Algebra II: Noncommutative Rings IdentitiesThe 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. |
Contents
Contents | 4 |
A Bokhut I V Lvov V K Kharchenko | 6 |
Kostrikin I R Shafarevich Steklov Mathematical Institute ul Vavilova | 42 |
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Common terms and phrases
abelian group algebra FG Algebra Logic Artinian ring associative algebra associative rings automorphisms base field basis of identities Brauer group called central simple algebra characteristic zero commutative ring construction contains countable cyclic denote direct sum division ring elements endomorphism English translation equals equivalent example exists field F field of characteristic finite basis finite group finite-dimensional algebra following theorem free algebra free group Frobenius group algebra group G homomorphism idempotents indecomposable infinite injective integer irreducible isomorphic Jacobson radical Krull dimension lattice left ideal left R-module Lie algebra locally finite mapping Math matrices Mn(F Morita multiplication Nauk Neumann nilpotent Noetherian ring non-commutative non-trivial non-zero obtained operations PI-algebra PI-ring polynomial prime rings problem R-module rank regular ring representations result ring theory Sect semigroup semiprime solvable subalgebra submodules subset subvarieties system of identities tensor product topological varieties of groups vector space zero divisors