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Table of Contents
Preface ix
Chapter 1. Groups and group representations 1
1.1 Groups and subgroups. Definitions and examples . . . . . . . . . . . . . . . . . 2
1.2 Symmetry. Symmetry groups 7
1.3 Quotient groups, homomorphisms and normal subgroups . . . . . . . . . . . . . . 10
1.4 Sylow theorems 14
1.5 Solvable and nilpotent groups 21
1.6 Group rings and group representations. Maschke theorem . . . . . . . . . 26
1.7 Properties of irreducible representations 35
1.8 Characters of groups. Orthogonality relations and their applications . 38
1.9 Modular group representations 47
1.10 Notes and references 49
Chapter 2. Quivers and their representations 53
2.1 Certain important algebras 53
2.2 Tensor algebra of a bimodule 60
2.3 Quivers and path algebras 67
2.4 Representations of quivers 74
2.5 Dynkin and Euclidean diagrams. Quadratic forms and roots . . . . . . .79
2.6 Gabriel theorem 93
2.7 K-species 99
2.8 Notes and references 100
Appendix to section 2.5. More about Dynkin and extended
Dynkin (= Eyclidean) diagrams 105
Chapter 3. Representations of posets and of finite dimensional
algebras 113
3.1 Representations of posets 114
3.2 Differentiation algorithms for posets 130
3.3 Representations and modules. The regular representations. . . . . . . .135
3.4 Algebras of finite representation type 140
v
vi TABLE OF CONTENTS
3.5 Roiter theorem 147
3.6 Notes and references 153
Chapter 4. Frobenius algebras and quasi-Frobenius rings 161
4.1 Duality properties 161
4.2 Frobenius and symmetric algebras 164
4.3 Monomial ideals and Nakayama permutations of semiperfect rings . . 166
4.4 Quasi-Frobenius algebras 169
4.5 Quasi-Frobenius rings 174
4.6 The socle of a module and a ring 177
4.7 Osofsky theorem for perfect rings 181
4.8 Socles of perfect rings 183
4.9 Semiperfect piecewise domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184
4.10 Duality in Noetherian rings 187
4.11 Semiperfect rings with duality for simple modules 190
4.12 Self-injective rings 193
4.13 Quivers of quasi-Frobenius rings 204
4.14 Symmetric algebras with given quivers 205
4.15 Rejection lemma 208
4.16 Notes and references 212
Chapter 5. Right serial rings 219
5.1 Homological dimensions of right Noetherian rings . . . . . . . . . . . . . . 219
5.2 Structure of right Artinian right serial rings 224
5.3 Quasi-Frobenius right serial rings 230
5.4 Right hereditary right serial rings 231
5.5 Semiprime right serial rings 233
5.6 Right serial quivers and trees 236
5.7 Cartan matrix for a right Artinian right serial ring 244
5.8 Notes and references 252
Chapter 6. Tiled orders over discrete valuation rings 255
6.1 Tiled orders over discrete valuation rings and exponent matrices . . . 255
6.2 Duality in tiled orders 270
6.3 Tiled ordersand Frobenius rings 276
6.4 Q-equivalent partially ordered sets 279
6.5 Indices of tiled orders 287
6.6 Finite Markovchains and reduced exponent matrices 292
6.7 Finite partially ordered sets, (0,1)-orders and finite Markov chains . . . 296
6.8 Adjacency matrices of admissible quivers without loops. . . . . . . . . 301
6.9 Tiled ordersand weakly prime rings 305
6.10 Global dimension of tiled orders 313
6.11 Notes and references 323
TABLE OF CONTENTS vii
Chapter 7. Gorenstein matrices 327
7.1 Gorenstein tiled orders. Examples 327
7.2 Cyclic Gorenstein matrices 338
7.3 Gorenstein (0,1)-matrices 346
7.4 Indices of Gorenstein matrices 356
7.5 d-matrices 364
7.6 Cayley tables of elementaryAbelian 2-groups 369
7.7 Quasi-Frobenius rings and Gorenstein tiled orders 379
7.8 Notes and references 381
Suggestions for further reading 385
389
Name index 397
Subject index
Preface
This book is the natural continuation of “Algebras, rings and modules. vol.I”.
The main part of it consists of the study of special classes of algebras and rings.
Topics covered include groups, algebras, quivers, partially ordered sets and their
representations, as well as such special rings as quasi-Frobenius and right serial
rings, tiled orders and Gorenstein matrices.
Representation theory is a fundamental tool for studying groups, algebras
and rings by means of linear algebra. Its origins are mostly in the work of
F.G.Frobenius, H.Weil, I.Schur, A.Young, T.Molien about century ago. The re-
sults of the representation theory of finite groups and finite dimensional algebras
play a fundamental role in many recent developments of mathematics and theoret-
ical physics. The physical aspects of this theory concern accounting for and using
the concepts of symmetry which appear in various physical processes.
We start this book with the main results of the theory of groups. For the
convenience of a reader in the beginning of this chapter we recall some basic
concepts and results of group theory which will be necessary for the next chapters
of the book.
Groups are a central object of algebra. The concept of a group is histori-
cally one of the first examples of an abstract algebraic system. Finite groups, in
particular permutation groups, are an increasingly important tool in many areas
of applied mathematics. Examples include coding theory, cryptography, design
theory, combinatorial optimization, quantum computing, and statistics.
In chapter I we give a short introduction to the theory of groups and their rep-
resentations. We consider the representation theory of groups from the module-
theoretical point of view using the main results about rings and modules as
recorded in volume I of this book. This theoretical approach was first used by
E.Noether who established a close connection between the theory of algebras and
the theory of representations. From that point of view the study of the repre-
sentation theory of groups becomes a special case of the study of modules over
rings. In the theory of representations of group a special role is played by the
famous Maschke theorem. Taking into account its great importance we give three
different proofs of this theorem following J P.Serre, I.N.Herstein and M.Hall. As
a consequence of the Maschke theorem, the representation theory of groups splits
into two different cases depending on the characteristic of a field k: classical and
modular (following L.E.Dickson). In “classical” representation theory one assumes
that the characteristic of k does not divide the group order |G| (e.g. k can be the
field of complex numbers). In “modular” representation theory one assumes that
the characteristic of k is a prime, dividing |G|. In this case the theory is almost
completely different from the classical case.
ix
x PREFACE
In this book we consider the results belonging to the classical representation
theory of finite groups, such as the characters of groups. We give the basic prop-
erties of irreducible characters and their connection with the ring structure of the
corresponding group algebras.
A central role in the theory of representations of finite dimensional algebras
and rings is played by quivers, which were introduced by P.Gabriel in connection
with problems of representations of finite dimensional algebras in 1972. The main
notions and result concerning the theory of quivers and their representations are
given in chapter 2.
A most remarkable result in the theory of representations of quivers is the
theorem classifying the quivers of finite representation type, which was obtained by
P.Gabriel in 1972. This theorem says that a quiver is of finite representation type
over an algebraically closed field if and only if the underlying diagram obtained
from the quiver by forgetting the orientations of all arrows is a disjoint union of
simple Dynkin diagrams. P.Gabriel also proved that there is a bijection between
the isomorphism classes of indecomposable representations of a quiver Q and the
set of positive roots of the Tits form corresponding to this quiver. A proof of this
theorem is given in section 2.6.
Another proof of this theorem in the general case, for an arbitrary field, us-
ing reflection functors and Coxeter functors has been obtained by I.N.Berstein,
I.M.Gel’fand, and V.A.Ponomarev in 1973. In their work the connection between
indecomposable representations of a quiver of finite type and properties of its Tits
quadratic form is elucidated.
Representations of finite partially ordered sets (posets, in short) play an im-
portant role in representation theory. They were first introduced by L.A.Nazarova
and A.V.Roiter. The first two sections of chapter 3 are devoted to partially ordered
sets and their representations. Here are given the main results of M.M.Kleiner on
representations of posets of finite type and results of L.A.Nazarova on representa-
tions of posets of infinite type. The most important result in this theory was been
obtained by Yu.A.Drozd who showed that there is a trichotomy between finite,
tame and wild representation types for finite posets over an algebraically closed
field.
One of the main problems of representation theory is to obtain information
about the possible structure of indecomposable modules and to describe the iso-
morphism classes of all indecomposable modules. By the famous theorem on tri-
chotomy for finite dimensional algebras over an algebraically closed field, obtained
by Yu.A.Drozd, all such algebras divide into three disjoint classes.
The main results on representations of finitely dimensional algebras are given
in section 3.4. Here we give structure theorems for some special classes of fi-
nite dimensional algebras of finite type, such as hereditary algebras and algebras
with zero square radical, obtained by P.Gabriel in terms of Dynkin diagrams.
Section 3.5 is devoted to the first Brauer-Thrall conjecture, of which a proof has
PREFACE xi
been obtained by A.V.Roiter for the case of finite dimensional algebra over an
arbitrary field.
Chapter 4 is devoted to study of Frobenius algebras and quasi-Frobenius rings.
The class of quasi-Frobenius rings was introduced by T.Nakayama in 1939 as a
generalization of Frobenius algebras. It is one of the most interesting and in-
tensively studied classes of Artinian rings. Frobenius algebras are determined
by the requirement that right and left regular modules are equivalent. And
quasi-Frobenius algebras are defined as algebras for which regular modules are
injective.
We start this chapter with a short study of duality properties for finite dimen-
sional algebras. In section 4.2 there are given equivalent definitions of Frobenius
algebras in terms of bilinear forms and linear functions. There is also a discussion
of symmetric algebras which are a special class of Frobenius algebras. The main
properties of quasi-Frobenius algebras are given in section 4.4.
The starting point in studying quasi-Frobenius rings in this chapter is the
Nakayama definition of them. The key concept in this definition is a permuta-
tion of indecomposable projective modules, which is naturally called Nakayama
permutation.
Quasi-Frobenius rings are also of interest because of the presence of a dual-
ity between the categories of left and right finitely generated modules over them.
The main properties of duality in Noetherian rings are considered in section 4.10.
Semiperfect rings with duality for simple modules are studied in section 4.11.
The equivalent definitions of quasi-Frobenius rings in terms of duality and semi-
injective rings are given 4.12. Quasi-Frobenius rings have many interesting equiv-
alent definitions, in particular, an Artinian ring A is quasi-Frobenius if and only
if A is a ring with duality for simple modules.
One of the most significant results in quasi-Frobenius ring theory is the theorem
of C.Faith and E.A.Walker. This theorem says that a ring A is quasi-Frobenius if
and only if every projective right A-module is injective and conversely.
Quivers of quasi-Frobenius rings are studied in section 4.13. The most impor-
tant result of this section is the Green theorem: the quiver of any quasi-Frobenius
ring is strongly connected. Conversely, for a given strongly connected quiver Q
there is a symmetric algebra A such that Q(A)=Q. Symmetric algebras with
given quivers are studied in section 4.14.
Chapter 5 is devoted to the study of the properties and structure of right serial
rings. Note that a module is called serial if it decomposes into a direct sum of
uniserial submodules, i.e., submodules with linear lattice of submodules. A ring is
called right serial if its right regular module is serial.
We start this chapter with a study of right Noetherian rings from the point of
view of some main properties of their homological dimensions.
In further sections we give the structure of right Artinian right serial rings in
terms of their quivers. We also describe the structure of particular classes of right
serial rings, suchas quasi-Frobenius rings, right hereditary rings, and semiprime
xii PREFACE
rings. In section 5.6 we introduce right serial quivers and trees and give their
description.
The last section of this chapter is devoted to the Cartan determinant conjecture
for right Artinian right serial rings. The main result of this section says that a
right Artinian right serial ring A has its Cartan determinant equal to 1 if and only
if the global dimension of A is finite.
In chapters 6 and 7 the theory of semiprime Noetherian semiperfect semidis-
tributive rings is developed (SPSD-rings). In view of the decomposition theorem
(see theorem 14.5.1, vol.I) it is sufficient to consider prime Noetherian SPSD-
rings, which are called tiled orders.
With any tiled order we can associate a reduced exponent matrix and its quiver.
This quiver Q is called the quiver of that tiled order. It is proved that Q is a simply
laced and strongly connected quiver. In chapter 6 a construction is given which
allows to form a countable set of Frobenius semidistributive rings from a tiled
order. Relations between finite posets and exponent (0,1)-matrices are described
and discussed. In particular, a finite ergodic Markov chain is associated with a
finite poset.
Chapter 7 is devoted to the study of Gorenstein matrices. We say that a
tiled order A is Gorenstein if r.inj.dim
A
A =1. Inthiscaser.inj.dim
A
A =
l.inj.dim
A
A = 1. Moreover, a tiled order is Gorenstein if and only if it is Morita
equivalent to a reduced tiled order with a Gorenstein exponent matrix.
Each chapter ends with a number of notes and references, some of which have
a bibliographical character and others are of a historical nature.
At the end of the book we give a literature list which can be considered as sug-
gestions for further reading to obtain fuller information concerning other aspects
of the theory of rings and algebras.
In closing, we would like to express our cordial thanks to a number of friends
and colleagues for reading preliminary versions of this text and offering valuable
suggestions which were taken into account in preparing the final version. We
are especially greatly indebted to Yu.A.Drozd, V.M.Bondarenko, S.A.Ovsienko,
M.Dokuchaev, V.Futorny, V.N.Zhuravlev, who made a large number of valu-
able comments, suggestions and corrections which have considerably improved the
book. Of course, any remaining errors are the sole responsibility of the authors.
Finally, we are most grateful to Marina Khibina for help in preparing the
manuscript. Her assistance has been extremely valuable to us.
1. Groups and group representations
Groups are a central subject in algebra. They embody the easiest concept of
symmetry. There are others: Lie algebras (for infinitesimal symmetry) and Hopf
algebras (quantum groups) who combine the two and more (see volume III). Fi-
nite groups, in particular permutation groups, are an increasingly important tool
in many areas of applied mathematics. Examples include coding theory, cryptog-
raphy, design theory, combinatorial optimization, quantum computing.
Representation theory, the art of realizing a group in a concrete way, usually
as a collection of matrices, is a fundamental tool for studying groups by means
of linear algebra. Its origins are mostly in the work of F.G.Frobenius, H.Weil,
I.Schur, A.Young, T.Molien about century ago. The results of the theory of repre-
sentations of finite groups play a fundamental role in many recent developments of
mathematics and theoretical physics. The physical aspects of this theory consist
in accounting for and using the concept of symmetry as present in various physical
processes – though not always obviously so. As understood at present, symme-
try rules physics and an elementary particle is the same thing as an irreducible
representation. This includes quantum physics. There is a seeming mystery here
which is explained by the fact that the representation theory of quantum groups
is virtually the same as that of their classical (Lie group) counterparts.
In this chapter we shall give a short introduction to the theory of groups
and their representations. We shall consider the representation theory of groups
from the module-theoretic point of view using the main results about rings and
modules as described in volume I of this book. This theoretical approach was
first used by E.Noether who established a close connection between the theory of
algebras and the theory of representations. From this point of view the study of
the representation theory of groups becomes a special case of the study of modules
over rings. At the end of this chapter we shall consider the characters of groups.
We shall give the basic properties of irreducible characters and their connection
with the ring structure of group algebras.
For the convenience of a reader in the beginning of this chapter we recall some
basic concepts and results of group theory which will be necessary for the next
chapters of the book.
1
2 ALGEBRAS, RINGS AND MODULES
1.1 GROUPS AND SUBGROUPS. DEFINITIONS AND EXAMPLES
The notion of an abstract group was first formulated by A.L.Cayley (1821-1895)
who used this to identify matrices and quaternions as groups. The first formal
definition of an abstract group in the modern form appeared in 1882. Before, a
group was exclusively a group of permutations of some set (or a group of matrices).
The famous book by Burnside (1905) illustrates this well.
Definition. A group is a nonempty set G together with a given binary
operation ∗ on G satisfying the following axioms:
(1) (a ∗b) ∗ c = a ∗(b ∗ c) for all a, b, c ∈ G; (associativity)
(2) there exists an element e ∈ G, called an identity of G, such that a ∗ e =
e ∗ a = a for every a ∈ G;
(3) for each a ∈ G there exists an element a
−1
∈ G, called an inverse of a,
such that a ∗ a
−1
= a
−1
∗ a = e.
From the axioms for a group G one can easily obtain the following properties:
(1) the identity element in G is unique;
(2) for each a ∈ G the element a
−1
is uniquely determined;
(3) (a
−1
)
−1
= a for every a ∈ G;
(4) (a ∗b)
−1
= b
−1
∗ a
−1
.
AgroupG is called Abelian (or commutative)ifa∗b = b ∗a for all a, b ∈ G.
For some commutative groups it is often convenient to use the additive symbol +
for the operation in a group and write x + y instead of x ∗y. In this case we call
this group additive. The identity of an additive group G is called the zero and
denoted by 0, and the inverse element of x is called its negative element and
denoted by −x.Inthiscasewewritex − y instead of x +(−y). Note that this
notation is almost never used for non-commutative groups.
For writing an operation of a group G we usually use the multiplicative sym-
bol · and write xy rather that x · y. In this case we say that the group G is
multiplicative and denote the identity of G by 1.
If G is a finite set G is called a finite group. The number of elements of a
finite group G is called the order of G and denoted by |G| or o(G)or#G.
Examples 1.1.1.
1. The sets Z
, Q, R and C are groups under the operation of addition + with
e =0anda
−1
= −a for all a. They are additive Abelian groups.
2. The sets Q \{0}, R \{0} and C \{0} are groups under the operation
of multiplication · with e =1anda
−1
=1/a for all a. They are multiplicative
Abelian groups. The set Z\{0} with the operation of multiplication · is not a group
because the inverse to n is 1/n, which is not integer if n =1. ThesetR
+
of all
positive rational numbers is a multiplicative Abelian group under multiplication.
3. The set of all invertible n × n matrices with entries from a field k forms a
group under matrix multiplication. This group is denoted by GL
n
(k) and called