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Axioms
Special Issues
Recent Advances in Representation Theory with Applications
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Recent Advances in Representation Theory with Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 30 June 2025 | Viewed by 5014

Special Issue Editor

Prof. Dr. Ivan Matić

E-Mail Website
Guest Editor
School of Applied Mathematics and Informatics, University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia
Interests: representations of classical p-adic groups; unitary dual; theta-correspondence; covering groups

Special Issue Information

Dear Colleagues,

The representation theory presents a well-studied field that has, for many years, influenced several branches in mathematics, ranging from mathematical physics to the number theory. In recent years, significant progress has been achieved in understanding both complex and Banach space representations of classical groups and of Lie algebras. This knowledge has been expanded through the development and application of several methods, such as Hecke algebra considerations, L-functions, endoscopic methods, and theta correspondence, among others. In general, several theoretical representation methods intertwine to provide a more comprehensive approach to the studied problems. Additionally, the results obtained occasionally provide a different insight into themes related to different important subjects in pure mathematics.

The main aim of this Special Issue is to provide an opportunity to present recent developments in the representation theory and its applications, and to show how the developed methods can be used and further upgraded in different situations. It covers all aspects of the representation theory, such as the structure of complex, l-adic and Banach representations, as well as those of related research areas, such as automorphic and modular forms, Lie groups, Lie algebras, and harmonic analysis on groups of the Lie type.

We invite high-quality original research papers as well as comprehensive reviews related to the proposed topic.

Prof. Dr. Ivan Matić
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • representations of reductive groups
  • automorphic forms
  • modular forms
  • L-functions
  • Lie algebras and their representations
  • representations of Clifford algebras
  • representations of finite groups
  • harmonic analysis on groups of the Lie type
  • unitary representations
  • applications of representation theory

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

15 pages, 299 KiB  
Article
RHS and Quantum Mechanics: Some Extra Examples
by Maria Blazquez, Manuel Gadella and Gerardo Jimenez-Trejo
Axioms 2024, 13(12), 868; https://doi.org/10.3390/axioms13120868 - 12 Dec 2024
Viewed by 734
Abstract
Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well as an abstract basis and the [...] Read more.
Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well as an abstract basis and the basis of special functions and representations of Lie algebras of symmetries are used by continuous operators. This is not possible in Hilbert spaces. In the present paper, we study a model showing all these features, based on the one-dimensional Pöschl–Teller Hamiltonian. Also, RHS supports representations of all kinds of ladder operators as continuous mappings. We give an interesting example based on one-dimensional Hamiltonians with an infinite chain of SUSY partners, in which the factorization of Hamiltonians by continuous operators on RHS plays a crucial role. Full article
(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)
21 pages, 359 KiB  
Article
Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity
by Viktor Abramov
Axioms 2024, 13(10), 687; https://doi.org/10.3390/axioms13100687 - 3 Oct 2024
Viewed by 753
Abstract
We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear [...] Read more.
We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed. Full article
(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)
20 pages, 356 KiB  
Article
Certain L2-Norms on Automorphic Representations of SL(2, R)
by Hongyu He
Axioms 2024, 13(2), 80; https://doi.org/10.3390/axioms13020080 - 25 Jan 2024
Viewed by 1188
Abstract
Let Γ be a non-uniform lattice in SL(2,R). In this paper, we study various L2-norms of automorphic representations of SL(2,R). We bound these norms with intrinsic norms [...] Read more.
Let Γ be a non-uniform lattice in SL(2,R). In this paper, we study various L2-norms of automorphic representations of SL(2,R). We bound these norms with intrinsic norms defined on the representation. Comparison of these norms can help us understand the growth of L-functions in a systematic way. Full article
(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)
13 pages, 358 KiB  
Article
Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group ▵*(3,q,r) for q < r Primes
by Sajida Younas, Sajida Kousar, Majed Albaity and Tahir Mahmood
Axioms 2023, 12(12), 1128; https://doi.org/10.3390/axioms12121128 - 15 Dec 2023
Viewed by 1306
Abstract
Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of [...] Read more.
Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics. Full article
(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)
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