The purpose of this paper is to explore some intriguing connections between Lie theory, relative homological algebra, invariant theory, and representations of finite dimensional algebras. In the process some results of Hilbert [20], Cartan and Eilenberg [7], Hochschild [21], Gelfand [ 171, and others are generalized, and a question of [Zl] is partly answered in Theorem 1.5. (Duflo seems to have stronger, unpublished, results on this problem.’ Sections 1 to 3 take place in a fairly general setting, whereas Section 4 focuses on semisimple Lie groups. The problem considered here goes back, in some sense, to Hilbert [20] (this paper was published 102 years ago!). To explain this, let me start by singling out two (out of many) fundamental contributions of the great mathematician. The first one is invariant theory, namely the study of the algebra (SV)n consisting of those polynomial functions on a finite dimensional vector space V* which are invariant under the action of a group H of linear transformations. The second one is the so-called syzygy theorem, which asserts, in today’s language, that the global dimension of the full symmetric algebra SV is dim V. This prompts the question: is there an algebra D, attached to V as naturally as SV is, on which H acts by automorphisms, such that the global dimension of the algebra B = DH of invariants is dim V? The most natural candidate is