arXiv subjects

Mohammad H. M Rashid

Publications and source records attributed to Mohammad H. M Rashid.

5 recordsLinked to original sources

arXiv2025

Structure and Invariants of Weyl-Type Algebras with Hyperbolic Sine Generators

This paper introduces and systematically studies a class of Weyl-type algebras enriched with hyperbolic sine and power generators over a field of characteristic zero, defined as $A_{p,t,\cA} = \Weyl{\sinh(\pm x^{p} \sinh(t)),\; \sinh(\cA x),\; x^{\cA}}$ in the associative setting and $\Nass{\sinh(\pm x^{p} \sinh(t)),\; \sinh(\cA x),\; x^{\cA}}$ in a non-associative framework. We establish fundamental structural properties, including the triviality of the center for the non-associative version and the explicit description $Z(A_{p,t,\cA}) = \FF[\sinh(\pm x^{p} \sinh(t))]$ for the associative one, proving that $A_{p,t,\cA}$ is an Azumaya algebra over its center and represents a nontrivial class in the Brauer group $\Br(\FF(y))$. Furthermore, we compute the Gelfand--Kirillov dimension for relevant examples and demonstrate its key properties, such as additivity under tensor products and the growth dichotomy. We completely characterize the automorphism group of $A_{p,t,\cA}$ as a semidirect product of a torus with a discrete group, and provide a sharp isomorphism criterion showing that the parameter $t$ is a complete invariant in the family. The paper concludes with two open problems concerning the GK dimension of non-associative hyperbolic sine algebras and the classification of their deformations, pointing toward future research directions in non-associative growth theory and deformation rigidity.

math.RA↗

arXiv2025

Structural and Classification Theorems for Weyl-Type Algebras over Expolynomial Rings

This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{αx}$, exponentials of exponentials $e^{\pm x^p e^{t}}$, and power functions $x^α$ for $α$ in an additive subgroup $\cA$ of a characteristic zero field $\FF$. We establish several fundamental structural results: scalar extensions preserve both the algebraic structure and simplicity; intermediate subalgebras associated with subgroups $\ZZ \subseteq \cB \subseteq \cA$ remain simple; the algebra of graded derivations is isomorphic to a semidirect product $\Weyl{e^{\pm x^p e^{t}},\; e^{\cA x},\; x^{\cA}} \rtimes \FF^n$; tensor products over disjoint variable sets decompose naturally into larger algebras; and a complete isomorphism criterion is given, showing that isomorphism depends precisely on the orbit of the parameter $p$ under the automorphism group of $\cA$ and the equality of the deformation parameter $t$. These theorems generalize classical results on Weyl and Witt algebras, provide new families of simple algebras, and offer a foundation for further research in deformation theory, representation theory, and cohomology.

math.RA↗

arXiv2025

Complete Structural Analysis of $q$-Heisenberg Algebras: Homology, Rigidity, Automorphisms, and Deformations

This paper establishes several fundamental structural properties of the $q$-Heisenberg algebra $\mathfrak{h}_n(q)$, a quantum deformation of the classical Heisenberg algebra. We first prove that when $q$ is not a root of unity, the global homological dimension of $\mathfrak{h}_n(q)$ is exactly $3n$, while it becomes infinite when $q$ is a root of unity. We then demonstrate the rigidity of its iterated Ore extension structure, showing that any such presentation is essentially unique up to permutation and scaling of variables. The graded automorphism group is completely determined to be isomorphic to $(\mathbb{C}^*)^{2n} \rtimes S_n$. Furthermore, $\mathfrak{h}_n(q)$ is shown to possess a universal deformation property as the canonical PBW-preserving deformation of the classical Heisenberg algebra $\mathfrak{h}_n(1)$. We compute its Hilbert series as $(1-t)^{-3n}$, confirming polynomial growth of degree $3n$, and establish that its Gelfand--Kirillov dimension coincides with its classical Krull dimension. These results are extended to a generalized multi-parameter version $\mathfrak{H}_n(\mathbf{Q})$, and illustrated through detailed examples and applications in representation theory and deformation quantization.

math.RA↗

arXiv2025

Weyl-Type and Witt-Type Algebras with Exponential Generators:Structure, Automorphisms, and Representation Theory

This paper introduces and systematically studies a new class of non-commutative algebras -- Weyl-type and Witt-type algebras -- generated by differential operators with exponential and generalized power function coefficients. We define the expolynomial ring $R_{p,t,\mathcal{A}} = \mathbb{F}\bigl[ e^{\pm x^{p} e^{t}},\; e^{\mathcal{A} x},\; x^{\mathcal{A}} \bigr]$ associated to an additive subgroup $\mathcal{A} \subset \mathbb{F}$, and investigate its Ore extension $A_{p,t,\mathcal{A}} = R_{p,t,\mathcal{A}}[\partial; δ]$ (Weyl-type) and its derivation algebra $\mathfrak{g}_{p,t,\mathcal{A}} = \operatorname{Der}_{\mathbb{F}}(R_{p,t,\mathcal{A}})$ (Witt-type). Our main results establish: (1) the automorphism group of $R_{p,t,\mathcal{A}}$ is isomorphic to $(\mathbb{F}^{\times})^{2r+1} \rtimes \operatorname{GL}(2r+1,\mathbb{Z})$; (2) a Galois descent theorem showing that fixed-point subalgebras under finite Galois actions recover the original Weyl-type algebra; (3) the non-existence of finite-dimensional simple modules for $A_{p,t,\mathcal{A}}$; (4) the Zariski density of isomorphism classes in moduli spaces as transcendental parameters vary; (5) the stability of simplicity under generic quantum deformation; and (6) a complete representation-theoretic framework including the classification of irreducible weight modules, the construction of Harish--Chandra modules with BGG-type resolutions, and the structure of category $\mathcal{O}$. These results unify and extend classical theories of Weyl algebras, Witt algebras, and generalized Weyl algebras, while opening new directions in deformation theory, non-commutative geometry, and the representation theory of infinite-dimensional algebras.

math.RA↗

arXiv2025

Second-Order $Λ$-Sets and Extensions to Non-Smooth, Hybrid, and Stochastic Optimal Control

This paper develops a comprehensive extension of the $Λ$-set framework for optimal control, introducing second-order $Λ$-sets and generalizing the theory to non-smooth, hybrid, and stochastic hybrid systems. We first establish second-order necessary conditions that incorporate curvature information of the reachable set, providing refined optimality criteria that bridge classical second-variation methods with the geometric $Λ$-set approach. The framework is then extended to Filippov systems with discontinuous dynamics and to hybrid dynamical systems with state-dependent switching, yielding new necessary conditions for optimality in these settings. Furthermore, we introduce stochastic $Λ$-sets for systems subject to both continuous diffusion and discrete random switching, connecting the framework to Peng's stochastic maximum principle. Throughout the paper, detailed examples -- including nonholonomic systems, mechanical systems with friction, and stochastic temperature control -- illustrate the theoretical developments and demonstrate the practical applicability of the extended $Λ$-set theory. The results unify and generalize existing maximum principles, offering a powerful geometric tool for analyzing optimal control problems across a broad spectrum of system classes, from classical smooth systems to modern stochastic hybrid systems.

math.OC↗