From the introduction: In his foundatory work [Ann. Math. (2) 79, 59–103 (1964; Zbl 0123.03101)], M. Gerstenhaber developed a theory of deformation of associative and Lie algebras. His theory links cohomologies of these algebras and the Gerstenhaber bracket giving ‘obstructions’ to deformations. Nijenhuis and Richardson noticed strong similarities between Gerstenhaber’s theory and the deformations of complex analytic structures on compact manifolds. They axiomatized the theory of deformations via the introduction of graded Lie algebras. The next step was to try to find more examples of structures entering under the scope of those ideas. One such example was given by the theory of deformations of homomorphisms [see A. Nijenhuis and R. W. Richardson jun., Bull. Am. Math. Soc. 73, 175–179 (1967; Zbl 0153.04402) cited as (*)].
The purpose of this Letter is to study another equivalence relation than the one used in (*) and to introduce a new type of cohomology. It enables us to deform simultaneously algebras and homomorphisms. The Letter is organised as follows:
In Section 2 we recall the concept of deformation for Lie algebras and the results obtained in (*) for the case of homomorphisms. In Section 3 we introduce a bundle which enables a more natural notion of equivalence. In Section 4 we explore the nature of the cohomology theory that should be associated with simultaneous deformations. In Section 5 we define the complex and give an explicit formula for the coboundary operator. We check fundamental properties and also that, in the particular case when the algebra structures are fixed, one recovers the classical notions of (*). Our approach differs from that of D. Arnal [in: Differential geometry and mathematical physics, Lect. Meet., Liège 1980, Leuven 1981, Math. Phys. Stud. 3, 3–15 (1983; Zbl 0523.17006)] in which the target Lie algebra is fixed. The idea of simultaneous deformations of algebra and their homomorphisms has already been exploited in [M. Gerstenhaber and S. D. Schack, Proc. Natl. Acad. Sci. USA 87, No. 1, 478–481 (1990; Zbl 0695.16005)] in the case of bialgebras. The algebras considered are associative, and the morphism is the coproduct. This led to the definition of a cohomology which agrees with ours to the order 1 but differs to the other orders. This is due to the fact that the deformations in the source and target spaces are strongly related.