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.

31 pages. Plain TeX. Typos corrected, minor changes done and section 3.5.6 partially rewritten. To appear in Journal of Algebra