Algebra: Noncommutative Rings Identities: 18 (Encyclopaedia of Mathematical Sciences) - Hardcover

Hardcover

ISBN 10: 3540181776 ISBN 13: 9783540181774

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1991

Synopsis

The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge- 1 bra * Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat- ics. Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap- plications in the classical mechanics of the past century. Later it turned out that quaternion analysis had important applications in field theory. The al- gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology. We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics. The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration. Clifford algebras, which include exterior algebras as a special case, have applications in rep- resentation theory and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with* polynomial coefficients) often appears in the representation theory of Lie algebras. In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le.

About this title

Synopsis

Algebra II is a two-part survey on the subject of non-commutative rings and algebras, with the second part focused on the theory of identities of these and other algebraic systems. It provides a broad overview of the most modern trends encountered in non-commutative algebra, as well as the numerous connections between algebraic theories and other areas of mathematics. A wide selection of examples of non-commutative rings is given at the start. Throughout the book, the authors include the historical background of the trends they are discussing. The authors, who are among the most prominent Soviet algebraists, share with their readers their knowledge of the subject, giving them a unique opportunity to learn the material from mathematicians who have made major contributions to it. This is especially true in the case of the theory of identities in varieties of algebraic objects where Soviet mathematicians have been a moving force behind this approach. This monograph on associative rings and algebras, group theory and algebraic geometry is intended for researchers and students.

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 1991
• Language: English
• ISBN 10: 3540181776
• ISBN 13: 9783540181774
• Binding: Hardcover
• Number of pages: 241
• Editor: Kostrikin A. I.