The Ideals of Free Differential Algebras

We consider the free $\mathbf{C}$-algebra ${\mathcal{B}}_q$ with $N$ generators $\{{\xi }_i{\}}_{i=1,\mathrm{.}\mathrm{.}\mathrm{.},N}$, together with a set of $N$ differential operators $\{{\mathrm{\partial }}_i{\}}_{i=1,\mathrm{.}\mathrm{.}\mathrm{.},N}$ that act as twisted derivations on ${\mathcal{B}}_q$ according to the rule ${\mathrm{\partial }}_i{\xi }_j={\delta }_{ij}+q_{ij}{\xi }_j{\mathrm{\partial }}_i$; that is, $\forall x \in {\cal B}q, \partial_i(\xi_jx) = \delta{ij}x + q_{ij}\xi_j\partial_i x,$and$\partial_i{\bf C} = 0$\mathrm{.}Thesuffix$q$on${\cal B}q$standsfor${q{ij}}_{i,j \in {1,…,N}}$ and is interpreted as a point in parameter space, $q=\{q_{ij}\}\in {\mathbf{C}}^{N^2}$. A constant $C \in {\cal B}_q$isanontrivialelementwiththeproperty$\partial_iC = 0, i = 1,…,N$. 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 ${\mathcal{I}}_q$ denote the ideal generated by the constants. We relate the quotient algebras ${\mathcal{B}}_q^{\mathrm{\prime }}={\mathcal{B}}_q\mathrm{/}{\mathcal{I}}_q$ 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.