Center and derivations of generalized Weyl algebras over $\mathbb{Z}/p^n\mathbb{Z}$

Akaki Tikaradze; Ruben Mamani-Velasco

arxiv: 2606.04183 · v2 · pith:JAZRQX6Bnew · submitted 2026-06-02 · 🧮 math.QA · math.RT

Center and derivations of generalized Weyl algebras over mathbb{Z}/p^nmathbb{Z}

Ruben Mamani-Velasco , Akaki Tikaradze This is my paper

Pith reviewed 2026-06-28 07:03 UTC · model grok-4.3

classification 🧮 math.QA math.RT

keywords centerderivationsgeneralized Weyl algebrasenveloping algebraWitt vectorsHochschild cohomologysl_2mixed characteristic

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The pith

The center of U(sl_2) over Z/p^n Z is generated by the Casimir element over the Witt vectors of length n of its p-center.

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

This paper computes the centers and derivations of classical generalized Weyl algebras and of U(sl_2) when these algebras are defined over the ring Z/p^n Z. For U(sl_2) it establishes that the full center is obtained by adjoining the Casimir element to the ring of Witt vectors of length n applied to the p-center. The explicit description of derivations then shows that the restriction map from the first Hochschild cohomology of A to the derivations of its center is an isomorphism whenever the base ring is a field of characteristic p greater than 2. A reader would care because these results give concrete algebraic presentations of the centers and outer derivations for two families of algebras that appear in representation theory and noncommutative algebraic geometry in mixed characteristic.

Core claim

Let A be either a classical generalized Weyl algebra or the enveloping algebra U(sl_2) over Z/p^n Z. The center of U(sl_2) is generated by the Casimir element over the ring of the Witt vectors of length n of its p-center. The paper's description of derivations of A implies that if the ground ring is a field k of characteristic p>2, then the restriction homomorphism HH^1_k(A) to Der_k(Z(A), Z(A)) is an isomorphism.

What carries the argument

The ring of Witt vectors of length n of the p-center, which together with the Casimir element generates the full center of U(sl_2) over Z/p^n Z.

If this is right

The first Hochschild cohomology of A over such a field k is isomorphic to the k-derivations of the center Z(A).
Derivations of these algebras over Z/p^n Z admit an explicit description in terms of the center.
The centers of the classical generalized Weyl algebras over Z/p^n Z are computable by the same methods used for U(sl_2).

Where Pith is reading between the lines

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

The generation statement may extend to other enveloping algebras of semisimple Lie algebras over the same base rings.
The isomorphism between Hochschild cohomology and derivations of the center could be used to classify outer derivations in mixed characteristic.
Direct verification of the center generators for small n and small p would provide an independent check of the Witt-vector construction.
The results restrict the possible deformations of these algebras that preserve the center structure.

Load-bearing premise

The algebras are taken to be classical generalized Weyl algebras or U(sl_2) defined over Z/p^n Z, and the Hochschild cohomology isomorphism is stated only for base fields of characteristic p greater than 2.

What would settle it

An explicit basis computation of the center of U(sl_2) for n=1 over a field of characteristic p>2 that produces an element outside the subalgebra generated by the Casimir and the p-center would falsify the generation claim.

read the original abstract

Let $A$ be either a classical generalized Weyl algebra (also known as a noncommutative deformation of type A Kleinian singularity) or the enveloping algebra $U(\mathfrak{sl}_{2})$ over $\mathbb{Z}/p^n\mathbb{Z}.$ In this paper we compute the center and derivations of $A.$ More specifically, we show that the center of $U(\mathfrak{sl}_2)$ is generated by the Casimir element over the ring of the Witt vectors (of length $n$) of its $p$-center. Our description of derivations of $A$ implies that if the ground ring is a field $k$ of characteristic $p>2,$ then the restriction homomorphism $HH^1_{k}(A)\to Der_{k}(Z(A), Z(A))$ from the first Hochschild cohomology of $A$ to $k$-derivations of the center is an isomorphism.

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.

Desk Editor's Note private letter to a colleague

The paper computes explicit generators for the center of U(sl_2) over Z/p^n Z as the Casimir over the length-n Witt vectors of the p-center, plus a derivation description that yields an HH^1 isomorphism over fields of char p>2.

read the letter

The core contribution is a direct computation of the center for these two families of algebras over Z/p^n Z. For U(sl_2) they show it is generated by the Casimir together with the Witt vector ring built from the p-center. They also give derivations for the generalized Weyl algebras and conclude that the natural map from first Hochschild cohomology to derivations of the center is an isomorphism when the base is a field of characteristic p>2.

This is the kind of explicit description that people working on representations or deformations in positive characteristic sometimes need but rarely get written down cleanly. The Witt vector step for lifting from the p-center looks like the right tool for the mixed-characteristic case, and the restriction to p>2 is stated up front rather than hidden.

The main soft spot is that the abstract gives the statements without visible proof sketches or intermediate lemmas, so a referee would have to check whether the lifting arguments from characteristic p to p^n introduce any extra conditions on the base ring. Nothing in the stated results suggests circularity or hidden fitting, but the verification burden sits on the calculations themselves.

This paper is for specialists in noncommutative algebra who already care about Weyl algebras or enveloping algebras in positive characteristic. A reader who needs concrete generators or wants to compute derivations on the center will get usable statements. It is worth sending to referees because the claims are narrow enough to check and the setting is standard enough that the computations can be validated or corrected without rewriting the whole paper.

Referee Report

0 major / 2 minor

Summary. The paper computes the center and derivations of classical generalized Weyl algebras (noncommutative deformations of type A Kleinian singularities) and of U(sl_2), both taken over the ring Z/p^n Z. The central result states that the center of U(sl_2) is generated by the Casimir element over the length-n Witt vector ring of the p-center. For a field k of characteristic p>2 the paper further shows that the restriction map HH^1_k(A) → Der_k(Z(A), Z(A)) is an isomorphism.

Significance. The explicit description of the center via the Casimir and Witt vectors, together with the isomorphism relating Hochschild cohomology to derivations of the center, supplies concrete structural information for these algebras over rings of p-power characteristic. Such results are useful for questions in modular representation theory and noncommutative deformation theory.

minor comments (2)

[Introduction] The abstract states the main theorems cleanly, but the introduction would benefit from a short paragraph outlining the strategy used to lift the p-center computations to the Witt-vector ring of length n.
[§2] Notation for the generalized Weyl algebra A should be fixed once at the beginning of §2 rather than re-introduced in each subsection.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; central claims rest on direct computation

full rationale

The paper computes the center of U(sl_2) over Z/p^n Z as generated by the Casimir over the Witt vector ring of the p-center, and describes derivations implying an isomorphism on HH^1 for char p>2 fields. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated results. The derivation is presented as a standard ring-theoretic computation without reduction to prior author results by construction. Minor self-citation at most is possible but not load-bearing for the main claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5693 in / 1145 out tokens · 22710 ms · 2026-06-28T07:03:08.945419+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references

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A. Stewart, V. Vologodsky, On the center of the ring of differential operators on a smooth variety over Z /p^n Z , Compos. Math. 149 (2013), no. 1, 63--80

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M. A. Farinati, A. Solotar and M. Suarez-Alvarez, Hochschild homology and cohomology of generalized Weyl algebras , Univ. de Grenoble. Ann. de l'Institut Fourier 53 (2003) 465--488

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X. Tang, Algebra endomorphisms and derivations of some localized down-up algebras , J. Algebra Appl. 14 (2015), no. 3, 1550034, 14 pp

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[1] [1]

Bavula, Generalized Weyl algebras and their representations , Algebra i Analiz 4 no

V. Bavula, Generalized Weyl algebras and their representations , Algebra i Analiz 4 no. 1 (1992), 75--97; English translation, St. Petersburg Math. Journal 4 no. 1 (1993), 71--92

1992

[2] [2]

Brown, K

K. Brown, K. Changtong, Symplectic reflection algebras in positive characteristic , Proceedings of the Edinburgh Mathematical Society (2010) 53, 61--81

2010

[3] [3]

Hodges, Noncommutative deformations of type-A Kleinian singularities , J

T. Hodges, Noncommutative deformations of type-A Kleinian singularities , J. Algebra 161 (1993), no. 2, 271--290

1993

[4] [4]

Lunts, A

V. Lunts, A. Rosenberg, Kashiwara Theorem for Hyperbolic Algebras , preprint MPIM

[5] [5]

A. N. Rudakov, I. R. Safarevich, Irreducible representations of a simple three-dimensional Lie algebra over a field of finite characteristic , Mat. Zametki 2 (1967), 439–454

1967

[6] [6]

Stewart, V

A. Stewart, V. Vologodsky, On the center of the ring of differential operators on a smooth variety over Z /p^n Z , Compos. Math. 149 (2013), no. 1, 63--80

2013

[7] [7]

Tikaradze, Hochschild cohomology of deformation quantizations over Z /p^n Z , Algebr

A. Tikaradze, Hochschild cohomology of deformation quantizations over Z /p^n Z , Algebr. Represent. Theory 19 (2016), no. 1, 209–214

2016

[8] [8]

Kitchin, Derivations of quantum and involution generalized Weyl algebras , J

A. Kitchin, Derivations of quantum and involution generalized Weyl algebras , J. Algebra Appl. 23 (2024), no. 4, Paper No. 2450076, 21 pp

2024

[9] [9]

M. A. Farinati, A. Solotar and M. Suarez-Alvarez, Hochschild homology and cohomology of generalized Weyl algebras , Univ. de Grenoble. Ann. de l'Institut Fourier 53 (2003) 465--488

2003

[10] [10]

Tang, Algebra endomorphisms and derivations of some localized down-up algebras , J

X. Tang, Algebra endomorphisms and derivations of some localized down-up algebras , J. Algebra Appl. 14 (2015), no. 3, 1550034, 14 pp

2015