Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bernstein-type inequalities for quantum algebras

Sanu Bera, Ashish Gupta, Sugata Mandal, Snehashis Mukherjee

TL;DR

The paper proves Bernstein-type inequalities for the multiparameter quantum algebras $K_n = K_{n,\Gamma}^{P,Q}(\mathbb{K})$, which encompass graded quantum Weyl algebras and several quantum spaces. It models $K_n$ as iterated ambiskew polynomial rings with normal elements $z_i$, develops explicit skew-commutator relations, and constructs localizations $\mathscr{B}_n$ and $\mathscr{C}_n$ leading to a quantum torus $\mathscr{C}_n$ whose dimension controls GK-dim bounds. By analyzing the dimension of $\mathscr{C}_n$ in cases where parameters equal 1 and are not roots of unity, the authors derive BI-1 and BI-2, establishing lower bounds on the GK-dimension of non-torsion modules and obtaining corollaries for quantum Weyl, symplectic, Euclidean, and Heisenberg quantum algebras. A simpler proof is provided for Bernstein inequalities in the quantum Weyl setting, illustrating the broad applicability of the ambiskew localization framework to noncommutative dimension theory.

Abstract

We establish Bernstein-type inequalities for the quantum algebras $K_{n,Γ}^{P,Q}(\mathbb{K})$ 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 $K_{n,Γ}^{P,Q}(\mathbb{K})$ are computed.

Bernstein-type inequalities for quantum algebras

TL;DR

The paper proves Bernstein-type inequalities for the multiparameter quantum algebras , which encompass graded quantum Weyl algebras and several quantum spaces. It models as iterated ambiskew polynomial rings with normal elements , develops explicit skew-commutator relations, and constructs localizations and leading to a quantum torus whose dimension controls GK-dim bounds. By analyzing the dimension of in cases where parameters equal 1 and are not roots of unity, the authors derive BI-1 and BI-2, establishing lower bounds on the GK-dimension of non-torsion modules and obtaining corollaries for quantum Weyl, symplectic, Euclidean, and Heisenberg quantum algebras. A simpler proof is provided for Bernstein inequalities in the quantum Weyl setting, illustrating the broad applicability of the ambiskew localization framework to noncommutative dimension theory.

Abstract

We establish Bernstein-type inequalities for the quantum algebras 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 are computed.
Paper Structure (12 sections, 18 theorems, 82 equations)

This paper contains 12 sections, 18 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Iterated Ambieskew polynomial structure of $K_n$
  3. The construction of ambiskew polynomial rings
  4. $K_n$ as an iterated ambiskew polynomial ring
  5. Skew commutator formulae for $K_n$
  6. The Localizations $\mathscr{B}_n$ and $\mathscr{C}_n$
  7. Modules over over a Quantum Torus
  8. The dimension of $\mathscr{C}_n$
  9. The dimension of $\mathscr{C}_n$ when $p_i=1$
  10. The dimension of $\mathscr{C}_n$ when $q_i=1$
  11. Proofs of the main results
  12. A simpler proof of the Bernstein inequality for the Quantum Weyl algebras $A_n^{\overline{q},\Lambda}$

Key Result

Corollary 1.1

Suppose that no $q_i$ is a root of unity $1\leq i \leq n$. If $M$ is a nonzero finitely generated module over $A_n^{Q,\Gamma}(\mathbb{K})$ that is not a $\mathscr{Z}$-torsion module, then

Theorems & Definitions (27)

  • Corollary 1.1: B.I. for the graded quantum Weyl algebra
  • Corollary 1.2: B.I for the quantum symplectic space
  • Corollary 1.3: B.I for the quantum Euclidean space
  • Corollary 1.4: B.I for the quantum Heisenberg space
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Theorem 5.1
  • ...and 17 more