Simple Modules over Second Quantum Weyl Algebra

Sanu Bera

REVIEW 3 major objections 5 minor 38 references

Simple Modules over Second Quantum Weyl Algebra

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read At roots of unity, every simple module of the second quantum Weyl algebra is finite-dimensional, and the paper gives the exact PI degree and a complete list up to isomorphism.

desk verdict The paper delivers a new PI degree formula and explicit Type-I/II classifications, but Section 9 does not actually deliver the advertised complete classification of simple modules. read the letter →

arxiv 2412.17261 v1 pith:GJQCKV36 submitted 2024-12-23 math.RT math.QA

Sanu Bera This is my paper · ORCID

classification math.RTmath.QA MSC 16D6016D7016S3616R2016T20

keywords quantumWeylalgebrasecondsimplemodulesPIdegreerootsofunitypolynomialidentityaffinespaceskewring

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

At roots of unity, the multiparameter second quantum Weyl algebra becomes a polynomial identity (PI) algebra, so every simple module is finite-dimensional and bounded by the PI degree. This paper proves that the PI degree is exactly $n_1 n_2 t_3$, where $n_1, n_2$ are the orders of $\varepsilon_1, \varepsilon_2$ and $t_3 = \operatorname{ord}(\lambda^{n_1 n_2})$, and then classifies all simple modules up to isomorphism. The classification is explicit: Type-I modules are the families $V_1(\mu), V_2(\mu), V_3(\mu), V_4$, Type-II modules are $V_5(\mu), V_6(\mu)$, and the remaining Type-III modules are described through the known simple modules of a quantum affine space. This gives a complete solution to Problem 2 of [23] for the second quantum Weyl algebra, covering parameter ranges where earlier work needed a divisibility condition.

What carries the argument

The argument is carried by three pieces. The algebra is presented as an iterated skew polynomial ring $K[y_1][x_1,\tau_1,\delta_1][y_2,\sigma_2][x_2,\tau_2,\delta_2]$, and a derivation-erasing theorem [14] is used to discard the skew derivations without changing the PI degree, reducing the computation to the quantum affine space $\mathcal{O}_\Lambda(K^4)$ associated with the matrix $\Lambda$ in (3.1). The PI degree of that quantum affine space is then computed from the invariant factors of the associated skew-symmetric integral matrix via [6] and [20], giving $n_1 n_2 t_3$. For the module classification, the normal elements $\omega_1 = x_1y_1-y_1x_1$ and $\omega_2 = x_2y_2-y_2x_2$ act on any simple module either as zero or invertibly, splitting the classification into Types I, II, and III; common eigenvectors of certain commuting elements produce the explicit families $V_1(\mu)$ through $V_6(\mu)$, and Type III is handled by the simple modules of the quantum affine space factor from [17].

What would settle it

For the parameter tuple with $\varepsilon_1 = -1$, $\varepsilon_2$ a primitive third root of unity, and $\lambda$ a primitive 18th root of unity, compute the invariant factors $h_1, h_2$ of the integral matrix $B$ from Section 3, Step 2, and evaluate $n/\gcd(h_1,n) \cdot n/\gcd(h_2,n)$ with $n = 18$; the formula predicts $18 = 2 \cdot 3 \cdot \operatorname{ord}(\lambda^6)$, so any other value would show the derivation-erasing step is wrong. Alternatively, build the algebra explicitly for these parameters and check that every simple module has dimension at most 18 and that the paper's families realize all isomorphism classes.

Watch

Extended reading notes

Core claim

The central claim is that for primitive $n_1$-th, $n_2$-th, and $n_3$-th roots of unity $\varepsilon_1, \varepsilon_2, \lambda$ over an algebraically closed field, the PI degree of $A(\varepsilon_1,\varepsilon_2,\lambda)$ is exactly $n_1 n_2 t_3$ with $t_3 = \operatorname{ord}(\lambda^{n_1 n_2})$, and that this number is the sharp upper bound for the dimension of a simple module. The paper further claims that every simple right module is isomorphic to exactly one of the explicitly constructed modules $V_1(\mu), V_2(\mu), V_3(\mu), V_4, V_5(\mu), V_6(\mu)$, or to a Type-III module obtained from the classification of simple modules over the factor of a quantum affine space, with the dimensions listed in Section 9. Together these claims resolve Problem 2 of [23] for the second quantum Weyl algebra, removing the divisibility condition $\operatorname{ord}(\lambda) \mid \operatorname{ord}(\varepsilon_1)$ assumed in earlier work [18] and recovering that earlier result as the special case where $t_3 = 1$.

Load-bearing premise

The load-bearing premise is that the theorem used to erase derivations from the skew-polynomial presentation actually applies to this algebra; the proof asserts this without checking the theorem's hypotheses, and the PI-degree formula would fail if it does not.

Editorial extensions

If this is right

The bound is sharp: the family $V_1(\mu)$ has dimension $n_1 n_2 t_3$, so the PI degree is attained by an explicit simple module.
The list is exhaustive: every simple right module is isomorphic to one of the six explicit families or to a Type-III module from the quantum affine space classification.
The earlier divisibility-condition result is recovered: when $\operatorname{ord}(\lambda)$ divides $\operatorname{ord}(\varepsilon_1)$, the factor $t_3$ equals 1 and the formula reduces to $n_1 n_2$.
This resolves Problem 2 of [23] for the second quantum Weyl algebra, giving a complete classification of irreducible representations in the root-of-unity setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same derivation-erasing reduction likely extends to higher-order multiparameter quantum Weyl algebras, where the PI degree should again be computed from invariant factors of a $2n \times 2n$ integral matrix.
Because the proof works over an arbitrary algebraically closed field, the classification holds in positive characteristic, so the finite-dimensional simple modules of these algebras are the same in modular settings.
The Type-III reduction indicates that the genuinely new module theory of the second quantum Weyl algebra lives in Types I and II, with the remaining simple modules inherited from quantum affine spaces.
A direct count of isomorphism classes from the parameter tuples $\mu$ would yield a dimension-by-dimension census of simple modules, which the paper does not spell out.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The paper has a real new result and a real hole. The PI degree formula (Theorem 3.4) — n1 n2 t3 with t3 = ord(lambda^{n1 n2}) — is derived cleanly from Leroy-Matczuk derivation erasing and the De Concini-Procesi invariant factor formula, and it correctly recovers the author's earlier divisibility-case result. The Type-I and Type-II classification (V1–V6) is explicit: modules are constructed, simplicity is proved, and isomorphism criteria are given. That is substantial work and, as far as I can see, correct.

The hole is Section 9. The paper claims a complete classification of all simple modules, but for the w1-torsion (Type-III) case it does not, in fact, classify anything. It asserts that such modules correspond to simple modules over O_Lambda(K^4)/<1+(epsilon1-1)y1x1> with invertible x1,y1, says that using the classification in [17] 'we can classify' them, lists four possible K-dimensions, and then declares the classification complete. No module family is constructed, no statement of which [17] modules survive the quotient relation, no isomorphism criterion. The stress-test note is right: the completeness claim is not supported as written. This is an internal omission, not a clash with external results, but it is the main claim of the paper's title and abstract.

Two smaller things. The paper repeatedly says 'we can easily verify' for module actions and isomorphisms; in this subject those verifications are sometimes delicate, and a referee should ask for at least representative details. And the application of the Leroy-Matczuk derivation erasing theorem is asserted rather than verified; the reader flagged this as the weakest assumption. It is plausible, but it should be checked line-by-line.

Bottom line: the PI degree computation and the Type-I/II classifications are worth publishing. The Type-III gap means the paper does not currently deliver the advertised complete classification. I would send it to a serious referee, with the request that the Type-III part be either written out properly or the claim reduced to what is actually proved. For a reader working on quantum Weyl algebras, the PI degree formula and V1–V6 families are useful; for the general reader, it's a paper to skim.

Referee Report

3 major / 5 minor

Summary. This paper studies the multiparameter second quantum Weyl algebra A(ε1,ε2,λ) over an algebraically closed field when the deformation parameters are roots of unity. The main results are an explicit PI-degree formula (Theorem 3.4): PI-deg A = n1 n2 t3 with t3 = ord(λ^{n1 n2}), and a classification of simple modules into Type-I (families V1–V4), Type-II (families V5–V6), and Type-III modules, the last being handled through the author's earlier classification of simple modules over quantum affine space [17]. The paper claims that this gives a complete solution to Walton's Problem 2 for the second quantum Weyl algebra.

Significance. If correct, the PI-degree formula is a clean, parameter-free extension of the author's previous divisibility-condition result, and the classification would be the first complete classification of simple modules for this algebra without extra hypotheses. The paper contains explicit module constructions and isomorphism criteria for Types I and II, with concrete dimension formulas, and the PI-degree derivation is a genuine application of established theorems. However, the Type-III part of the classification is not actually carried out in the manuscript, and one proof in Section 6 relies on an incorrect centrality claim. These issues affect the central completeness claim, so the significance is conditional on a completed and corrected treatment.

major comments (3)

[Section 9] The Type-III classification is not carried out. The section asserts that a w1-torsion simple module corresponds to a simple module over O_Λ(K^4)/<1+(ε1-1)y1x1> with invertible x1,y1, invokes the classification of [17], lists four possible K-dimensions, and then concludes the classification. No Type-III module family is explicitly constructed, no statement is given of which simple modules over O_Λ(K^4) survive the quotient relation, and no isomorphism criterion is provided. Since completeness of the classification is one of the two central claims of the paper, this is a load-bearing gap. The section should either give the explicit Type-III modules and their isomorphism classes, or provide a precise reduction to [17] with a bijection on isomorphism classes and a proof of exhaustiveness.
[Section 6 (and Corollary 2.2)] The opening of Section 6 states that x1^{n1}, y1^{n1}, x2^{n2}, y2^{n2} are central elements and uses Schur's lemma to conclude that each acts as a scalar. This is false in general: for instance, x1^{n1} y2 = λ^{−n1} y2 x1^{n1} by the relations in (2.1), which equals y2 x1^{n1} only if λ^{n1}=1, a condition not assumed when n3 does not divide n1. Consequently the stated proof of the dichotomy 'each of x1,y1,x2,y2 is nilpotent or invertible' is invalid. The dichotomy can likely be recovered by proving that these powers are normal elements and using the fact that the kernel of a normal element on a simple module is a submodule, but this argument is not given and would also require correcting Corollary 2.2, which is false as stated.
[Section 3, Step 1] The derivation-erasing step is not fully justified. The paper asserts that all hypotheses of [14, Theorem 7] are satisfied, but only verifies the q-skew relation δi τi = εi τi δi for i=1,2. The hypotheses of Leroy–Matczuk's theorem concern the structure of iterated Ore extensions satisfying a polynomial identity, and the manuscript does not spell out or check them. Since the equality PI-deg A = PI-deg O_Λ(K^4) is the foundation of Theorem 3.4, the verification should be written out explicitly, or the theorem should be quoted with its hypotheses and a point-by-point check.

minor comments (5)

[Theorem 5.1] In the proof of Theorem 5.1, Case 2, the range '1 ≤ j ≠ s ≤ n1 − 1' should presumably be '1 ≤ j ≠ s ≤ n2 − 1', since the second index ranges over 0, ..., n2 − 1.
[Section 6.4] There is a duplicated word in 'we can determine the constant values values'; this should read 'constant values'.
[Section 9] In item (3) of the list of possible dimensions, the phrase 'If the action of is y2 invertible' is missing a variable; it should read 'If the action of y2 is invertible and x2 is nilpotent'.
[References] Reference [18] is listed with only 'DOI' in place of full publication data; please provide the complete journal and article details.
[General] The version supplied for review contains many encoding artifacts in the abstract and section headings (e.g., 'A/b.pc/s.pc/t.pc/r.pc/...' and numerous '/u1D...' strings). Please ensure the published PDF is clean.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the PI-degree formula is derived from external theorems and invariant-factor arithmetic, and Types I–II are constructed and classified explicitly; the Type-III classification is deferred to the author's prior work [17], a load-bearing self-citation that is independent rather than circular.

full rationale

The PI-degree computation in Section 3 is self-contained. It applies the derivation-erasing theorem of Leroy-Matczuk [14, Theorem 7] and the quantum-affine-space PI-degree formula [20, Lemma 5.7] to the algebra's iterated skew polynomial presentation, then computes invariant factors of an explicit integral matrix. No equation in this chain defines the PI degree in terms of the quantity being predicted, and the result is not assumed as an input. Corollary 3.5 recovers the earlier divisibility-condition formula from [18] as a special case, but that earlier result is not used to prove Theorem 3.4. The Type-I and Type-II classifications in Sections 5-8 are direct: the modules V1-V6 are given with explicit bases and generator actions, simplicity is proved by submodule arguments, and isomorphism criteria are stated and proved. No fitted parameter is relabeled as a prediction, and no quantity is defined in terms of the classification it is supposed to produce. The only passage that approaches a circularity concern is Section 9, where Type-III simple modules are reduced to simple modules over a quotient of quantum affine space and then 'based on' the classification in [17] the paper asserts that they can be classified. This is a genuine reliance on prior work by the same authors, and the text does not explicitly describe the Type-III module families or their isomorphism classes. However, [17] is a separate published classification for quantum affine space, not a restatement of the present theorem, so the reliance is independent support rather than a circular reduction. The completeness gap in Section 9 is a correctness or exposition concern, not a case of the derivation collapsing into its own inputs. Overall, the central PI-degree result and the Types I-II classifications are independent of the target claim, and the self-citations present are minor or load-bearing but not circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; the module labels μ are representation parameters, not adjustable constants. The axioms are standard theorems plus the explicit hypotheses of the root-of-unity setting. No new entities are postulated.

assumptions (6)

domain assumption K is an algebraically closed field of arbitrary characteristic.
Used throughout; Kaplansky's theorem and Schur's lemma require an algebraically closed field (Section 2.2).
domain assumption ε1, ε2, λ are primitive roots of unity of orders n1, n2, n3.
The root of unity setting is the whole paper; stated in the Assumptions section after the introduction.
standard math Leroy-Matczuk derivation erasing theorem [14, Theorem 7] applies to the skew polynomial presentation (2.1).
Section 3, Step 1 asserts the hypotheses are satisfied but does not verify them. This theorem is critical for reducing PI-deg A to PI-deg O_Λ(K^4).
standard math De Concini-Procesi formula for PI degree of quantum affine spaces [6, Proposition 7.1] as stated in [20, Lemma 5.7].
Used in Step 3, Equation (3.3), to express PI degree in terms of invariant factors.
standard math Kaplansky's theorem: a prime affine PI algebra over an algebraically closed field has simple modules of dimension at most its PI degree.
Proposition 2.4, the foundational bound for the classification.
standard math Classification of simple modules over the quantum affine space O_Λ(K^4) from [17].
Type-III classification in Section 9 reduces to this external result by the same author.

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Pith. "Pith review of Simple Modules over Second Quantum Weyl Algebra." pith.science (2026). https://pith.science/paper/GJQCKV36

@misc{pith2026241217261,
  author       = {Pith},
  title        = {Pith review of: Simple Modules over Second Quantum Weyl Algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJQCKV36}},
  note         = {Machine review of arXiv:2412.17261}
}

abstract

In this article, we study the multiparameter second quantum Weyl algebra at roots of unity. In this setting, the algebra is a polynomial identity (PI) algebra, and the dimension of its simple modules is bounded above by its PI degree. We explicitly determine the PI degree and provide a complete classification of simple modules. This classification offers a comprehensive solution to $\href{https://doi.org/10.1007/978-3-030-19486-4_23}{\text{Problem 2: C, Walton (2019)- An Invitation to Noncommutative Algebra}}$ for the second quantum Weyl algebra.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

[17]

Akhavizadegan and D

M. Akhavizadegan and D. Jordan, Prime ideals of quantize d Weyl algebras, Glasgow Math. J., 38(3), (1996), 283–297

work page 1996
[1]

I/n.pc/t.pc/r.pc/o.pc/d.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc Let K be a ﬁeld and K∗ denote the multiplicative group of nonzero elements of K and let /u1D45B be a positive integer. Let Λ := (/u1D706/u1D456 /u1D457 ) be an /u1D45B× /u1D45Bmultiplicatively antisymmetric matrix over K, that is, /u1D706/u1D456/u1D456= 1 and /u1D706/u1D456 /u1D457/u1D706/u1D457/u1D...

work page 2020
[2]

SECOND QUANTUM WEYL ALGEBRA 3 2.1

P/r.pc/e.pc/l.pc/i.pc/m.pc/i.pc/n.pc/a.pc/r.pc/i.pc/e.pc/s.pc In this section, we recall essential facts about the second q uantum Weyl algebras and polynomial identity algebras, which will be applied to comp ute the PI degree (an invariant) and to classify simple modules. SECOND QUANTUM WEYL ALGEBRA 3 2.1. Commutation Relations. First recall that the alg...

work page
[3]

In [18, Theorem 3.1], we determined the PI degree under the divisibil- ity condition

PI D/e.pc/g.pc/r.pc/e.pc/e.pc /f.pc/o.pc/r.pc /s.pc/e.pc/c.pc/o.pc/n.pc/d.pc Q/u.pc/a.pc/n.pc/t.pc/u.pc/m.pc W/e.pc/y.pc/l.pc /a.pc/l.pc/g.pc/e.pc/b.pc/r.pc/a.pc In this section, we compute an explicit expression of the PI degree for second quantum Weyl algebra at roots of unity. In [18, Theorem 3.1], we determined the PI degree under the divisibil- ity c...

work page
[4]

Suppose /u1D440is a simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module

S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc /o.pc/v.pc/e.pc/r.pc/u1D434(/u1D45E1, /u1D45E2, /u1D706) In the root of unity context, the second quantum Weyl algebra /u1D434(/u1D45E1, /u1D45E2, /u1D706) is classiﬁed as a prime aﬃne PI algebra. Suppose /u1D440is a simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module. Then /u1D440is ﬁnite- dimensi...

work page
[5]

C/o.pc/n.pc/s.pc/t.pc/r.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-I S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc In this section we wish to construct Type-I simple modules ov er /u1D434(/u1D45E1, /u1D45E2, /u1D706). 5.1. Simple Modules V1(/u1D707). For /u1D707= (/u1D7071, /u1D7072, /u1D7073, /u1D7074) ∈ ( K∗)4, let V1(/u1D...

work page
[6]

Suppose /u1D713: V1(/u1D707 ) → V 1(/u1D707′) is a module isomorphism

(5.1) Proof. Suppose /u1D713: V1(/u1D707 ) → V 1(/u1D707′) is a module isomorphism. Observe that /u1D452(/u1D44E1, /u1D44E2) = /u1D707−/u1D44E1 1 /u1D707−/u1D44E2 2 /u1D452(0, 0)/u1D465/u1D44E2 2 /u1D465/u1D44E1 1 holds in V1(/u1D707). Then /u1D713can be uniquely determined by the image /u1D713(/u1D452(0, 0)) , i,e., say /u1D713(/u1D452(0, 0)) = /summatio...

work page
[7]

/u1D44E (/u1D707−1 2 /u1D707′ 2)/u1D44F (/u1D45E1/u1D706)/u1D45F /u1D44F when /u1D44E∔ /u1D45F >0 (/u1D707−1 1 /u1D707′

work page

Show all 38 references

[8]

Note that /u1D719is a bĳection

/u1D44E (/u1D707−1 2 /u1D707′ 2)/u1D44F (/u1D45E1/u1D706)/u1D45F /u1D44F/u1D706−/u1D460/u1D4591 /u1D43F3 when /u1D44E∔ /u1D45F= 0. Note that /u1D719is a bĳection. Then using the relations ( 5.1), we can easily verify that /u1D719is an /u1D434(/u1D45E1, /u1D45E2, /u1D706)-modul...
[9]

It can be easily verify that /u1D719is an /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module isomorphism

/u1D44E/u1D706−/u1D45F /u1D44F/u1D452(/u1D44E⊕ /u1D45F, /u1D44F), where ⊕ denotes addition modulo /u1D4591. It can be easily verify that /u1D719is an /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module isomorphism. □ SECOND QUANTUM WEYL ALGEBRA 11 5.3. Simple Modules V3(/u1D707). For ...
[10]

This means the actions of /u1D4671 and /u1D4672 on /u1D440are invertible

C/l.pc/a.pc/s.pc/s.pc/i.pc/f.pc/i.pc/c.pc/a.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-I S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc Let /u1D440be a /u1D4671, /u1D4672-torsionfree simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module. This means the actions...
[11]

Denote /u1D451:= lcm(/u1D4591, /u1D4593)

C/o.pc/n.pc/s.pc/t.pc/r.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-II S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc In this section, we wish to construct Type-II simple modules over /u1D434(/u1D45E1, /u1D45E2, /u1D706). Denote /u1D451:= lcm(/u1D4...
[12]

/u1D451, /u1D7072 = (/u1D45E1/u1D706)/u1D45F/u1D707′ 2 and /u1D7073 = /u1D45E/u1D45F 1 /u1D707′ 3. Proof. The proof is parallel to Theorem 5.2. □ 7.2. Simple Modules V6(/u1D707). For each /u1D707∈ K∗, let V6(/u1D707) be the K-vector space with basis {/u1D463/u1D458: 0 ≤ /u1D45...
[13]

This means that the operator /u1D4671 is invertible on /u1D440, while the operator /u1D4672 is zero on /u1D440

C/l.pc/a.pc/s.pc/s.pc/i.pc/f.pc/i.pc/c.pc/a.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-II S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc Let /u1D440be a /u1D4671-torsionfree and /u1D4672-torsion simple module over /u1D434(/u1D45E1, /u1D45E2, /u1D706). This ...
[14]

Thus by Schur’s lemma, /u1D7195 becomes a module isomorphism

We can verify that /u1D7195 is a nonzero /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module homomorphism. Thus by Schur’s lemma, /u1D7195 becomes a module isomorphism. Case 2: Assume that /u1D6FD≠ 0. This gives that /u1D4661 is an invertible operator on /u1D440. In this case, the sim...
[15]

This means that the action of /u1D4671 on /u1D440 is zero

S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-III Let /u1D440be a /u1D4671-torsion simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module. This means that the action of /u1D4671 on /u1D440 is zero. The relation /u1D4671 = 1 + ( /u1D45E1 − 1...
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Alev and F

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Akhavizadegan and D

M. Akhavizadegan and D. Jordan, Prime ideals of quantize d Weyl algebras, Glasgow Math. J., 38(3), (1996), 283–297

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[2] [1]

I/n.pc/t.pc/r.pc/o.pc/d.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc Let K be a ﬁeld and K∗ denote the multiplicative group of nonzero elements of K and let /u1D45B be a positive integer. Let Λ := (/u1D706/u1D456 /u1D457 ) be an /u1D45B× /u1D45Bmultiplicatively antisymmetric matrix over K, that is, /u1D706/u1D456/u1D456= 1 and /u1D706/u1D456 /u1D457/u1D706/u1D457/u1D...

work page 2020

[3] [2]

SECOND QUANTUM WEYL ALGEBRA 3 2.1

P/r.pc/e.pc/l.pc/i.pc/m.pc/i.pc/n.pc/a.pc/r.pc/i.pc/e.pc/s.pc In this section, we recall essential facts about the second q uantum Weyl algebras and polynomial identity algebras, which will be applied to comp ute the PI degree (an invariant) and to classify simple modules. SECOND QUANTUM WEYL ALGEBRA 3 2.1. Commutation Relations. First recall that the alg...

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[4] [3]

In [18, Theorem 3.1], we determined the PI degree under the divisibil- ity condition

PI D/e.pc/g.pc/r.pc/e.pc/e.pc /f.pc/o.pc/r.pc /s.pc/e.pc/c.pc/o.pc/n.pc/d.pc Q/u.pc/a.pc/n.pc/t.pc/u.pc/m.pc W/e.pc/y.pc/l.pc /a.pc/l.pc/g.pc/e.pc/b.pc/r.pc/a.pc In this section, we compute an explicit expression of the PI degree for second quantum Weyl algebra at roots of unity. In [18, Theorem 3.1], we determined the PI degree under the divisibil- ity c...

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[5] [4]

Suppose /u1D440is a simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module

S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc /o.pc/v.pc/e.pc/r.pc/u1D434(/u1D45E1, /u1D45E2, /u1D706) In the root of unity context, the second quantum Weyl algebra /u1D434(/u1D45E1, /u1D45E2, /u1D706) is classiﬁed as a prime aﬃne PI algebra. Suppose /u1D440is a simple /u1D434(/u1D45E1, /u1D45E2, /u1D706)-module. Then /u1D440is ﬁnite- dimensi...

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[6] [5]

C/o.pc/n.pc/s.pc/t.pc/r.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc /o.pc/f.pc T/y.pc/p.pc/e.pc-I S/i.pc/m.pc/p.pc/l.pc/e.pc M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc In this section we wish to construct Type-I simple modules ov er /u1D434(/u1D45E1, /u1D45E2, /u1D706). 5.1. Simple Modules V1(/u1D707). For /u1D707= (/u1D7071, /u1D7072, /u1D7073, /u1D7074) ∈ ( K∗)4, let V1(/u1D...

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[7] [6]

Suppose /u1D713: V1(/u1D707 ) → V 1(/u1D707′) is a module isomorphism

(5.1) Proof. Suppose /u1D713: V1(/u1D707 ) → V 1(/u1D707′) is a module isomorphism. Observe that /u1D452(/u1D44E1, /u1D44E2) = /u1D707−/u1D44E1 1 /u1D707−/u1D44E2 2 /u1D452(0, 0)/u1D465/u1D44E2 2 /u1D465/u1D44E1 1 holds in V1(/u1D707). Then /u1D713can be uniquely determined by the image /u1D713(/u1D452(0, 0)) , i,e., say /u1D713(/u1D452(0, 0)) = /summatio...

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[8] [7]

/u1D44E (/u1D707−1 2 /u1D707′ 2)/u1D44F (/u1D45E1/u1D706)/u1D45F /u1D44F when /u1D44E∔ /u1D45F >0 (/u1D707−1 1 /u1D707′

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