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Finite Dimensional Algebras

2012-12-06 Mathematics
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Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 441/5 ( reviews)

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Book Synopsis Finite Dimensional Algebras by : Yurj A. Drozd

Download or read book Finite Dimensional Algebras written by Yurj A. Drozd. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: This English edition has an additional chapter "Elements of Homological Al gebra". Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras.

Finite-Dimensional Division Algebras over Fields

2009-12-09 Mathematics
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Release : 2009-12-09
Genre : Mathematics
Kind : eBook
Book Rating : 297/5 ( reviews)

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Book Synopsis Finite-Dimensional Division Algebras over Fields by : Nathan Jacobson

Download or read book Finite-Dimensional Division Algebras over Fields written by Nathan Jacobson. This book was released on 2009-12-09. Available in PDF, EPUB and Kindle. Book excerpt: Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called "multiplication algebras of Riemann matrices". Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.

$\textrm {C}^*$-Algebras and Finite-Dimensional Approximations

2008 Mathematics
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Release : 2008
Genre : Mathematics
Kind : eBook
Book Rating : 818/5 ( reviews)

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Book Synopsis $\textrm {C}^*$-Algebras and Finite-Dimensional Approximations by : Nathanial Patrick Brown

Download or read book $\textrm {C}^*$-Algebras and Finite-Dimensional Approximations written by Nathanial Patrick Brown. This book was released on 2008. Available in PDF, EPUB and Kindle. Book excerpt: $\textrm{C}*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\textrm{C}*$-approximation theory.

Representations of Finite-Dimensional Algebras

1997-09-12 Mathematics
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Release : 1997-09-12
Genre : Mathematics
Kind : eBook
Book Rating : 900/5 ( reviews)

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Book Synopsis Representations of Finite-Dimensional Algebras by : Peter Gabriel

Download or read book Representations of Finite-Dimensional Algebras written by Peter Gabriel. This book was released on 1997-09-12. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "... [Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ..." --The Mathematical Gazette

Finite Dimensional Algebras and Quantum Groups

2008 Mathematics
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Release : 2008
Genre : Mathematics
Kind : eBook
Book Rating : 866/5 ( reviews)

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Book Synopsis Finite Dimensional Algebras and Quantum Groups by : Bangming Deng

Download or read book Finite Dimensional Algebras and Quantum Groups written by Bangming Deng. This book was released on 2008. Available in PDF, EPUB and Kindle. Book excerpt: "The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.

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