Representations of Algebras, Geometry and Physics.

Format:

Book

Author/Creator:

Igusa, Kiyoshi.

Contributor:

Martsinkovsky, A. (Alex)

Todorov, G. (Gordana)

Series:

Contemporary Mathematics

Contemporary Mathematics ; v.769

Language:

English

Subjects (All):

Representations of algebras--Congresses.

Representations of algebras.

Geometry, Algebraic--Congresses.

Geometry, Algebraic.

Commutative algebra--Congresses.

Commutative algebra.

Physical Description:

1 online resource (252 pages)

Edition:

1st ed.

Other Title:

Representations of algebras, geometry and physics

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

This volume contains selected expository lectures delivered at the 2018 Maurice Auslander Distinguished Lectures and International Conference, held April 25-30, 2018, at the Woods Hole Oceanographic Institute, Woods Hole, MA.Reflecting recent developments in modern representation theory of algebras, the selected topics include an introduction to a new class of quiver algebras on surfaces, called "geodesic ghor algebras", a detailed presentation of Feynman categories from a representation-theoretic viewpoint, connections between representations of quivers and the structure theory of Coxeter groups, powerful new applications of approximable triangulated categories, new results on the heart of a t-structure, and an introduction to methods of constructive category theory.

Contents:

Cover

Title page

Contents

Preface

Examples of geodesic ghor algebras on hyperbolic surfaces

1. Introduction

2. Ghor algebras: Background and main results

3. Examples

References

Feynman categories and representation theory

Introduction

1. Representations from a categorical viewpoint

2. Feynman categories

3. Constructions and examples

4. Modules and enriched Feynman categories

5. Bar, co-bar, Feynman transforms, &amp

master equations

6. W-construction and cubical structures

7. Outlook

Appendix A. Graph glossary and graphical Feynman categories

Appendix B. Graph description of ⁺, ^{+ } and ^{ }

Appendix C. Double categories, 2-categories and monoidal categories

Appendix D. Model structures

Acknowledgments

Preprojective roots and graph monoids of Coxeter groups

1. Graph monoid

2. Preprojective roots

3. Preprojective roots and finite Coxeter groups

4. Reduced -admissible words

Approximable triangulated categories

2. Background

3. Approximability-the intuition, which comes from ( )

4. Measuring the complexity of an object

5. The formal definition of approximability

6. The main theorems

7. More about the strong generation of \dperf and ^{ }_{ }( )

8. More about finite -linear functors :\big[\dperf \big]^{ }⟶\Mod and ̃ : ^{ }_{ }( )⟶\Mod

9. The categories \dperf and ^{ }_{ }( ) determine each other

Appendix A. Some dumb maps in _{}ℭ^{ℭ'}( ), and the proof that the third map of the triangle is a cochain map

Appendix B. The assumption that the short exact sequences of cochain complexes are degreewise split is harmless

Appendix C. Translating the approach to derived categories we presented here to the more standard one in the literature.

Acknowledgments

Methods of constructive category theory

1. Category constructors

1.1. Computable categories

1.2. \Ab-categories

1.3. Additive closure

1.4. Homomorphism structures

1.5. Freyd category

1.6. Computing with Freyd categories

1.6.1. Equality of morphisms

1.6.2. Cokernels

1.6.3. Kernels

1.6.4. The abelian case

1.6.5. Homomorphisms

1.7. Computing natural transformations

2. Constructive diagram chases

2.1. Additive relations

2.2. Category of generalized morphisms

2.3. Computation rules

2.4. Cohomology

2.5. Snake lemma

2.6. Generalized homomorphism theorem

2.7. Computing spectral sequences

The HRS tilting process and Grothendieck hearts of t-structures

2. Preliminaries

3. Projective and injective objects in the heart. Quasi-(co)tilting torsion pairs

4. When is the heart of a torsion pair a Grothendieck category?

5. Beyond the HRS case: Some recent results

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

ISBN:

9781470464257

147046425X

OCLC:

1256254526