On the Cohomology Theory for Associative Algebras

Abstract

The cohomology theory for associative algebras may be regarded as a superstructure built upon the classical representation theory. Let us consider an algebra 21 over a field 4. Suppose that we are given a finite dimensional linear space 3 over &1 which is simultaneously a representation space and an antirepresentation space for W. Then, to each element a of 21, there correspond two linear transformations u -* u -a and u a -u of A. If, in addition to the usual requirements, these transformations commute with each other, i.e. if a,(ua2) = (a,-u)-a2, for all a1, a2 e 21 and all u e $, we say that $ is an 2-module. The notions of the cohomology theory are based on this set-up, as follows: The set of all m-linear mappings of 21 into $, when regarded as an additive group in the usual way, is called the group of the m-dimensional $-cochains (or (m, $)-cochains) of W. We shall denote it by Cm(2f, p). The group C'(21, p) is identified with the additive group of p. The coboundary operator, 6, as defined by Eilenberg and MacLane, maps each Cm(21, p) linearly into Cm'+(2t, p), as follows: Let f e Cz'(, p), a,, *, am e W; then (6f) {a, , amA} = a, f{aI, * * 7 am?, + Z-m~n (-l~tf{ai, ** *, a~aai +, *, a.+,} + (1)m+lf{ai, am am,+. For any cochain f, we have 6(6f) = 0. Hence the cohomology groups may be defined by the well known procedure of combinatorial topology. An element f E Cm(?2, p) is called an m-dimensional cocycle if cf = 0. An element of the form (5g, where g e Cml(2[, p) is called an m-dimensional coboundary. These form a subgroup B'(2, p) of the group Zm(2, p) of the m-dimensional cocycles. The m-dimensional cohomology group Hm(2, p) is defined as the quotient group Zm(21, p)/Bm(21, p). For m = 1 and m = 2, these groups have been interpreted with reference to classical notions of structure in a previous paper.' Although no such interpretation has as yet been found for the higher dimensional groups it is possible to make a rough forecast of the r6le which the cohomology theory will play in the study of the structure of algebras. There is good evidence that it will provide appropriate methods for investigating a variety of structural characteristics which seem to lie outside the domain of the classical representation theory. The trend which takes its logical departure in Wedderburn's structure theorems has been to intensity and refine the structural analysis only at the level of maximal regularity.2 Thus, the radical of an algebra is split off by Wedder-