Study Of Representation Theory Research Articles

Investigation of Representation Theory for Algebraic and Geometric Applications

Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking algebraic structures with linear transformations.The representation theory has significant effects on algebra. Through the examination of the representations that go along with a group, it allows us to examine its composition and behaviour. One can learn more about the internal structures and underlying symmetries of groups by breaking representations down into irreducible parts. This has implications for number theory, combinatorics, and quantum physics, among other fields.Understanding symmetries and transformations of geometric objects in geometry depends critically on representation theory. The investigation of shape and space symmetry is made possible by representation theory, which links geometric objects with linear transformations. This has uses in a variety of disciplines, including physics' study of symmetry groups, differential geometry, and crystallography.A crucial area of research, representation theory has extensive uses in both algebra and geometry. It is indispensable in many branches of mathematics and opens up new directions for research and discovery because it can reveal the underlying structures and symmetries of abstract algebraic objects.

ALMOST SPLIT SEQUENCES FOR COMPLEXES VIA RELATIVE HOMOLOGY

Abstract. We use methods of relative homological algebra on the cate-gory C(modΛ), of complexes of finitely generated modules over an artinalgebra Λ, to give some characterizations of almost split sequences. 1. IntroductionThroughout this article, we let Λ be an R-algebra which is a finitely gener-ated R-module, where R is a commutative artin ring. We denote by modΛ thecategory of finitely generated left Λ-modules, and J(Λ) the Jacboson radical ofΛ.The concept of an almost split sequence (or Auslander-Reiten sequence) inmodΛ was introduced by Auslander and Reiten [3], and it plays an impor-tant role in the study of representation theory. The theory of almost split se-quences developed further, for example, some results for subcategories of modΛ[5, 14, 18], the corresponding almost split exact triangles were introduced byHappel [11, 12], and studied extensively [7, 13, 15, 19]. The existence of almostsplit sequences was studied in more general abelian categories [16, 17], andin monomorphism categories [21]. In [19], by letting the Nakayama functoract degree-wise, Salarian and Vahed defined a translation τ in the categoryC(modΛ) of complexes of finitely generated left Λ-modules. They investigatedthe existence of almost split sequences in C

Quantized Universal Enveloping Algebras andq-de Rham Cocycles

In this work a close connection is established between certain cohomology spaces of quantized universal enveloping algebras and a twisted q-de Ž . Rham Jackson]Aomoto cohomology of configuration spaces. In the Lie algebra case, the idea of such a connection belongs to V. w x w x Ginzburg and V. Schechtman. In 4 and 5 , these authors have constructed canonical morphisms between the de Rham homology of certain local systems over configuration spaces and Ext-spaces between Fock-type modules over Kac]Moody and Virasoro Lie algebras. This construction is, in turn, a generalization of the classical Feigin]Fuchs construction. In their study of representation theory of Virasoro algebras, Feigin and Fuchs have discovered a way of obtaining intertwiners between Fock modules over the Virasoro algebra from the top homology of certain one-dimensional local systems. We investigate this connection between the geometry of configuration spaces and the representation theory in the case of quantum groups. The representations considered here are Verma modules over the quantized enveloping algebras of semisimple Lie algebras. The existence and the w x uniqueness of these modules were established by Lusztig 9 . We consider a family of operators between the Verma modules that satisfy certain difference equations and certain cocycle conditions. These equations are built using a family of q-difference operators that generate a flat connec-

Harish-Chandra's Method of Descent

1. One of the main tools in the study of representation theory of reductive groups is Harish-Chandra's method of descent. Let G be a real reductive group. Given a semisimple element s of G, let H be the centralizer of s in G. Then the behavior of an invariant eigendistribution in a neighborhood of s may be described in terms of an invariant distribution of H in a neighborhood of the identity. This leads to the notion of descent, taking an invariant eigendistribution of G, defined on a neighborhood of s, to an invariant distribution of H defined in a neighborhood of the identity in H. The descent of any standard character of

Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].