Quiver Representations And Quiver Varieties

Book Detail:
Author: Alexander Kirillov Jr.
Publisher: American Mathematical Soc.
ISBN: 1470423073
Size: 63.83 MB
Format: PDF, ePub, Docs
Category : Algebraic geometry -- Cycles and subschemes -- Parametrization (Chow and Hilbert schemes)
Languages : en
Pages : 295
View: 7219
Book Description: This book is an introduction to the theory of quiver representations and quiver varieties, starting with basic definitions and ending with Nakajima's work on quiver varieties and the geometric realization of Kac–Moody Lie algebras. The first part of the book is devoted to the classical theory of quivers of finite type. Here the exposition is mostly self-contained and all important proofs are presented in detail. The second part contains the more recent topics of quiver theory that are related to quivers of infinite type: Coxeter functor, tame and wild quivers, McKay correspondence, and representations of Euclidean quivers. In the third part, topics related to geometric aspects of quiver theory are discussed, such as quiver varieties, Hilbert schemes, and the geometric realization of Kac–Moody algebras. Here some of the more technical proofs are omitted; instead only the statements and some ideas of the proofs are given, and the reader is referred to original papers for details. The exposition in the book requires only a basic knowledge of algebraic geometry, differential geometry, and the theory of Lie groups and Lie algebras. Some sections use the language of derived categories; however, the use of this language is reduced to a minimum. The many examples make the book accessible to graduate students who want to learn about quivers, their representations, and their relations to algebraic geometry and Lie algebras.

An Introduction To Quiver Representations

Book Detail:
Author: Harm Derksen
Publisher: American Mathematical Soc.
ISBN: 1470425564
Size: 18.38 MB
Format: PDF, Docs
Category : Directed graphs
Languages : en
Pages : 344
View: 1296
Book Description: 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.

Two Algebraic Byways From Differential Equations Gr Bner Bases And Quivers

Book Detail:
Author: Kenji Iohara
Publisher: Springer Nature
ISBN: 3030264548
Size: 17.92 MB
Format: PDF, ePub, Mobi
Category : Mathematics
Languages : en
Pages : 371
View: 5854
Book Description: This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

Representations Of Finite Dimensional Algebras And Related Topics In Lie Theory And Geometry

Book Detail:
Author: Vlastimil Dlab
Publisher: American Mathematical Soc.
ISBN: 0821834169
Size: 60.38 MB
Format: PDF, ePub, Mobi
Category : Mathematics
Languages : en
Pages : 479
View: 838
Book Description: These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional instructional'' workshop preceding the conference, there were also workshops on Commutative Algebra, Algebraic Geometry and Representation Theory'', Finite Dimensional Algebras, Algebraic Groups and Lie Theory'', and Quantum Groups and Hall Algebras''. These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated. The book is recommended for graduate students and researchers in algebra and geometry.

Geometrische Methoden In Der Invariantentheorie

Book Detail:
Author: Hanspeter Kraft
Publisher: Springer-Verlag
ISBN: 3663101436
Size: 68.61 MB
Format: PDF, ePub
Category : Technology & Engineering
Languages : de
Pages : 308
View: 6771
Book Description: In dieser Einführung geht es vor allem um die geometrischen Aspekte der Invariantentheorie. Die hauptsächliche Motivation bildet das Studium von Klassifikations- und Normalformenproblemen, die auch historisch der Ausgangspunkt für invariantentheoretische Untersuchungen waren.

Quiver Representations And Their Dense Orbits

Book Detail:
Author: Danny Lara
Publisher:
ISBN:
Size: 30.32 MB
Format: PDF, Mobi
Category : Directed graphs
Languages : en
Pages : 87
View: 3251
Book Description: We can view quiver representations of a fixed dimension vector as an algebraic variety over an algebraically closed field $K$. There is an action of the product of general linear groups on each of these varieties where the orbits of the action correspond to isomorphism classes of quiver representation. A $K$-algebra $A$ is said to have the dense orbit property if for each dimension vector, the product of the general linear group acts on each irreducible component of the module variety with a dense orbit. Under certain conditions, a $K$ algebra $A$ is representation finite if and only if it $A$ has the dense orbit property. The implication representation finite implies the dense orbit property is always true. The converse is not true in general, as shown by Chindris, Kinser, and Weyman in \cite{ryan}. Our main theorem of this thesis builds on their work to give a family of representation infinite algebras with the dense orbit property. We also give a conjectured classification of indecomposables with dense orbits. \par In the future, we hope the work presented here can be used to find even more examples of representation infinite algebra with the dense orbit property to then develop deeper theory to classify algebras with the dense orbit property that are representation infinite.

On Categories O Of Quiver Varieties Overlying The Bouquet Graphs

Book Detail:
Author: Boris Tsvelikhovskiy
Publisher:
ISBN:
Size: 56.44 MB
Format: PDF, ePub
Category : Algebra, Homological
Languages : en
Pages : 67
View: 1510
Book Description: "We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations $\overline{\mathcal{A}}_{\lambda}(n, \ell)$ if both $\mbox{dim}~V=n$ and the number of loops $\ell$ are greater than $1$. We show that when $n\leq 3$ there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ and compute the multiplicities of simples in standards for $n=2$ in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension"--Author's abstract.

Regular Solids And Isolated Singularities

Book Detail:
Author: Klaus Lamotke
Publisher: Vieweg+Teubner Verlag
ISBN: 9783528089580
Size: 52.74 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 224
View: 3522
Book Description: The last book XIII of Euclid's Elements deals with the regular solids which therefore are sometimes considered as crown of classical geometry. More than two thousand years later around 1850 Schl~fli extended the classification of regular solids to four and more dimensions. A few decades later, thanks to the invention of group and invariant theory the old three dimensional regular solid were involved in the development of new mathematical ideas: F. Klein (Lectures on the Icosa hedron and the Resolution of Equations of Degree Five, 1884) emphasized the relation of the regular solids to the finite rotation groups. He introduced complex coordinates and by means of invariant theory associated polynomial equations with these groups. These equations in turn describe isolated singularities of complex surfaces. The structure of the singularities is investigated by methods of commutative algebra, algebraic and complex analytic geometry, differential and algebraic topology. A paper by DuVal from 1934 (see the References), in which resolutions play an important rele, marked an early stage of these investigations. Around 1970 Klein's polynomials were again related to new mathematical ideas: V. I. Arnold established a hierarchy of critical points of functions in several variables according to growing com plexity. In this hierarchy Kleinls polynomials describe the "simple" critical points.

Vorlesungen Ber Das Ikosaeder

Book Detail:
Author: Felix Klein
Publisher: Springer DE
ISBN:
Size: 70.69 MB
Format: PDF, Kindle
Category : Equations, Quintic
Languages : de
Pages : 260
View: 3825
Book Description:

Quantized Multiplicative Quiver Varieties And Actions Of Higher Genus Braid Groups

Book Detail:
Author: David Andrew Jordan
Publisher:
ISBN:
Size: 25.11 MB
Format: PDF, ePub, Docs
Category :
Languages : en
Pages : 112
View: 5116
Book Description: In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain "moment map" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n.