The Ideals of Free Differential Algebras
| dc.creator | Fronsdal, C. | |
| dc.creator | Galindo, A. | |
| dc.date | 1998-06-12 | |
| dc.date | 1999-11-17 | |
| dc.date.accessioned | 2026-07-07T05:25:01Z | |
| dc.date.available | 2026-07-07T05:25:01Z | |
| dc.description | 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. | |
| dc.description | 31 pages. Plain TeX. Typos corrected, minor changes done and section 3.5.6 partially rewritten. To appear in Journal of Algebra | |
| dc.identifier | https://arxiv.org/abs/math/9806069 | |
| dc.identifier | http://arxiv.org/abs/math/9806069 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/77036 | |
| dc.subject | Quantum Algebra | |
| dc.title | The Ideals of Free Differential Algebras | |
| dc.type | text |