Algebraic deformation theory

The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations are also considered. In this paper, we consider general questions on deformations of Lie algebras over a field of characteristic zero, and related problems of computing cohomology with coefficients in adjoint representations. We consider the construction of a versal family and the nature of obstructions to the extension of deformations. Our aim is to carry over general constructions of the modern theory of deformations and related properties of the cohomology of (local) commutative algebras to Lie algebras in parallel with the papers [3], [10] and [11]. 1. We shall require some information on the Harrison cohomology of commutative rings (see [12] and [8]). Harrison cohomology is the cohomology in the category of commutative rings. We shall only require the 1-dimensional and 2-dimensional cohomology, and restrict ourselves to their explicit definition. (In contrast with the traditional indexing, we consider Harrison cohomology with the indices increased by 1.) Let A be a commutative k-algebra, where k is a field of characteristic zero, and let N be an A-module. We write down a cochain complex N d0 → K1 d1 → K2, whereK1 = Hom k(A,N) andK 2 is the subspace of Hom k(S 2A,N) consisting of the maps φ for which aφ(b, c) − φ(ab, c) − cφ(a, b) + φ(a, bc) = 0 for any three elements a, b, c ∈ A. The differentials d0 and d1 are arranged so that d0(n)(a) = an, a ∈ A, n ∈ N, d1θ(a, b) = aθ(b)−θ(ab)+bθ(a), a, b ∈ A. The spaces H1 Harr(A;N) and H 2 Harr(A;N) of 1-dimensional and 2-dimensional cohomology are by definition Ker d1/Im d0 and K 2/Im d1, respectively. From the definition one can see that 1-cocycles are derivations. Let A be an algebra, m a maximal ideal, and A/m ∼= k. Then H1 Harr(A;k) ∼= (m/m2)∗. In other words, H1 Harr(A;k) is isomorphic to the space of homomorphisms A → k[t]/(t2) for which the kernel of the composition A → k[t]/(t2) → k is m. The 2-dimensional cohomology is interpreted as extension (see [9]). An extension of the algebra A by a module N is an exact sequence 0→ N i → B π → A → 0, where B is a commutative algebra and i(N) is an ideal in B with trivial multiplication such that bi(n) = π(b)n for b ∈ B and n ∈ N . To an 1980 Mathematics Subject Classification (1985 Revision). Primary 17B56; Secondary 13D03.