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arxiv: 2606.09687 · v1 · pith:DMO4YNQ6new · submitted 2026-06-08 · 🧮 math.QA

Codifferential Calculi on Quantum Homogeneous Spaces

Julius Benner This is my paper

Pith reviewed 2026-06-27 14:02 UTC · model grok-4.3

classification 🧮 math.QA
keywords codifferential calculiquantum homogeneous spacesquantum tangent spacesmaximal prolongationquantized projective spacesPodleś sphereHeckenberger-Kolb calculiequivariant calculi
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The pith

Equivariant first-order codifferential calculi on quantum homogeneous spaces correspond to quantum tangent spaces.

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

The paper develops the theory of first- and higher-order codifferential calculi over coalgebras, introducing an explicit maximal prolongation that forms a differential graded coalgebra satisfying a universal property. For module coalgebras C of the form U ⊗_H k where U is faithfully flat as a left and right H-module, it establishes that equivariant first-order codifferential calculi correspond to certain right coideals T inside ker(ε), termed quantum tangent spaces. When H is a sub bialgebra and the right C-coaction on T is trivial, the maximal prolongation is described by a quadratic coalgebra. The framework yields explicit computations on the coalgebra preduals of the Podleś sphere and quantized projective spaces, including a new proof that the antiholomorphic Heckenberger-Kolb calculi have classical dimension.

Core claim

Equivariant first-order codifferential calculi on C = U ⊗_H k correspond to certain right coideals T ⊆ ker(ε) called quantum tangent spaces. When H is a sub bialgebra and the right C-coaction on T is trivial, the maximal prolongation is described in terms of a quadratic coalgebra. The theory relates codifferential calculi to differential calculi and Cartan pairs over the dual algebra C^*, and is applied to compute codifferential calculi on the coalgebra preduals of the Podleś sphere and the quantized projective spaces, giving a new proof that the antiholomorphic Heckenberger-Kolb calculi on quantized projective spaces have classical dimension.

What carries the argument

The bijection between equivariant first-order codifferential calculi and quantum tangent spaces (specific right coideals T ⊆ ker(ε)).

If this is right

  • Explicit constructions of equivariant calculi follow directly from selecting suitable right coideals T.
  • Under the trivial coaction condition the maximal prolongation is given explicitly by the associated quadratic coalgebra.
  • Codifferential calculi on the preduals of the Podleś sphere and quantized projective spaces admit concrete descriptions.
  • The antiholomorphic Heckenberger-Kolb calculi on quantized projective spaces have classical dimension.
  • Codifferential calculi relate to differential calculi on the dual algebra C^* via the established correspondence.

Where Pith is reading between the lines

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

  • The same tangent-space description may classify calculi on other coalgebras that are not faithfully flat homogeneous spaces.
  • Quadratic coalgebra presentations could simplify the study of higher-order prolongations in noncommutative settings.
  • The relation to Cartan pairs on dual algebras opens a route to dualizing known results from differential calculus to the codifferential side.

Load-bearing premise

The coalgebra C must be of the form U ⊗_H k with U faithfully flat as a left- and right H-module, and the base field must have characteristic not equal to 2.

What would settle it

A direct calculation on the coalgebra predual of the Podleś sphere that produces an equivariant first-order codifferential calculus with no corresponding right coideal T in ker(ε) would show the correspondence fails.

read the original abstract

We develop the theory of first- and higher-order codifferential calculi over coalgebras $C$ over fields $k$ with characteristic $\mathrm{char}(k)\neq 2$. For a given first-order codifferential calculus, we introduce its maximal prolongation by means of an explicit construction that associates to it a differential graded coalgebra, satisfying a universal property. For module coalgebras over a Hopf algebra $U$, we introduce the notion of an equivariant codifferential calculus. If $C$ is of the form $U\otimes_H k$ for a Hopf algebra $U$ and a right coideal subalgebra $H$ such that $U$ is faithfully flat as a left- and right $H$-module, we show that equivariant first-order codifferential calculi correspond to certain right coideals $T\subseteq \ker(\varepsilon\colon C\rightarrow k)$ called quantum tangent spaces. If $H$ is a sub bialgebra and the right $C$-coaction on $T$ is trivial, then the maximal prolongation is described in terms of a quadratic coalgebra. We further relate codifferential calculi to differential calculi and Cartan pairs over the dual algebra $C^\ast$, or more generally subalgebras thereof. We explicitly compute codifferential calculi on the coalgebra pre duals of the Podle\'s sphere and the quantized projective spaces. As an application, we give a new proof that the antiholomorphic Heckenberger--Kolb calculi on quantized projective spaces have classical dimension.

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 develops the theory of first- and higher-order codifferential calculi on coalgebras C over fields of characteristic not 2. It introduces maximal prolongations via an explicit construction yielding a differential graded coalgebra with a universal property. For module coalgebras C = U ⊗_H k with U a Hopf algebra and H a right coideal subalgebra, it defines equivariant codifferential calculi and proves (under the hypothesis that U is faithfully flat as left and right H-module) that these correspond bijectively to certain right coideals T ⊆ ker(ε) called quantum tangent spaces. When H is a sub-bialgebra and the right C-coaction on T is trivial, the maximal prolongation is described by a quadratic coalgebra. The work relates codifferential calculi to differential calculi and Cartan pairs on the dual algebra C^*, and computes explicit examples on the coalgebra preduals of the Podleś sphere and quantized projective spaces, obtaining the antiholomorphic Heckenberger-Kolb calculi and proving they have classical dimension.

Significance. If the correspondence and universal properties hold under the stated hypotheses and the explicit computations are valid, the paper supplies a new systematic approach to codifferential calculi on quantum homogeneous spaces, including explicit constructions, universal properties, and concrete applications that recover known calculi with classical dimension. The explicit computations on standard examples constitute a concrete strength.

major comments (1)
  1. [§3] §3 (Correspondence Theorem, likely around the statement following the faithful-flatness hypothesis): The bijection between equivariant first-order codifferential calculi on C = U ⊗_H k and right coideals T ⊆ ker(ε) (and the universal property of the maximal prolongation) is proved only under the assumption that U is faithfully flat as a left- and right H-module (plus char(k) ≠ 2). The paper performs explicit computations in later sections on the coalgebra preduals of the Podleś sphere and quantized projective spaces to obtain the Heckenberger-Kolb calculi and their classical dimension, but does not verify that the faithful-flatness hypothesis holds for the specific U and H arising in these examples. If the hypothesis fails for these U and H, the correspondence does not apply and the dimension claim is unsupported.
minor comments (1)
  1. Notation for the coalgebra C = U ⊗_H k and the coaction on T should be cross-referenced explicitly when the examples are introduced, to make clear which instance of the general theorem is being invoked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for highlighting this important point about the hypotheses in our correspondence theorem. We address the comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Correspondence Theorem, likely around the statement following the faithful-flatness hypothesis): The bijection between equivariant first-order codifferential calculi on C = U ⊗_H k and right coideals T ⊆ ker(ε) (and the universal property of the maximal prolongation) is proved only under the assumption that U is faithfully flat as a left- and right H-module (plus char(k) ≠ 2). The paper performs explicit computations in later sections on the coalgebra preduals of the Podleś sphere and quantized projective spaces to obtain the Heckenberger-Kolb calculi and their classical dimension, but does not verify that the faithful-flatness hypothesis holds for the specific U and H arising in these examples. If the hypothesis fails for these U and H, the correspondence does not apply and the dimension claim is unsupported.

    Authors: We agree that the correspondence theorem requires the faithful flatness hypothesis and that this must be confirmed for the examples if the theorem is to be invoked there. In the revised manuscript we will add an explicit verification (or reference to known results on faithful flatness for these standard quantum homogeneous spaces) that U is faithfully flat as a left and right H-module in the cases of the Podleś sphere and the quantized projective spaces. With this addition the identification of the computed codifferential calculi with the Heckenberger–Kolb calculi and the subsequent dimension argument will be fully justified under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity; theorems stated under explicit hypotheses with independent proofs and explicit example computations.

full rationale

The paper states the faithful flatness hypothesis upfront, proves the correspondence and prolongation results under that hypothesis using Hopf algebra axioms, then performs direct computations on coalgebra preduals of the Podleś sphere and quantized projective spaces. No step reduces a claimed result to a fitted parameter, self-definition, or unverified self-citation chain. The derivation chain is self-contained against the stated assumptions and external Hopf-algebraic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Builds on standard coalgebra and Hopf-algebra theory; no free parameters or invented entities; domain assumptions are the characteristic restriction and faithful flatness needed for the main correspondence.

axioms (2)
  • domain assumption char(k) eq 2
    Stated as necessary for the theory of first- and higher-order codifferential calculi over coalgebras.
  • domain assumption U is faithfully flat as left- and right H-module
    Required for the bijection between equivariant calculi and quantum tangent spaces when C = U ⊗_H k.

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