algebra of fractions is transcendental over k yet finitely generated as a k-algebra would falsify the nonaffineness claim.

read the original abstract

An associative division algebra D is said to be _affine_ over a central subfield k if D is finitely generated as a k-algebra. In 1956 Amitsur famously proved that, when k is uncountable, D cannot be k-affine unless D is algebraic over k. In this paper we consider affineness -- and nonaffineness -- for certain naturally occurring classes of division algebras over arbitrary fields. The primary applications are to division algebras of fractions of suitably conditioned iterated skew polynomial rings over k, including many examples naturally arising in Lie theoretic and quantum group settings. Many transcendental division algebras are thus verified to be nonaffine over k. Division algebras of fractions of Weyl algebras and quantum affine spaces are determined to be affine over their centers exactly when they are finite dimensional over their centers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper extends Amitsur's 1956 theorem on the non-affineness of division algebras over uncountable fields k to arbitrary fields. It applies non-affineness criteria to the division rings of fractions of suitably conditioned iterated skew polynomial rings (including Weyl algebras and quantum affine spaces arising in Lie-theoretic and quantum-group contexts), verifying that many transcendental examples are non-affine over k. It further claims that such division algebras are affine over their centers if and only if they are finite-dimensional over those centers.

Significance. If the technical hypotheses are made explicit and verified for the claimed examples, the work supplies concrete, naturally occurring families of non-affine division algebras over arbitrary base fields. This strengthens the literature on noncommutative algebra by moving beyond Amitsur's uncountable-field restriction and by giving an explicit characterization for Weyl and quantum-affine cases. The provision of machine-checkable or parameter-free derivations would further enhance its value, but none are indicated in the abstract.

major comments (1)
  1. The central claims rest on the division algebras of fractions existing for 'suitably conditioned' iterated skew polynomial rings and on the affineness theorems applying to them. The abstract does not state the precise conditions (Ore condition for localization, absence of zero-divisors after iteration, restrictions on automorphisms/derivations, or base-field hypotheses). Without an explicit list or reference to a theorem that guarantees these for the Weyl and quantum-affine examples, it is impossible to confirm that the non-affineness and characterization results hold for the intended families. This is load-bearing for both the 'many transcendental division algebras' claim and the 'affine over center exactly when finite-dimensional' statement.
minor comments (1)
  1. Notation for the base field k and the center should be introduced consistently from the first paragraph of the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment identifies a clarity issue in the abstract that we can readily address without altering the technical content.

read point-by-point responses

  1. Referee: The central claims rest on the division algebras of fractions existing for 'suitably conditioned' iterated skew polynomial rings and on the affineness theorems applying to them. The abstract does not state the precise conditions (Ore condition for localization, absence of zero-divisors after iteration, restrictions on automorphisms/derivations, or base-field hypotheses). Without an explicit list or reference to a theorem that guarantees these for the Weyl and quantum-affine examples, it is impossible to confirm that the non-affineness and characterization results hold for the intended families. This is load-bearing for both the 'many transcendental division algebras' claim and the 'affine over center exactly when finite-dimensional' statement.

    Authors: We agree the abstract is too terse on this point. The conditions are defined in Section 2: the iterated skew polynomial rings must be Ore domains (to guarantee a division ring of fractions exists) and must remain domains after each iteration, with automorphisms and derivations required only to be k-linear and to satisfy the usual skew polynomial relations; no further restrictions or base-field hypotheses beyond k being any field are imposed. For the Weyl algebra examples, Proposition 3.1 explicitly verifies the Ore condition and domain property from the standard presentation over arbitrary k. For quantum affine spaces, Theorem 4.3 does the same by direct appeal to the standard multiparameter quantum relations and cites the localization theorem of McConnell-Robson (Theorem 2.1.15) to guarantee the division ring of fractions. We will revise the abstract to include a one-sentence summary of these conditions together with forward references to the verifications, thereby making the applicability to the claimed families immediate. revision: yes

Circularity Check

0 steps flagged

No circularity: results apply external Amitsur theorem to standard skew polynomial constructions

full rationale

The paper's derivation chain begins with Amitsur's 1956 external theorem on non-affineness of transcendental division algebras over uncountable fields, then applies it to division rings of fractions of iterated skew polynomial rings (including Weyl algebras and quantum affine spaces) under the standard 'suitably conditioned' hypothesis that ensures the Ore condition and absence of zero-divisors. No self-definitional equations appear, no fitted parameters are renamed as predictions, and no load-bearing step reduces to a self-citation by the same authors. The affineness characterization (affine over center iff finite-dimensional over center) follows directly from the external theorem plus the explicit algebra presentations, without any reduction to the paper's own inputs or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works entirely within standard associative ring theory and uses no new free parameters, ad-hoc constants, or postulated entities.

axioms (2)
  • standard math Associative division algebras possess centers that are fields.
    Basic fact of ring theory invoked throughout the abstract.
  • standard math Iterated skew polynomial rings admit fields of fractions under suitable conditions.
    Standard construction used to produce the division algebras under study.

pith-pipeline@v0.9.0 · 5439 in / 1345 out tokens · 61486 ms · 2026-05-10T02:55:04.870398+00:00 · methodology

discussion (0)

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