Algebra

Introduction

algebraic representation theory is a new branch of the early 1970s, the rise of algebra. The basic content is visible mold on a study Artin algebra.

algebraic representation theory is the study of a given type Artin algebra is finite or infinite type. If the type is limited, which determines all indecomposable module; Infinity if, given mode distribution. We are familiar with the standard can be seen as Jordan is a univariate polynomial ring quotient ring.

In fact, let A be any nxn matrix over the complex C, then C [A] is a finite dimensional vector algebra C, C [A} is molded on a finite complex field dimensional vector space V, having a linear transformation to itself A. Indecomposable A mold having a number of different values ​​in the order of the maximum if and when a block C is isomorphic [A] on. If the classical structure theory is directly portray structure algebra, modern algebra representation theory is the study of the structure of an algebraic method of module theory.

In the nearly twenty five years, this theory has been greatly developed and gradually perfected. Main contents include basic algebra theory and method of Hall, and highlighted using methods to achieve this theory and Kac-Moody Lie algebra and the corresponding quantum enveloping algebra represented by algebraic theory; and represents quasi-hereditary algebra theory, and Contact this theory and representation theory of complex semi-simple Lie algebra and algebraic group and the like.

Background

origin

In the early twentieth century, the famous theorem Wdederburn would be completely characterize the structure of the finite-dimensional semi-simple algebra, algebraic this isomorphic to a limited number of other linear algebra full matrix and the ring, on which the mold half is a single mode. So, the structure of non-semisimple algebra and how? Classical theory is an algebraic structure is divided into two parts roots and semisimple, algebraic it is seen as a means of expanding its root semisimple part. Nilpotent root development by a variety of different properties of harmonic root zero roots, Jacobson roots. In general, a single half portion can give a better characterization, but the structure is very complex roots. Specially developed from the "root" theory, carry out research in this area.

1945, the American mathematician Baruer and Tharn raised speculation about the two finite-dimensional algebra. First, "Bounded algebraic type is limited." Second, "to represent any type of algebraic infinite, there exist infinitely many natural number d, such that the dimension d equal to have an unlimited number of die." These conjectures become algebraic theory of origins.

is the limited algebraic called a type, which means only a limited plurality of (up to isomorphism) indecomposable module, otherwise referred to as infinite type. It is well known, represents a mold with Algebra Algebra, Algebraic homomorphism i.e. a thing like a full matrix algebra. If we put such a state as seen with a photo of the original algebra is finite representation type algebra is finite photos can reveal a clear algebra, of course, is relatively simple. The Infinity algebraic expression need to use an unlimited number of photos.

start

In Brauer-Throl made twenty years after they guess in an attempt to solve these two guess work has been no substantial progress. Until 1968, the Soviet mathematician Rojetr article: "infinite algebraic representation type of indecomposable modules unbounded dimensionality" of finite-dimensional algebra over a field proven Brauer Tharl first guess. This article can be referred to the beginning of algebraic theory.

breakthrough and progress

1969 Nian, Kac-Mody is defined as generalized Cartan matrix, the theory of Lie algebra has been a great breakthrough. As this matter is inspired by Swiss mathematician Gabriel published in 1972 and 1973 "indecomposable representations Ⅰ and Ⅱ", using maps and quadratic approach to the algebraically closed field of finite dimensional path algebra representation model has been completely Classification.

1974- 1977, the American mathematician Ausladner and Norwegian mathematician Rieten published their series of articles: "Airtn algebra representation theory Ⅰ-Ⅳ", using the same tone technique research indecomposable modules, made almost this important concept split sequence, laid the theoretical foundation algebraic theory.

Hall Algebra

to a homogeneous class of finite p- group consisting of Groups Abel group can impart a multiplier, it is a structure constant certain limited groups of p- the number of kinds of filter chain. Obtained in this manner has a binding unit membered ring (Z _p ), called Hall algebraic p-adic ring Zp of an integer. It is a commutative ring and plays an important role in the theoretical and algebraic combinations. This is first algebraic E. Hall Steinitz, was later Ph. The study Hall.

1990 years, C. M. Ringel will promote the Hall algebra built on the rather arbitrary ring ──finitary ring (in particular, finite-dimensional algebra over finite fields) on. At this time, Hall Algebra generally do not exchange its corresponding Lie algebra caught his attention. His series of studies show, Hall algebra is a very important class of algebra, it can be achieved with many Kac-Moody Lie algebra and the corresponding quantum enveloping algebra. This established through the Hall algebra and algebraic relations of the Li theory is worthy of further study.

restriction Hall algebra is the Hall algebra of finite dimensional algebra over a finite field, and agreed: k is a finite field, A finite dimensional algebra over a k, which is bonded and with unit; modA remember all finite dimensional mold left visible, ind, full subcategory a is composed of representatives of the class of all homogeneous indecomposable A- mold element; for any M∈modA, denoted [M] is the same M type configuration; Further, Z and Q each represent an integer of the rational number field ring, and the cardinality of set S by S.

Hall Algebraic Synthesis Algebra

to M, N ₁ , N ₂ , ..., N _{t < / sub> ∈modA, provided F ^M Nt, Nt-1, ..., N1 as is the number of filtrations M:}

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such that, 1≤i≤t. (Note: if N ₁ , N ₂ , ..., N _t is single-mode, F ^M < sub> Nt, Nt-1, ..., N1synthetic exactly the number of columns having predetermined factor M of the synthesis.)

in particular, for M, L, n∈ modA, F ^M _{Nt, Nt-1, ..., N1} is the number of sub die U of M: and.

(integer) Hall algebras H (A) algebra A defined as follows: it is a group having a {u [M]} M∈modA mold consisting of Z- and has a multiplication

Note a base is limited. Therefore, for a fixed L, N∈modA, almost all of the F ^M _{L N} is zero. Thus the formula is a finite and right. In this manner, H (A) into a binding loop and with unit u _[0] .

Representation Theory of Lie algebra

K provided (modA) is about modA Grothendieck group splittable exact sequences, which are all the same mold indecomposable A- Z- configuration class consisting of the mold base. We naturally recognizable K (modA) is H (A) ₁ sub-additive group.

Theorem: A is provided with a Hall polynomial. Then the subgroup K (modA) is H (A) ₁ of a Subalgebras, and after they are expanded in the Rational Field, Hall algebraic H (A) ₁ * _Z Q precisely its Subalgebras K (modA) _Z Q of the universal enveloping algebra.

Ringel proved in his first article about the Hall algebra in the Hall he has either been to algebraic polynomial. Guo gives a formula expression either cyclic serial algebra Hall polynomials, thereby solving the problem of the existence of Hall polynomials such algebra.

This type of Two Classes are limited and is expressed algebraically in two extreme cases, one is indecomposable modules each are straight, the other is not a straight die each indecomposable Towards. Based on this observation, Ringel guess: a finite representation of any type of algebraic polynomial has Hall.

Quasi hereditary algebra

concept was proposed by E. Genetic algebra Cline, B. And L. Parshall , Which aims to Scott proposed to study appeared in the highest weight category complex semi-simple Lie algebra and algebraic group representation theory. The results show that the quasi-hereditary algebra is quite common, many algebraic naturally occurring proved to be quasi-hereditary algebra, such as hereditary algebra, the Schur algebras, Auslander algebra

Let A be a finite dimensional algebraic the field k . If [chi] A- is a class module, we denote by modA F (χ) full subcategory, wherein each object has a chain such that each submodule is isomorphic to the factor [chi] of an object.

set E (i) (i∈Λ) is a single mode A- all isomorphic, where Λ is a set of indicators with a partial order ≤ set of partial order. For each i∈Λ, set P (i) and Q (i) are the E (i) the lid projection and the inner bag exit. We △ (i) referred to P (i) such that it has the form factor of the synthesis of E (j) (j≤i) Maximum quotient module. Dual ground, with (i) referred to Q (i) such that its synthetic form factor having great submodule E (j) (j≤i), and referred with a △ △ with (i) and (i), i∈ Λ, mold type configuration.

Algebra A (more precisely, binary pair (A, Λ)) is called quasi-hereditary algebra, if

(i) End (△ (i)) k, for all i∈Λ;

(ii) each projection module belonging to F (△).

For the quasi-hereditary algebra (A, Λ), we call for the right to set A Λ, Λ A of the right elements. For each weight i∈Λ, said △ (i) is the standard mode (in special cases, also called Verma mold, or mold Weyl), said (i) is more than the standard mode. Quasi hereditary algebra (A, Λ) visible mold set to its standard mode of △ be set at a right Cline, Parshall and Scott Lambda is defined as the "highest weight category." In contrast, the highest weight category with any one of a limited set of rights can be seen as a quasi-hereditary algebra visible mold. Because of duality, a proposed standard mode and standard mode than genetic algebra algebraic also proposed anti-genetic.

situation and problems to be solved

At present study algebraic theory emphasis on Taoe type algebra. This is due to the limited type of algebraic are believed to have known about more thorough. This means that only a finite number of arbitrary dimension has been given, the limited type d-dimensional algebra.

On the other hand, the wild type comprising simultaneous algebraic unsolvable problem of two linear transformation, is considered to be difficult to start with. There are thus only some individual conclusions on certain specific wild-type algebra. Tame and is the simplest algebraic infinite represents the algebraic type, it can be considered only by a number of visible mold (finite, countable or uncountable infinity infinity) finite dimensional k [x] fight together to form a visible mode.

Bognart quadratic had employed to determine whether the positive definiteness of a class of algebraic type is the limited thereby, deIaPena tried using quadratic qualitative positive semi determines whether a class of algebraic Tame representation type. He found that the second type of semi-positive definiteness is expressed Tame type of necessary but not sufficient condition. So he gave a series of algebraic additional restrictions to consider this issue. Way

using algebraic theory Tame Bocs is a natural, but very difficult. Currently Brener, Butler and their students discuss Bocs represent the category of almost split sequences. Representation Theory of Beijing Normal University research team is also attempting to use Bocs solve algebraic structure Tame AR- quiver.