Quiver Representations And Gabriel S Theorem
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Persistence Theory: From Quiver Representations to Data Analysis
Author: Steve Y. Oudot
language: en
Publisher: American Mathematical Soc.
Release Date: 2017-05-17
Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work. The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.
Quiver Representations
This book is intended to serve as a textbook for a course in Representation Theory of Algebras at the beginning graduate level. The text has two parts. In Part I, the theory is studied in an elementary way using quivers and their representations. This is a very hands-on approach and requires only basic knowledge of linear algebra. The main tool for describing the representation theory of a finite-dimensional algebra is its Auslander-Reiten quiver, and the text introduces these quivers as early as possible. Part II then uses the language of algebras and modules to build on the material developed before. The equivalence of the two approaches is proved in the text. The last chapter gives a proof of Gabriel’s Theorem. The language of category theory is developed along the way as needed.
Computational Methods for Representations of Groups and Algebras
Author: P. Dräxler
language: en
Publisher: Springer Science & Business Media
Release Date: 1999
I Introductory Articles.- 1 Classification Problems in the Representation Theory of Finite-Dimensional Algebras.- 2 Noncommutative Gröbner Bases, and Projective Resolutions.- 3 Construction of Finite Matrix Groups.- II Keynote Articles.- 4 Derived Tubularity: a Computational Approach.- 5 Problems in the Calculation of Group Cohomology.- 6 On a Tensor Category for the Exceptional Lie Groups.- 7 Non-Commutative Gröbner Bases and Anick's Resolution.- 8 A new Existence Proof of Janko's Simple Group J4.- 9 The Normalization: a new Algorithm, Implementation and Comparisons.- 10 A Computer Algebra Approach to sheaves over Weighted Projective Lines.- 11 Open Problems in the Theory of Kazhdan-Lusztig polynomials.- 12 Relative Trace Ideals and Cohen Macaulay Quotients.- 13 On Sims' Presentation for Lyons' Simple Group.- 14 A Presentation for the Lyons Simple Group.- 15 Reduction of Weakly Definite Unit Forms.- 16 Decision Problems in Finitely Presented Groups.- 17 Some Algorithms in Invariant Theory of Finite Groups.- 18 Coxeter Transformations associated with Finite Dimensional Algebras.- 19 The 2-Modular Decomposition Numbers of Co2.- 20 Bimodule and Matrix Problems.