The cohomology theory of associative algebras has been developed by G. liochschild [1], [2], [3], and the 1-, 2-, and 3-dimensional cohomology groups have been interpreted with reference to classical notions of structure in his papers. Recently M. Ikeda has obtained, by a detailed analysis of Hochschild’s modules, an interesting structural characterization of the class of algebras whose 2-dimensional cohomology groups are all zero [5].
Cohomology theory for (associative) algebras was first established in general higher dimensionalities by G. Hochschild [3], [4], [5]. Algebras with vanishing 1-cohomology groups are separable semisimple algebras ([3], Theorem 4.1). On extending and refining our recent results [6], [8], [12], we establish in the present paper the following:Let n ≧ 2. Let A be an (associative) algebra (of finite rank) possessing a unit element 1 over a field Ω, and N be its radical.
In an earlier paper (3), a non-abelian cohomology theory (in the dimensions 0 and 1) for associative algebras was developed. One of the objectives was to obtain equivalence classes of crossed homomorphisms by considering inner automorphisms of the coefficient-algebra. This paper is an adaptation of the methods employed there to the case of Lie algebras. Throughout, all our Lie algebras will be over the field of real numbers, and finite-dimensional.
AbstractIn this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.
Note on the Cohomology Groups of Associative Algebras
