自由微分代数的理想
The Ideals of Free Differential Algebras
摘要：
我们考虑自由的${\bf C}$ -代数${\cal B}_q$与$N$生成元$\{ξ_i\}_{i = 1,...,N}$，以及一组$N$微分算子$\{\partial_i\}_{i = 1,...,N}$，根据规则$\partial_iξ_j = δ_{ij} + q_{ij}ξ_j\partial_i$，即$\forall x \in {\cal B}_q, \partial_i(ξ_jx) = δ_{ij}x + q_{ij}ξ_j\partial_i x,$和$\partial_i{\bf C} = 0$作为${\cal B}_q$上的扭曲导子。${\cal B}_q$上的后缀$q$代表$\{q_{ij}\}_{i,j \in \{1,...,N\}}$，并被解释为参数空间$q = \{q_{ij}\}\in {\bf C}^{N^2}$中的点。常量$C \in {\cal B}_q$是具有属性$\partial_iC = 0, i = 1,...,N$的非平凡元素。参数空间中的每一点都对应着一组唯一的常数和一个微分复形。当参数$q_{ij}$处于常规位置时，没有常量。我们得到了关于参数空间中存在常数的代数曲面的一些精确结果。令${\cal I}_q$表示由常数生成的理想。我们将商代数${\cal B}_q' = {\cal B}_q/{\cal I}_q$与杨-巴克斯特代数，特别是与量子化卡茨-穆迪代数联系起来。微分复形是用塞尔生成元描述的量子化卡茨-穆迪代数的微分复形的推广。$q$ -微分方程的可积性条件与Hochschild上同调有关。结果表明，$p \geq 1$的$H^p({\cal B}_q',{\cal B}_q') = 0$。广义的，量化的卡茨-穆迪代数的亲密关系表明，这些代数的分类问题的一种方法。
英文摘要：
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.
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