Beyond Supersymmetry and Quantum Symmetry (an introduction to braided groups and braided matrices)
Beyond Supersymmetry and Quantum Symmetry (an introduction to braided groups and braided matrices)
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超越超对称和量子对称（编织群和编织矩阵简介）
DOI：
10.1142/9789814503761_0007
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
S. Majid
中科院分区：
文献类型：
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作者：
S. Majid
这是一个系统的介绍，为物理学家的理论代数和群与辫子统计，作为发展在过去三年的作者。有编织线，编织平面，编织矩阵和编织群都类似于超线，超平面等的主要思想是，格拉斯曼坐标之间的玻色费米统计现在被一般的编织统计取代，通常由杨巴克斯特矩阵R$。大多数代数证明最好通过画纽结和缠结图来完成，然而超对称的大多数构造似乎都能很好地推广。辫子统计的粒子是存在的，并且可以预期以这种方式来描述。与此同时，我们发现许多应用到普通量子群论：如何使量子群协变（辫子）张量积和自旋链，量子群的作用角变量，q$-Minkowski空间上的向量加法和半直积q-Poincar\'e群是迄今为止的主要应用。每个量子群都可以看作是一个辫群，所以这个理论包含了量子群论和超对称性。似乎也有一个丰富的编织几何理论，比超几何更一般，包括量子几何的方面。辫子导子服从辫子-莱布尼茨规则，并恢复通常的杰克逊$q$-导数作为1维的情况。
This is a systematic introduction for physicists to the theory of algebras and groups with braid statistics, as developed over the last three years by the author. There are braided lines, braided planes, braided matrices and braided groups all in analogy with superlines, superplanes etc. The main idea is that the bose-fermi $\pm1$ statistics between Grassmannn coordinates is now replaced by a general braid statistics $\Psi$, typically given by a Yang-Baxter matrix $R$. Most of the algebraic proofs are best done by drawing knot and tangle diagrams, yet most constructions in supersymmetry appear to generalise well. Particles of braid statistics exist and can be expected to be described in this way. At the same time, we find many applications to ordinary quantum group theory: how to make quantum-group covariant (braided) tensor products and spin chains, action-angle variables for quantum groups, vector addition on $q$-Minkowski space and a semidirect product q-Poincar\'e group are among the main applications so far. Every quantum group can be viewed as a braided group, so the theory contains quantum group theory as well as supersymmetry. There also appears to be a rich theory of braided geometry, more general than super-geometry and including aspects of quantum geometry. Braided-derivations obey a braided-Leibniz rule and recover the usual Jackson $q$-derivative as the 1-dimensional case.