Algebra II
Noncommutative Rings Identities
Editors: Kostrikin, A.I., Shafarevich, I.R. (Eds.)
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The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the bestknown example of a noncommutative 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 noncommutative 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 socalled microlocal analysis. The theory of operator algebras (Le.
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Bibliographic Information
 Bibliographic Information

 Book Title
 Algebra II
 Book Subtitle
 Noncommutative Rings Identities
 Series Title
 Encyclopaedia of Mathematical Sciences
 Series Volume
 18
 Copyright
 1991
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783642728990
 DOI
 10.1007/9783642728990
 Softcover ISBN
 9783642729010
 Series ISSN
 09380396
 Edition Number
 1