Vladimir V. Kirichenko

Finite dimensional algebras

1. Introduction.- 1.1 Basic Concepts. Examples.- 1.2 Isomorphisms and Homomorphisms. Division Algebras.- 1.3 Representations and Modules.- 1.4 Submodules and Factor Modules. Ideals and Quotient Algebras.- 1.5 The Jordan-Holder Theorem.- 1.6 Direct Sums.- 1.7 Endomorphisms. The Peirce Decomposition.- Exerises to Chapter 1.- 2. Semisimple Algebras.- 2.1 Schurs Lemma.- 2.2 Semisimple Modules and Algebras.- 2.3 Vector Spaces and Matrices.- 2.4 The Wedderburn-Artin Theorem.- 2.5 Uniqueness of the Decomposition.- 2.6 Representations of Semisimple Algebras.- Exercises to Chapter 2.- 3. The Radical.- 3.1 The Radical of a Module and of an Algebra.- 3.2 Lifting of Idempotents. Principal Modules.- 3.3 Projective Modules and Projective Covers.- 3.4 The Krull-Schmidt Theorem.- 3.5 The Radical of an Endomorphism Algebra.- 3.6 Diagram of an Algebra.- 3.7 Hereditary Algebras.- Exercises to Chapter 3.- 4. Central Simple Algebras.- 4.1 Bimodules.- 4.2 Tensor Products.- 4.3 Central Simple Algebras.- 4.4 Fundamental Theorems of the Theory of Division Algebras.- 4.5 Subfields of Division Algebras. Splitting Fields.- 4.6 Brauer Group. The Frobenius Theorem.- Exercises to Chapter 4.- 5. Galois Theory.- 5.1 Elements of Field Theory.- 5.2 Finite Fields. The Wedderburn Theorem.- 5.3 Separable Extensions.- 5.4 Normal Extensions. The Galois Group.- 5.5 The Fundamental Theorem of Galois Theory.- 5.6 Crossed Products.- Exercises to Chapter 5.- 6. Separable Algebras.- 6.1 Bimodules over Separable Algebras.- 6.2 The Wedderburn-Malcev Theorem.- 6.3 Trace, Norm, Discriminant.- Exercises to Chapter 6.- 7. Representations of Finite Groups.- 7.1 Maschkes Theorem.- 7.2 Number and Dimensions of Irreducible Representations.- 7.3 Characters.- 7.4 Algebraic Integers.- 7.5 Tensor Products of Representations.- 7.6 Burnsides Theorem.- Exercises to Chapter 7.- 8. The Morita Theorem.- 8.1 Categories and Functors.- 8.2 Exact Sequences.- 8.3 Tensor Products.- 8.4 The Morita Theorem.- 8.5 Tensor Algebras and Hereditary Algebras.- Exercises to Chapter 8.- 9. Quasi-Frobenius Algebras.- 9.1 Duality. Injective Modules.- 9.2 Lemma on Separation.- 9.3 Quasi-Frobenius Algebras.- 9.4 Uniserial Algebras.- Exercises to Chapter 9.- 10. Serial Algebras.- 10.1 The Nakayama-Skornjakov Theorem.- 10.2 Right Serial Algebras.- 10.3 The Structure of Serial Algebras.- 10.4 Quasi-Frobenius and Hereditary Serial Algebras.- Exercises to Chapter 10.- 11. Elements of Homological Algebra.- 11.1 Complexes and Homology.- 11.2 Resolutions and Derived Functors.- 11.3 Ext and Tor. Extensions.- 11.4 Homological Dimensions.- 11.5 Duality.- 11.6 Almost Split Sequences.- 11.7 Auslander Algebras.- Exercises to Chapter 11.- References.- A.1 Preliminaries. Standard and Costandard Modules.- A.3 Basic Properties.- A.4 Canonical Constructions.- A.6 Final Remarks.- References to the Appendix.

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.

Semi-Perfect Semi-Distributive Rings

The present paper is devoted to the study of semi-perfect semi-distributive rings (SPSD-rings). In particular, the concept of a prime quiver of a semi-perfect ring and a quiver of an SPSD-ring is widely used. The description of semi-hereditary SPSD-rings is reduced to the case of prime semi-hereditary serial rings and finite posets without rhombuses. For semi-hereditary, semi-perfect, semi-distributive rings we prove the existence of classical quotient rings.

Elements of Homological Algebra

The present chapter has been written for the English edition. The aim of this extension is to present an introduction to homological methods, which play an increasingly important role in the theory of algebras, and in this way to make the book more suitable as a textbook. Besides the fundamental concepts of a complex, resolutions and derived functors, we shall also briefly examine three special topics: homological dimension, almost split sequences and Auslander algebras.

Quasi-Frobenius Algebras

The duality which exists between the categories of the right and left modules plays an important role in the theory of finite dimensional algebras. In the present chapter we shall introduce this duality, investigate its properties and apply the obtained results to the study of two classes of algebras, viz. to quasi-Frobenius algebras introduced into the theory by T. Nakayama and to serial algebras, or principal ideal algebras, which were studied first by K. Asano.

Central Simple Algebras

The Wedderburn-Artin theorem reduces the study of semisimple algebras to the description of division algebras over a field K. If D is a finite dimensional division algebra over K and C its center, then C is a field (an extension of the field K) and D can be considered as an algebra over the field C. In this way, the investigation is divided into two steps: the study of the extensions of the field K and the study of central division algebras, i. e. of division algebras whose center coincides with the ground field. It turns out that these are two separate problems. However, one can conveniently apply common methods of investigation based on the concept of a bimodule and tensor product of algebras.

Representations of Finite Groups

In this chapter we shall apply the general theory of semisimple algebras and their representations to obtain basic results of the classical theory of representations of finite groups.

The Morita Theorem

In Sect. 2.3 we have noted that modules over a division algebra D and modules over the simple algebra M n (D) are “equally structured”. Results of Sect. 2.6 show that, in general, modules over isotypic semisimple algebras possess the same properties: such modules have isomorphic endomorphism rings, etc. In Sect. 3.5 these results have been extended to projective modules over arbitrary isotypic algebras (Lemma 3.5.5). It turns out that one can remove the requirement of projectivity: All modules over isotypic algebras are equally structured. However, in order to formulate this statement properly, it is necessary to introduce a number of concepts which presently play an important role in various areas of mathematics. Above all, it is the concept of a category and a functor, as well as the notion of an equivalence of categories, which appears to be a mathematical formulation of the expression “equally structured”.