辫子矩阵的标准分解和对角化：显式构造及应用
Canonical factorization and diagonalization of Baxterized braid matrices: Explicit constructions and applications
摘要：
对应于矢量表示的辫状矩阵$\hat{R}(θ)$被频谱分解，获得出现在分解中的每个投影仪$P_{i}$的系数的比率$f_{i}(θ)/f_{i}(-θ)$。这直接产生一个分解$(F(-θ))^{-1}F(θ)$的编织矩阵，也意味着关系$\hat{R}(-θ)\hat{R}(θ)=I$。这是实现$GL_{q}(n),SO_{q}(2n+1),SO_{q}(2n),Sp_{q}(2n)$的所有$n$，也为其他各种有趣的情况下，包括8-顶点矩阵。我们解释如何限制$θ\to \pm \infty$可以解释为提供标准（非分解）编织矩阵的分解。一个系统的方法对角化的投影仪，因此辫子矩阵的显式构造$GL_{q}(2),GL_{q}(3),SO_{q}(3),SO_{q}(4),Sp_{q}(4)$和其他各种情况下，如8-顶点之一。对于一个特定的嵌套序列的投影仪对角化得到的所有维度。在每个因子$F(θ)$中，我们的对角化再次分解出对作为对角矩阵的谱参数$θ$的所有依赖性。在对角化器中实现的规范属性是行的相互正交性。指出了我们的形式主义在构造$L-$算子和转移矩阵中的应用。在附录中，我们的类型的因式分解相比，其他作者提出的另一个。
英文摘要：
Braid matrices $\hat{R}(θ)$, corresponding to vector representations, are spectrally decomposed obtaining a ratio $f_{i}(θ)/f_{i}(-θ)$ for the coefficient of each projector $P_{i}$ appearing in the decomposition. This directly yields a factorization $(F(-θ))^{-1}F(θ)$ for the braid matrix, implying also the relation $\hat{R}(-θ)\hat{R}(θ)=I$.This is achieved for $GL_{q}(n),SO_{q}(2n+1),SO_{q}(2n),Sp_{q}(2n)$ for all $n$ and also for various other interesting cases including the 8-vertex matrix.We explain how the limits $θ\to \pm \infty$ can be interpreted to provide factorizations of the standard (non-Baxterized) braid matrices. A systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for $GL_{q}(2),GL_{q}(3),SO_{q}(3),SO_{q}(4),Sp_{q}(4)$ and various other cases such as the 8-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. In each factor $F(θ)$ our diagonalization again factors out all dependence on the spectral parameter $θ$ as a diagonal matrix. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. Applications of our formalism to the construction of $L-$operators and transfer matrices are indicated. In an Appendix our type of factorization is compared to another one proposed by other authors.
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