The Ideals of Free Differential Algebras

We consider the free C-algebra q with N generators {ξi}i = 1,…,N, together with a set of N differential operators {∂i}i = 1,…,N that act as twisted derivations on q according to the rule ∂iξj = δij + qijξj∂i; that is, x , ∂i(ξjx) = δijx + qijξj∂ix, and ∂iC = 0. The subscript q on q stands for {qij}i, j {1,…,N} and is interpreted as a point in parameter space, q = {qij} CN2. A constant C q is a nontrivial element with the property ∂iC = 0, i = 1,…,N. To each point in parameter space there corresponds a unique set of constants and a differential complex. There are no constants when the parameters qij are in general position. We obtain some precise results concerning the algebraic surfaces in parameter space on which constants exist. Let q denote the ideal generated by the constants. We relate the quotient algebras ′q = q/ q to Yang–Baxter algebras and, in particular, to quantized Kac–Moody algebras. The differential complex is a generalization of that of a quantized Kac–Moody algebra described in terms of Serre generators. Integrability conditions for q-differential equations are related to Hochschild cohomology. It is shown that Hp( ′q, ′q) = 0 for p ≥ 1. The intimate relationship to generalized, quantized Kac–Moody algebras suggests an approach to the problem of classification of these algebras.