Advances in Algebraic Structures: Representation Theory and Lie Groups

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 24 November 2025 | Viewed by 1202

Special Issue Editor


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Guest Editor
Department of Mathematics, Georgia College & State University, Milledgeville, GA, USA
Interests: lie algebras; leibniz algebras and their representations; leibniz algebra cohomology

Special Issue Information

Dear Colleagues,

Understanding the structure of Lie groups using representation theory has become an important topic of research in modern mathematics with a wide variety of applications in many fields, ranging from differential geometry, algebraic geometry, or Riemannian geometry to mechanics, robotics, or gauge theory in physics. Consequently, authors publishing papers in these areas may be better served by a platform that tracks recent development and applications of Lie groups and their representations.

This Special Issue aims to collect reviews and original research papers on various topics concerning the recent progress in the theory of Lie groups, Lie algebras, and their representation. This includes research on other disciplines that involves information of and relationship with Lie groups.

Dr. Guy Biyogmam
Guest Editor

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Keywords

  • algebraic structures
  • algebraic groups
  • lie groups
  • lie algebras
  • representation theory
  • root systems
  • coxeter groups
  • characteristic classes
  • flag varieties
  • quantum groups
  • holonomy
  • variational methods

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Published Papers (2 papers)

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Research

10 pages, 243 KiB  
Article
Relative Vertex-Source-Pairs of Modules of and Idempotent Morita Equivalences of Rings
by Morton E. Harris
Mathematics 2025, 13(15), 2327; https://doi.org/10.3390/math13152327 - 22 Jul 2025
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Abstract
Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module [...] Read more.
Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module is a finite direct sum of indecomposable R-modules. Finite Group Modular Representation Theory concerns the study of left finitely generated OG-modules where G is a finite group and O is a complete discrete valuation ring with O/J(O) a field of prime characteristic p. Thus OG is a noetherian O-algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple OG-module) a theory of vertices and sources invariants. The vertices are derived from the set of p-subgroups of G. As suggested by the above, in Basic Definition and Main Results for Rings Section, let Σ be a fixed subset of subrings of the ring R and we develop a theory of Σ-vertices and sources for finitely generated R-modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R. For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that R=ReR. Set B=eRe so that B is a subring of R with identity e. Then, the functions eRR:RmodBmod and ReB:BmodRmod form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras. Full article
8 pages, 218 KiB  
Article
A Zassenhaus Lemma for Digroups
by Guy Roger Biyogmam
Mathematics 2024, 12(19), 2972; https://doi.org/10.3390/math12192972 - 25 Sep 2024
Viewed by 686
Abstract
In this paper, we construct a quotient structure on digroups. This construction yields a new functor from the category of digroups to the category of groups. We obtain a modular property for digroups and use it to prove an analogue of the Zassenhaus [...] Read more.
In this paper, we construct a quotient structure on digroups. This construction yields a new functor from the category of digroups to the category of groups. We obtain a modular property for digroups and use it to prove an analogue of the Zassenhaus lemma in this framework. Full article
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