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Quiver Representations

Author : Ralf Schiffler
Publisher : Springer
Page : 230 pages
File Size : 21,1 MB
Release : 2014-09-04
Category : Mathematics
ISBN : 3319092049


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.

Persistence Theory: From Quiver Representations to Data Analysis

Author : Steve Y. Oudot
Publisher : American Mathematical Soc.
Page : 218 pages
File Size : 46,6 MB
Release : 2017-05-17
Category :
ISBN : 1470434431


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.

An Introduction to Quiver Representations

Author : Harm Derksen
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 32,62 MB
Release : 2017-11-29
Category : Directed graphs
ISBN : 1470425564


This book is an introduction to the representation theory of quivers and finite dimensional algebras. It gives a thorough and modern treatment of the algebraic approach based on Auslander-Reiten theory as well as the approach based on geometric invariant theory. The material in the opening chapters is developed starting slowly with topics such as homological algebra, Morita equivalence, and Gabriel's theorem. Next, the book presents Auslander-Reiten theory, including almost split sequences and the Auslander-Reiten transform, and gives a proof of Kac's generalization of Gabriel's theorem. Once this basic material is established, the book goes on with developing the geometric invariant theory of quiver representations. The book features the exposition of the saturation theorem for semi-invariants of quiver representations and its application to Littlewood-Richardson coefficients. In the final chapters, the book exposes tilting modules, exceptional sequences and a connection to cluster categories. The book is suitable for a graduate course in quiver representations and has numerous exercises and examples throughout the text. The book will also be of use to experts in such areas as representation theory, invariant theory and algebraic geometry, who want to learn about applications of quiver representations to their fields.

Quiver Grassmannians of Extended Dynkin Type D Part I: Schubert Systems and Decompositions into Affine Spaces

Author : Oliver Lorscheid
Publisher : American Mathematical Soc.
Page : 78 pages
File Size : 35,39 MB
Release : 2019-12-02
Category : Education
ISBN : 1470436477


Let Q be a quiver of extended Dynkin type D˜n. In this first of two papers, the authors show that the quiver Grassmannian Gre–(M) has a decomposition into affine spaces for every dimension vector e– and every indecomposable representation M of defect −1 and defect 0, with the exception of the non-Schurian representations in homogeneous tubes. The authors characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for M. The method of proof is to exhibit explicit equations for the Schubert cells of Gre–(M) and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution the authors develop the theory of Schubert systems. In Part 2 of this pair of papers, they extend the result of this paper to all indecomposable representations M of Q and determine explicit formulae for the F-polynomial of M.

Arrows from my Quiver

Author : James Caughey
Publisher : BoD – Books on Demand
Page : 482 pages
File Size : 27,57 MB
Release : 2021-11-04
Category : Fiction
ISBN : 3752530464


Reprint of the original, first published in 1868.

In a Warrior's Quiver

Author : D.L. Crager
Publisher : Christian Faith Publishing, Inc.
Page : 320 pages
File Size : 49,14 MB
Release : 2021-12-16
Category : Fiction
ISBN : 1639034757


Before Journey's End, there was a beginning. The wealthiest and most powerful reigning king--King Solomon--may have died three thousand years ago, but because of his heavenly blessed wisdom, the scientific and religious world as we know it today is about to erupt as guarded truths of the beginning of times are now soon to be exposed. At the beginning, King Solomon wisely gave his general--General Shuriah--a mission of epic proportions. The older general was Solomon's most trusted guardian since Solomon's birth to Bathsheba and King David. General Shuriah and thousands of his Israelite soldiers are now sailing across the Mediterranean Sea to a land of their enemies to exhume and safely move the greatest treasures of treasures he himself stumbled upon a year earlier. During the starting days of the journey, the old general is consumed with reflecting back to his youth. Memories come alive of his struggles in relationships, intense training, and many combat situations as he fought hard to someday become a mighty warrior for King David. Young Shuriah is very unpredictable, yet he is extremely admired and feared at the same time because he possesses extraordinary, almost supernatural, abilities to kill and the mental intuition to strategize for war. At the age of fifteen, Shuriah has become the youngest ever to achieve the coveted honor of being a mighty warrior, having killed more than three hundred enemy soldiers in one battle. But this happened only after his stepfather Uriah the Hittite, is murdered in battle. Through certain events, it is revealed that King David secretly ordered the murder of Uriah, after discovering Bathsheba, Shuriah's stepmother and Uriah's wife, is pregnant with his child. Now, the new mighty warrior must face the toughest and most painful battle in his life, which could bring his king to his knees or even his grave.