Bernstein-type inequalities for quantum algebras

Ashish Gupta; Sanu Bera; Snehashis Mukherjee; Sugata Mandal

REVIEW 3 major objections 4 minor 21 references

Bernstein-type inequalities for quantum algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Every nonzero finitely generated module of Horton's multiparameter quantum algebras that avoids the torsion subcategory has Gelfand–Kirillov dimension at least half (or half minus one) the algebra's own dimension.

desk verdict A correct-looking but under-polished extension of Jordan's method giving Bernstein inequalities for Horton's multiparameter quantum algebras; worth refereeing once the gaps are fixed. read the letter →

arxiv 2501.11399 v1 pith:G3RQHAMQ submitted 2025-01-20 math.QA math.RA

Sanu Bera , Ashish Gupta , Sugata Mandal , Snehashis Mukherjee This is my paper · ORCID

classification math.QAmath.RA MSC 16D2516D6016D7016S8516T2016R20

keywords BernsteininequalityGelfand-KirillovdimensionAmbiskewpolynomialringQuantumtorusQuantizedWeylalgebraMultiparametersymplecticspaceEuclidean

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

This paper establishes Bernstein-type inequalities—lower bounds on the Gelfand–Kirillov dimension of modules—for the multiparameter quantum algebras $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ introduced by Horton, a family that contains graded quantum Weyl algebras, quantum symplectic spaces, quantum Euclidean spaces, and quantum Heisenberg algebras. When $p_i=1$ and no $q_i$ is a root of unity, every nonzero finitely generated module that is not $\mathcal{Z}$-torsion has GK-dimension at least $n$, which is exactly half the GK-dimension of the algebra $K_n$. When $q_i=1$ and no $p_i$ is a root of unity, the same module class has GK-dimension at least $n-1$, one less than half. The proof reduces the whole problem to computing the dimension of a quantum torus obtained by localizing $K_n$, namely the size of its largest family of independent commuting monomials, and that dimension is shown to be $n$ (or $n+1$) precisely because any larger commuting family would force a parameter to be a root of unity. As a by-product the paper gives new inequalities for symplectic, Euclidean, and Heisenberg quantum spaces and a simplified proof of the known inequality for quantum Weyl algebras.

What carries the argument

The machinery is the iterated ambiskew polynomial ring structure of $K_n$: following the construction of [7], $K_n$ is built step by step so that the elements $z_i = \sum_{j\le i}(q_j-p_j)y_jx_j$ are normal, pairwise commuting, and generate an Ore set $\mathcal{Z}$. Localizing at $\mathcal{Z}$ yields $\mathcal{B}_n$, and inverting an appropriate choice of the $x_i$ or $y_i$ produces the quantum torus $U_n$, whose dimension is computed by Theorem 5.1 (taken from [1]) as the rank of the largest commutative subgroup of monomials—equivalently, the size of a maximal system of independent commuting monomials. Theorems 5.2 and 5.3 evaluate this number as $n$ (when $p_i=1$) or $n+1$ (when $q_i=1$) under the non-root-of-unity hypotheses. The proof of the Bernstein inequality then runs through Theorem 6.1, which bounds the GK-dimension of any module over a rank-$2n$ quantum torus below by $2n$ minus the torus dimension, and an embedding argument that transfers the bound from the torus back to $K_n$.

What would settle it

For the case $p_i=1$ with no $q_i$ a root of unity, write out the commutation matrix of $U_n$ (displayed as (5.2)) and search for a rank-$(n+1)$ subgroup of the monomial lattice on which the alternating bicharacter vanishes; equivalently, exhibit two independent monomials in $U_n$ that commute with each other and with all the $z_i$'s but are not powers of the $z_i$'s. Either construction would force $\dim(U_n) \ge n+1$, contradicting Theorem 5.2 and weakening Theorem BI-1 from $n$ to $n-1$.

Watch

Extended reading notes

Core claim

The central claim is that the classical Bernstein inequality admits a uniform quantum analogue whose sharp constant is the dimension of a fully localized quantum torus. For the algebra $K_n := K_{n,\Gamma}^{P,Q}(\mathbb{K})$ with $p_i = 1$ and no $q_i$ a root of unity, the localization $U_n$ that inverts every $y_i$ alongside the normal elements $z_j$ has dimension $\dim(U_n) = n$, and Theorem 6.2 yields $\operatorname{GK-dim}(M) \ge 2n - \dim(U_n) = n = \operatorname{GK-dim}(K_n)/2$ for every nonzero finitely generated $\mathcal{Z}$-torsionfree module $M$. In the case $q_i = 1$ with no $p_i$ a root of unity, the same computation gives $\dim(U_n) = n+1$, so $\operatorname{GK-dim}(M) \ge 2n - (n+1) = n-1 = \operatorname{GK-dim}(K_n)/2 - 1$. Since $\operatorname{GK-dim}(K_n) = 2n$, these bounds are exactly half (or half minus one) the algebra's growth. The dimension of the torus is the load-bearing number: by the dimension theorem for quantum tori it is the largest rank of a commutative subgroup of monomials, and Theorems 5.2 and 5.3 show that any additional independent commuting monomial would make a parameter a root of unity.

Load-bearing premise

The proof depends on the dimension computation $\dim(U_n) = n$ or $n+1$: if a commuting family of monomials one larger than the known one existed in the fully localized algebra without any parameter being a root of unity, the Bernstein bound would drop by one.

Editorial extensions

If this is right

Corollaries 1.2–1.4 establish Bernstein-type inequalities for quantum symplectic spaces, quantum Euclidean spaces, and quantum Heisenberg spaces; the lower bounds are $n$, $n-1$, and $n-1$ respectively.
Theorem 6.2 provides a uniform recipe: under the standing assumption on $p_i q_i^{-1}$, every nonzero finitely generated $\mathcal{Z}$-torsionfree module over $K_n$ satisfies $\operatorname{GK-dim}(M) \ge 2n - \dim(U_n)$, so computing a localization's dimension immediately yields the Bernstein bound.
The proof recovers, with a shorter argument, the previously known Bernstein inequalities for quantized Weyl algebras $A_n^{q,\Lambda}$ and for graded quantum Weyl algebras.
The equality $\operatorname{GK-dim}(K_n) = 2n$ is established for every member of the family under the standing assumption, so all these algebras share the same growth rate as the $2n$-dimensional quantum torus.

Reading between the lines

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

The dimension formulas in Theorems 5.2 and 5.3 are statements about the kernel of the alternating bicharacter on $\mathbb{Z}^{2n}$; reformulating the induction as a lattice argument could yield a closed-form dimension for other parameter regimes, including partial roots of unity, and would make the 'suitable relabeling' steps fully explicit.
The general bound $\operatorname{GK-dim}(M) \ge 2n - \dim(U_n)$ suggests that any algebra in the ambiskew family whose fully localized torus has dimension $d$ satisfies a Bernstein-type inequality with rate $2n-d$; testing other multiparameter deformations where $d$ can be computed would map out exactly which quantum algebras inherit the phenomenon.
One can ask whether the bounds are sharp: constructing explicit $\mathcal{Z}$-torsionfree modules of GK-dimension exactly $n$ (or $n-1$), for instance by inducing from the commutative subalgebra generated by the $z_i$'s, would show the inequalities cannot be improved within this class.
The obstruction to a stronger bound is purely multiplicative: a rank-$n+1$ commuting family of monomials in $U_n$ would exist exactly when a product of the $q_i$'s (respectively $p_i$'s) is a root of unity, so the theorem is essentially a fact about the multiplicative independence of the parameters.

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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

Quick take: this is a useful, workmanlike extension of Jordan's ambiskew-polynomial-ring route to Bernstein inequalities. The authors apply it to Horton's multiparameter algebras K_n and get the expected bounds: n for the p_i=1 case, n-1 for the q_i=1 case, plus Krull and global dimensions for the further localizations. The results look right; a symplectic-form check of the commuting-subgroup arguments supports dim(U_n)=n and n+1. This is solid subfield progress, not a paradigm shift.

What is genuinely new: Bernstein-type inequalities for the quantum symplectic, quantum Euclidean, and quantum Heisenberg spaces, and the unified treatment of Horton's family. The proof that GK-dim(E_n)=2n via the quantum affine subalgebra is clean, and the quick reproof of the known quantized Weyl algebra case is a nice bonus.

Soft spots, in rough order of importance. First, Theorem 6.2 reduces to the case of a simple module by saying a quotient's GK-dimension bounds M's. That is true, but the simple quotient of a Z-torsionfree module need not be Z-torsionfree, and the localization argument needs that. This is patchable, for instance by pulling back a maximal submodule of the localization, but as written it is a real gap. Second, Theorems 5.2 and 5.3 contain the steps "by a suitable relabeling" and "it is clear" inside the induction on commuting monomials. Those steps are load-bearing because they are where the non-root-of-unity hypotheses enter; a referee will want them written out. Third, there are typos: Theorem BI-2 states the hypothesis with i up to n-1 while Corollary 6.2 uses i up to n; the two type descriptions in the introduction are duplicated (both say p_i=1); and Corollaries 1.3 and 1.4 display n = GKdim/2 - 1 where n-1 is meant. These are easy fixes but confusing.

The citation pattern is sound and the key external results (Brookes, McConnell-Pettit, Jordan) are the right ones. No circularity, no fitted parameters.

This paper is for people working on GK-dimension and module theory of quantum algebras. It deserves a serious referee, who should ask for the combinatorial steps to be made explicit and the simple-reduction gap to be patched. I would accept it for review.

Referee Report

3 major / 4 minor

Summary. The paper studies the multiparameter quantum algebra K_n^{P,Q}(Γ) introduced by Horton, viewed as an iterated ambiskew polynomial ring, and proves Bernstein-type lower bounds for the Gelfand-Kirillov dimension of finitely generated non-Z-torsion modules. The strategy is to localize K_n at the Ore set generated by the normal elements z_i, then further localize to a quantum torus U_n, compute the Krull/global dimension of U_n as n in the case p_i = 1, q_i non-roots of unity and as n+1 in the case q_i = 1, p_i non-roots of unity, and then apply Brookes's inequality GK-dim(M) ≥ 2n − dim(U_n). The paper also gives corollaries for graded quantum Weyl algebras, quantum symplectic and Euclidean spaces, and quantum Heisenberg spaces, and sketches a simplified proof of Fukuda's theorem for quantized Weyl algebras.

Significance. If the technical gaps are repaired, the main results are a useful unified treatment: they recover and extend Bernstein-type inequalities for several well-known quantum algebras from the same localization-based framework, and they supply dimension computations for the relevant quantum tori. The paper is largely free of ad hoc parameters: the bounds follow from the external structural results of Brookes and of McConnell-Pettit, and no circular use is made of the target inequalities. The exposition is generally clear, but the proof of the crucial dimension computation and the reduction step in Section 6 need additional justification before the claims can be considered established.

major comments (3)

[Section 6, proof of Theorem 6.2] The reduction 'Since M has a maximal proper submodule and the GK-dimension of a factor of M bounds the GK-dimension of M it suffices to assume that M is simple' is not valid for the subsequent argument. A simple quotient M/N of a Z-torsionfree module need not be Z-torsionfree, so the assertion 'M z = M for all z in Z' and the later claims that M embeds in its localization at V and is torsionfree over the quantum affine space F'_q are not inherited by the quotient. This is load-bearing because the proof of the inequality depends on these torsion-freeness properties. The gap is repairable: localize M at Z to obtain M_B, choose a maximal B-submodule of M_B, and replace M by the image of M in the resulting simple B-module; this image is a quotient of M, is Z-torsionfree, and embeds in a simple B-module, so the rest of the proof can run on it. The manuscript should implement this or provide an equivalent argument.
[Section 5, Theorems 5.2 and 5.3] The induction in Theorems 5.2 and 5.3 contains two unproved combinatorial assertions: 'by a suitable relabeling we may assume' in Step 2 and around (5.3) and (5.5), and 'it is clear that taking suitable combinations of the v_j as in Step 2' after (5.3). These steps are exactly what force the commuting subgroup H to have a large intersection with the D-subgroup; if a larger commutative subgroup survived, dim(U_n) would increase and the final bounds in Theorems BI-1 and BI-2 would weaken. The assertions should be replaced by a rigorous argument, for example an explicit GL_k(Z) column-reduction of the exponent matrix of h_1,...,h_k showing that a mutually independent set with strictly increasing indices can be chosen, or by a direct proof using the alternating form λ. As written, the proofs of dim(U_n) = n and dim(U_n) = n+1 are incomplete.
[Section 6, Corollaries 6.1 and 6.2] Theorem 6.2 is stated for Z-torsionfree modules, while Theorems BI-1 and BI-2 only assume that the module is not Z-torsion. The corollaries state that the result is 'immediate', but this requires the standard step of passing to M/T, where T is the Z-torsion submodule, so that M/T is nonzero and Z-torsionfree and GK-dim(M) ≥ GK-dim(M/T). This step should be written out, since Theorem 6.2, as stated, does not apply to M directly.

minor comments (4)

[Section 1, Theorem BI-2] Theorem BI-2 in the introduction states the hypothesis on p_i for 1 ≤ i ≤ n−1, while Theorem 5.3 and Corollary 6.2 require 1 ≤ i ≤ n; please make the statements consistent.
[Section 1, Corollaries 1.3 and 1.4] In Corollaries 1.3 and 1.4 the displayed inequality 'GK-dim(M) ≥ n = GK-dim/2 − 1' is inconsistent; the right-hand side should be n−1 = GK-dim/2 − 1.
[Section 3, paragraph after Theorem 3.1] The sentence 'As K_n ⊆ U_n by [11, Lemma 8.1.13]' cites the wrong lemma for the embedding; the inclusion is a property of localization, and Lemma 8.1.13 is used later for monotonicity of GK-dimension.
[Section 4, Proposition 4.1] In the statement of Proposition 4.1, the phrase 'of 2t generators' is garbled; the subset in (i) should be of cardinality 2t.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Bernstein bounds follow from independent Brookes theorems and a self-contained quantum-torus dimension computation; the noted gaps are rigor issues, not circular reductions.

full rationale

The derivation chain is: GK-dim(K_n)=2n (Section 3, via the quantum torus U_n); dim(U_n)=n in case (A) and n+1 in case (B) (Theorems 5.2 and 5.3, based on [1, Theorem A]); GK-dim(M) >= 2n - dim(U_n) for localizable modules (Theorem 6.2, using [1, Theorem 2]); then BI-1 and BI-2 follow by substituting d=n or d=n+1. Every load-bearing input is external: [1] is Brookes' theorem on crossed products and quantum tori, [6] is Horton's definition of K_n, and [7] is Jordan's ambiskew construction; none is authored by the present paper, and none assumes the Bernstein inequality being proved. No parameter is fitted to the target modules; the non-root-of-unity hypotheses enter only through the dimension computation and the Ore/torsion arguments. The paper contains no self-citations. The under-specified relabeling claims around (5.3) and (5.5) and the quotient-to-simple reduction in Theorem 6.2 (where a simple quotient need not inherit B-torsionfreeness) are omissions in justification, not circular reductions: even if those steps were gaps, the conclusion would not be an input of the argument. Hence no circularity is present, and the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The algebra parameters p_i, q_i, gamma_ij are inputs defining K_n. The proof leans on prior theorems about quantum tori and GK dimension, none authored by the present authors. No new particles or entities are introduced.

assumptions (5)

standard math The dimension of a quantum torus equals the supremum of the ranks of commutative subgroups (Theorem 5.1, [1, Theorem A], [12]).
Used in Theorems 5.2 and 5.3 to bound dim(U_n) from above and below.
standard math For a finitely generated module over a quantum torus A of rank m, GK-dim(M) ≥ m - dim(A) (Theorem 6.1, [1, Theorem 2]).
This is the quantitative engine of the main Bernstein-type inequalities.
domain assumption The algebras K_{n,Γ}^{P,Q}(K) form iterated ambiskew polynomial rings with the stated normal elements z_i and commutation relations (Section 2.2, from [7] and [6]).
This structural fact is inherited from Horton's work and is used throughout the localization arguments.
domain assumption p_i q_i^{-1} is not a root of unity for each i (assumed throughout, as in [6]).
Needed in Lemma 6.1 and in the torus dimension computations to rule out unintended commutations.
standard math Standard properties of GK dimension for subalgebras and factor modules (Proposition 4.2, [11, Lemma 8.1.13]).
Used to transfer GK-dimension bounds between a module and its localizations.

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Cite this review

Pith. "Pith review of Bernstein-type inequalities for quantum algebras." pith.science (2026). https://pith.science/paper/G3RQHAMQ

@misc{pith2026250111399,
  author       = {Pith},
  title        = {Pith review of: Bernstein-type inequalities for quantum algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3RQHAMQ}},
  note         = {Machine review of arXiv:2501.11399}
}

abstract

We establish Bernstein-type inequalities for the quantum algebras $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ introduced by K. L. Horton that include the graded quantum Weyl algebra, the quantum symplectic space, the quantum Euclidean space, and quantum Heisenberg algebra etc., obtaining new results and as well as simplified proofs of previously known results. The Krull and global dimensions of certain further localizations of $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ are computed.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

21 extracted references · 21 canonical work pages

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It plays a key role in the theory of /u1D437-modules ([9])

I/n.pc/t.pc/r.pc/o.pc/d.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc The Gelfand–Kirillov dimension of a nonzero ﬁnitely genera ted /u1D434/u1D45B(C)-module is bounded below by/u1D45B= 1 2 (GK-dim(/u1D434/u1D45B(C)) where/u1D434/u1D45B(C) is the/u1D45B-th Weyl algebra over the ﬁeld C - a fact commonly known as the Bernstein’s inequality (B.I) [ 15]. It plays a key ro...

work page 2020
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We recall the construction here referring to [ 7] for further details

I/t.pc/e.pc/r.pc/a.pc/t.pc/e.pc/d.pc A/m.pc/b.pc/i.pc/e.pc/s.pc/k.pc/e.pc/w.pc /p.pc/o.pc/l.pc/y.pc/n.pc/o.pc/m.pc/i.pc/a.pc/l.pc /s.pc/t.pc/r.pc/u.pc/c.pc/t.pc/u.pc/r.pc/e.pc /o.pc/f.pc/u1D43E/u1D45B The simplicity of our approach relies on the fact that our alg ebras ﬁt into the framework of an iterated ambiskew polynomial ring as deﬁned in [ 7]. We rec...

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[3]

For the algebra /u1D43E/u1D45Banalogous localizations can be deﬁned in a similar manner

T/h.pc/e.pc L/o.pc/c.pc/a.pc/l.pc/i.pc/z.pc/a.pc/t.pc/i.pc/o.pc/n.pc/s.pcℬ/u1D45B/a.pc/n.pc/d.pc/u1D49E/u1D45B Certain important localizations of the /u1D45B-th quantum multiparameter Weyl algebra /u1D434/u1D45E,Λ /u1D45Bwhich is an iterated ambiskew polynomial ring were discusse d in [ 7]. For the algebra /u1D43E/u1D45Banalogous localizations can be deﬁn...

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a further localization

M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc /o.pc/v.pc/e.pc/r.pc /o.pc/v.pc/e.pc/r.pc /a.pc Q/u.pc/a.pc/n.pc/t.pc/u.pc/m.pc T/o.pc/r.pc/u.pc/s.pc As seen in Section 3 “a further localization”, namely, /u1D49E/u1D45Bof /u1D49C/u1D45B, is a quantum torus. Recall that a quantum torus K(/u1D706/u1D456 /u1D457 ) [/u1D44B±1 1 , · · ·,/u1D44B±1 /u1D45B] of rank /u1D45Bis de...

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The dimension of the quantum torus /u1D49E/u1D45Bthat arises as a further localization of/u1D43E/u1D45Bwill be used in our proof of the Bernstein-type inequality for/u1D43E/u1D45B

T/h.pc/e.pc /d.pc/i.pc/m.pc/e.pc/n.pc/s.pc/i.pc/o.pc/n.pc /o.pc/f.pc/u1D49E/u1D45B It was shown in [12] that the Krull and the global dimensions for a quantum torus A coincide and so we will write dim (A) to denote either of these. The dimension of the quantum torus /u1D49E/u1D45Bthat arises as a further localization of/u1D43E/u1D45Bwill be used in our pr...

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

It plays a key role in the theory of /u1D437-modules ([9])

I/n.pc/t.pc/r.pc/o.pc/d.pc/u.pc/c.pc/t.pc/i.pc/o.pc/n.pc The Gelfand–Kirillov dimension of a nonzero ﬁnitely genera ted /u1D434/u1D45B(C)-module is bounded below by/u1D45B= 1 2 (GK-dim(/u1D434/u1D45B(C)) where/u1D434/u1D45B(C) is the/u1D45B-th Weyl algebra over the ﬁeld C - a fact commonly known as the Bernstein’s inequality (B.I) [ 15]. It plays a key ro...

work page 2020

[2] [2]

We recall the construction here referring to [ 7] for further details

I/t.pc/e.pc/r.pc/a.pc/t.pc/e.pc/d.pc A/m.pc/b.pc/i.pc/e.pc/s.pc/k.pc/e.pc/w.pc /p.pc/o.pc/l.pc/y.pc/n.pc/o.pc/m.pc/i.pc/a.pc/l.pc /s.pc/t.pc/r.pc/u.pc/c.pc/t.pc/u.pc/r.pc/e.pc /o.pc/f.pc/u1D43E/u1D45B The simplicity of our approach relies on the fact that our alg ebras ﬁt into the framework of an iterated ambiskew polynomial ring as deﬁned in [ 7]. We rec...

work page

[3] [3]

For the algebra /u1D43E/u1D45Banalogous localizations can be deﬁned in a similar manner

T/h.pc/e.pc L/o.pc/c.pc/a.pc/l.pc/i.pc/z.pc/a.pc/t.pc/i.pc/o.pc/n.pc/s.pcℬ/u1D45B/a.pc/n.pc/d.pc/u1D49E/u1D45B Certain important localizations of the /u1D45B-th quantum multiparameter Weyl algebra /u1D434/u1D45E,Λ /u1D45Bwhich is an iterated ambiskew polynomial ring were discusse d in [ 7]. For the algebra /u1D43E/u1D45Banalogous localizations can be deﬁn...

work page

[4] [4]

a further localization

M/o.pc/d.pc/u.pc/l.pc/e.pc/s.pc /o.pc/v.pc/e.pc/r.pc /o.pc/v.pc/e.pc/r.pc /a.pc Q/u.pc/a.pc/n.pc/t.pc/u.pc/m.pc T/o.pc/r.pc/u.pc/s.pc As seen in Section 3 “a further localization”, namely, /u1D49E/u1D45Bof /u1D49C/u1D45B, is a quantum torus. Recall that a quantum torus K(/u1D706/u1D456 /u1D457 ) [/u1D44B±1 1 , · · ·,/u1D44B±1 /u1D45B] of rank /u1D45Bis de...

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[5] [5]

The dimension of the quantum torus /u1D49E/u1D45Bthat arises as a further localization of/u1D43E/u1D45Bwill be used in our proof of the Bernstein-type inequality for/u1D43E/u1D45B

T/h.pc/e.pc /d.pc/i.pc/m.pc/e.pc/n.pc/s.pc/i.pc/o.pc/n.pc /o.pc/f.pc/u1D49E/u1D45B It was shown in [12] that the Krull and the global dimensions for a quantum torus A coincide and so we will write dim (A) to denote either of these. The dimension of the quantum torus /u1D49E/u1D45Bthat arises as a further localization of/u1D43E/u1D45Bwill be used in our pr...

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[6] [6]

Let /u1D45D/u1D456/u1D45E−1 /u1D456not be a root of unity for each 1 ≤ /u1D456≤ /u1D45B

P/r.pc/o.pc/o.pc/f.pc/s.pc /o.pc/f.pc /t.pc/h.pc/e.pc /m.pc/a.pc/i.pc/n.pc /r.pc/e.pc/s.pc/u.pc/l.pc/t.pc/s.pc Lemma 6.1. Let /u1D45D/u1D456/u1D45E−1 /u1D456not be a root of unity for each 1 ≤ /u1D456≤ /u1D45B. Then any nonzero simple ℬ/u1D45B-module/u1D440is either /u1D4B3/u1D456-torsion-free or /u1D4B4/u1D456-torsion-free for all 1 ≤/u1D456≤/u1D45B. Pro...

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Brookes, Crossed products and ﬁnitely presented groups, J

C.J.B. Brookes, Crossed products and ﬁnitely presented groups, J. Group Theory 3(4), (2000), 433-444

work page 2000

[8] [8]

E. E. Demidov, Modules over a Weyl quantum algebra., Moscow Univ. Math. Bull. 48, (1993), 49-51

work page 1993