Bøger i Algebra and Applications serien

Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory.

This book explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory.

This book offers a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of degree 3. It links a number of the algebras to the polynomial irreducible representations of the group SL3(C).

The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and their basic properties. The second half focuses specifically on the ring theoretic properties of polycyclic groups and their applications.

This monograph presents a new model of mathematical structures called weak n-categories. These structures find their motivation in a wide range of fields, from algebraic topology to mathematical physics, algebraic geometry and mathematical logic.While strict n-categories are easily defined in terms associative and unital composition operations they are of limited use in applications, which often call for weakened variants of these laws. The author proposes a new approach to this weakening, whose generality arises not from a weakening of such laws but from the very geometric structure of its cells; a geometry dubbed weak globularity. The new model, called weakly globular n-fold categories, is one of the simplest known algebraic structures yielding a model of weak n-categories. The central result is the equivalence of this model to one of the existing models, due to Tamsamani and further studied by Simpson. This theory has intended applications to homotopy theory, mathematical physics and to long-standing open questions in category theory.As the theory is described in elementary terms and the book is largely self-contained, it is accessible to beginning graduate students and to mathematicians from a wide range of disciplines well beyond higher category theory. The new model makes a transparent connection between higher category theory and homotopy theory, rendering it particularly suitable for category theorists and algebraic topologists. Although the results are complex, readers are guided with an intuitive explanation before each concept is introduced, and with diagrams showing the interconnections between the main ideas and results.

This monograph presents combinatorial and numerical issues on integral quadratic forms as originally obtained in the context of representation theory of algebras and derived categories. Some of these beautiful results remain practically unknown to students and scholars, and are scattered in papers written between 1970 and the present day. Besides the many classical results, the book also encompasses a few new results and generalizations.The material presented will appeal to a wide group of researchers (in representation theory of algebras, Lie theory, number theory and graph theory) and, due to its accessible nature and the many exercises provided, also to undergraduate and graduate students with a solid foundation in linear algebra and some familiarity on graph theory.

This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig's seminal work on the subject was published in 2002.

The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of 'nearby' calculations in K-theory.

Here is a comprehensive treatment of the main results and methods of the theory of Noetherian semigroup algebras. These results are applied and illustrated in the context of important classes of algebras that arise in a variety of areas and have recently been intensively studied.

Group rings play a central role in the theory of representations of groups and are very interesting algebraic objects in their own right. In their study, many branches of algebra come to an interplay. This book takes the reader from beginning to research level. It is suitable for mathematicians working in the area of group rings.