|
|
Milestone Year
1971
Advances in Polynomial Identity Theory
The algebra of n-by-n matrices is non-commutative: In general AB is not equal to BA. Nevertheless, these (dimension-dependent) algebras satisfy some polynomial identities. For example any two-by-two matrices x, y and z satisfy the polynomial identity (xy-yx)(xy-yx)z - z(xy-yx)(xy-yx) = 0. The study of the polynomial identities of various algebras is a major subarea of Non-Commutative Ring Theory, called Polynomial Identities Theory.
In 1971 a WIS scientist introduced, for each such algebra, an infinite sequence of dimensions, called the sequence of codimensions, and proved that the growth rate of elements in this sequence is upper bounded by an exponential function. This result was used in solving several of the main open problems in the area of polynomial Identities. For example, in 1971 using the exponential growth of codimentions, a WIS scientist proved that, for two algebras, if each satisfies some polynomial identities, then their tensor product satisfies some polynomial identities. This problem was open since the 1940's.
In 1975, through that sequence of codimensions, a WIS scientist found a connection between these algebras on the one hand, and The Representation Theory of the Symmetric Group on the other hand. In 1976 this connection allowed the WIS scientist to construct certain explicit identities for such algebras, answering a classical problem that was open since 1956.
In general, the sequence of codimentions method found numerous other applications, also in non-associative algebras, and is now recognized as a basic tool in the study of polynomial identities.