The Ideals of Free Differential Algebras

Abstract

We consider the free ${\bf C}$-algebra ${\cal B}_q$ with $N$ generators $\{ξ_i\}_{i = 1,...,N}$, together with a set of $N$ differential operators $\{\partial_i\}_{i = 1,...,N}$ that act as twisted derivations on ${\cal B}_q$ according to the rule $\partial_iξ_j = δ_{ij} + q_{ij}ξ_j\partial_i$; that is, $\forall x \in {\cal B}_q, \partial_i(ξ_jx) = δ_{ij}x + q_{ij}ξ_j\partial_i x,$ and $\partial_i{\bf C} = 0$. The suffix $q$ on ${\cal B}_q$ stands for $\{q_{ij}\}_{i,j \in \{1,...,N\}}$ and is interpreted as a point in parameter space, $q = \{q_{ij}\}\in {\bf C}^{N^2}$. A constant $C \in {\cal B}_q$ is a nontrivial element with the property $\partial_iC = 0, i = 1,...,N$. To each point in parameter space there correponds a unique set of constants and a differential complex. There are no constants when the parameters $q_{ij}$ are in general position. We obtain some precise results concerning the algebraic surfaces in parameter space on which constants exist. Let ${\cal I}_q$ denote the ideal generated by the constants. We relate the quotient algebras ${\cal B}_q' = {\cal B}_q/{\cal I}_q$ 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 $H^p({\cal B}_q',{\cal B}_q') = 0$ for $p \geq 1$. The intimate relationship to generalized, quantized Kac-Moody algebras suggests an approach to the problem of classification of these algebras.