arXiv subjects

Hichem Gargoubi

Publications and source records attributed to Hichem Gargoubi.

6 recordsLinked to original sources

arXiv2026

Bicomplex Algebraic Numbers

We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct classes of extensions depending on the nature of the generating element. We further establish a key decomposition property for bicomplex extensions, which serves as a foundation for studying their rings of integers. We also observe that prime elements in the ring of integers of a number field may become semiprime in the rings of integers of suitable bicomplex extensions. Finally, we present two explicit examples of finite bicomplex extensions.

math.NT↗

arXiv2022

Bicomplex numbers as a normal complexified f-algebra

The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show that D-norms generate the same topology in B. We develop the D-trigonometric form of a bicomplex number which leads us to a geometric interpretation of the nth roots of a bicomplex number in terms of polyhedral tori. We use the concepts developed, in particular that of Riesz subnorm of a D-norm, to study the uniform convergence of the bicomplex zeta and gamma functions. The main result of this paper is the generalization to the bicomplex case of the Riemann functional equation and Euler's reflection formula.

math.FA↗

arXiv2008

Supertransvectants and symplectic geometry

We consider the $osp(1|2)$-invariant bilinear operations on weighted densities on the supercircle $S^{1|1}$ called the supertransvectants. These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that these operations coincide with the iterated Poisson and ghost Poisson brackets on ${\mathbb R}^{2|1}$ and apply this result to construct star-products involving the supertransvectants.

math-ph↗

arXiv2006

Differential operators on supercircle: conformally equivariant quantization and symbol calculus

We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist.

math-ph↗

arXiv2005

Symmetries of modules of differential operators

Let ${\cal F}\_λ(S^1)$ be the space of tensor densities of degree (or weight) $λ$ on the circle $S^1$. The space ${\cal D}^k\_{λ,μ}(S^1)$ of $k$-th order linear differential operators from ${\cal F}\_λ(S^1)$ to ${\cal F}\_μ(S^1)$ is a natural module over $\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the algebra of symmetries of the modules ${\cal D}^k\_{λ,μ}(S^1)$, i.e., the linear maps on ${\cal D}^k\_{λ,μ}(S^1)$ commuting with the $\mathrm{Diff}(S^1)$-action. We also solve the same problem in the case of straight line $\mathbb{R}$ (instead of $S^1$) and compare the results in the compact and non-compact cases.

math-ph↗

arXiv2003

Algebra gl(λ) inside the algebra of differential operators on the real line

The Lie algebra gl(λ) with λ\in \mathbb C, introduced by B.L.Feigin, can be embedded into the Lie algebra of differential operators on the real line. We give an explicit formula of the embedding of gl(λ) into the algebra D_λ of differential operators on the space of tensor densities of degree λon \mathbb R. Our main tool is the notion of projectively equivariant symbol of a differential operator.

math.QA↗