关于非齐次量子群的几何
On the Geometry of Inhomogeneous Quantum Groups
摘要：
本文首先对q-群上的微分学进行了教学介绍，分析了微分学与q-李代数的关系。研究了双协变微分的等价定义，并解释了它们的几何解释。定义了向量场、压缩算子和李导数，并讨论了它们的性质.在回顾了（多参数）线性q群$GL_{q,r}(N)$的几何之后，我们构造了非齐次q群$IGL_{q,r}(N)$作为$GL_{q,r}(N+1)$的投影，即作为$GL_{q,r}(N+1)$关于合适的Hopf代数理想的商。由$GL_{q,r}(N)$ q-子群乘平移给出的$IGL_{q,r}(N)$的半直积结构易于分析. $IGL_{q,r}(N)$上的一个双协变演算被显式地得到为$GL_{q,r}(N+1)$上的一个双协变演算的投影。$IGL_{q,r}(N)$的泛包络代数及其$R$ -矩阵公式是沿着相同的线构造的。我们在正交和辛的情况下进行类似的。通过从$B_{n+1}, C_{n+1},D_{n+1}$的投影，我们找到了$B_n,C_n,D_n$系列的非齐次多参数q群。我们给出了一个$R$ -矩阵公式，并讨论了实形式。我们研究了它们的泛包络代数和微分演算。特别地，我们得到了Poincaré群$ISO_q(3,1)$的无扰动最小变形上的双协变演算.投影过程也被用来构造微分演算的多参数q-正交平面上的任何尺寸$N$。研究了实形式，特别地，我们得到了一个q-Minkowski空间和它的q-变形相空间，它们具有厄米特算子$x^a$和$p_a$ .
英文摘要：
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their geometrical interpretation is explained. Vectorfields, contraction operator and Lie derivative are defined and their properties discussed. After a review of the geometry of the (multiparametric) linear q-group $GL_{q,r}(N)$ we construct the inhomogeneous q-group $IGL_{q,r}(N)$ as a projection from $GL_{q,r}(N+1)$, i.e. as a quotient of $GL_{q,r}(N+1)$ with respect to a suitable Hopf algebra ideal. The semidirect product structure of $IGL_{q,r}(N)$ given by the $GL_{q,r}(N)$ q-subgroup times translations is easily analized. A bicovariant calculus on $IGL_{q,r}(N)$ is explicitly obtained as a projection from the one on $GL_{q,r}(N+1)$. The universal enveloping algebra of $IGL_{q,r}(N)$ and its $R$-matrix formulation are constructed along the same lines. We proceed similarly in the orthogonal and symplectic case. We find the inhomogeneous multiparametric q-groups of the $B_n,C_n,D_n$ series via a projection from $B_{n+1}, C_{n+1},D_{n+1}$. We give an $R$-matrix formulation and discuss real forms. We study their universal enveloping algebras and differential calculi. In particular we obtain the bicovariant calculus on a dilatation-free minimal deformation of the Poincaré group $ISO_q(3,1)$. The projection procedure is also used to construct differential calculi on multiparametric q-orthogonal planes in any dimension $N$. Real forms are studied and in particular we obtain a q-Minkowski space and its q-deformed phase-space with hermitian operators $x^a$ and $p_a$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
