标题： T-Minkowski非交换时空I：庞加莱群、微分演算与辫积 显示英文标题

标题： T-Minkowski noncommutative spacetimes I: Poincaré groups, differential calculi and braiding

摘要： 本文介绍并研究了一类非交换时空，我将其称为“T-闵可夫斯基”，其量子庞加莱群的等距对称性表现出独特且物理上有意义的特性。 值得注意的是，洛伦兹子群上的坐标保持交换性，而变形仅限于平移（因此名称中的T），它们在洛伦兹群上表现为一组可积的矢量场。 这类似于Majid的双交叉乘积构造，尽管我的方法允许描述具有包括常数矩阵以及坐标线性项的对易关系的时空（所得结构为一个中心扩展的李代数）。 此外，我要求可以定义量子庞加莱群的协变编织张量积表示，描述N点代数。 这也意味着在非交换时空上存在4维双协变微分演算。 所得模型都可以通过RTT关系（以及齐次时空的RXX、RXY和RXdX关系，编织结构和微分演算）用数值三角形R矩阵来描述。 我发现的R矩阵与庞加莱群上的三角形r矩阵一一对应，洛伦兹生成元中不含二次项。 这些已由Zakrzewski分类，总计有16个不等价模型。 本文是系列文章的第一篇，重点在于识别所有满足我假设条件的量子庞加莱群及其相关的量子齐次时空、微分演算和编织构造。 显示英文摘要

摘要： This paper introduces and investigates a class of noncommutative spacetimes that I will call ``T-Minkowski,'' whose quantum Poincar\'e group of isometries exhibits unique and physically motivated characteristics. Notably, the coordinates on the Lorentz subgroup remain commutative, while the deformation is confined to the translations (hence the T in the name), which act like an integrable set of vector fields on the Lorentz group. This is similar to Majid's bicrossproduct construction, although my approach allows the description of spacetimes with commutators that include a constant matrix as well as terms that are linear in the coordinates (the resulting structure is that of a centrally-extended Lie algebra). Moreover, I require that one can define a covariant braided tensor product representation of the quantum Poincar\'e group, describing the algebra of N-points. This also implies that a 4-dimensional bicovariant differential calculus exists on the noncommutative spacetime. The resulting models can all be described in terms of a numerical triangular R-matrix through RTT relations (as well as RXX, RXY and RXdX relations for the homogeneous spacetime, the braiding and the differential calculus). The R-matrices I find are in one-to-one correspondence with the triangular r-matrices on the Poincar\'e group without quadratic terms in the Lorentz generators. These have been classified, up to automorphisms, by Zakrzewski, and amount to 16 inequivalent models. This paper is the first of a series, focusing on the identification of all the quantum Poincar\'e groups that are allowed by my assumptions, as well as the associated quantum homogeneous spacetimes, differential calculi and braiding constructions.