We study five different types of the homology a Lie algebra over commutative ring which are naturally isomorphic fields. show that they not rings, even Z, and connections between them. In particular, we in case is flat as module. As an auxiliary result prove Koszul complex module M principal ideal domain connects exterior symmetric powers 0→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 purely acyclic.

Let g be a Lie algebra over a field F of characteristic zero, let C be a certain tensor category of representations of g, and C a certain category of duals. In [M 4] we associated to C and C by a Tannaka reconstruction a monoid M with a coordinate ring F [M ] of matrix coefficients, as well as a Lie algebra Lie(M). We interpreted the Tannaka monoid M algebraic geometrically as a weak algebraic ...