论文标题
非理性的$θ$ - 定型汽车的分类$ c^*$ - 代数
Classification of irrational $Θ$-deformed CAR $C^*$-algebras
论文作者
论文摘要
给定一个偏斜的对称的真实$ n \ times n $矩阵$θ$，我们考虑了$* $* $* ngebra的通用$ c^* $ - 代数$ \ mathsf {car}_θ$由$ a_1，\ ldots，\ ldots，\ ldots，a_n $ cyvessed \ a_n $ cyverty \ [a_n $ \ [a_i^* a_j = e^{2πiθ_{i，j}} a_j a_i^*，\] \ [a_i a_j = e^{ - 2πiθ_{i，j}} a_j a_i_i。 \]我们证明$ \ mathsf {car}_θ$具有$ c（k_n）$ - 结构，其中$ k_n = \ left [0，\ frac {1} {2} {2} {2} \ right]^n $是hypercube并描述纤维。我们将$ \ mathsf {car}_θ$的不可约表示，以较高维度的非惯用态度的不可约表示。我们证明，对于给定的非理性偏压$θ_1$，只有有限的$θ_2$有限的，因此$ \ Mathsf {car} _ {θ_1} \ simeq \ simeq \ Mathsf {car} _ {θ_2} $。也就是说，$ \ Mathsf {car} _ {θ_1} \ simeq \ simeq \ Mathsf {car} _ {θ_2} $ insum $（θ_1）_ {ij} = \ pm（θ_2） $ \ {（i，j）：i <j，\ i，j = 1，\ ldots，n \} $。对于$ n = 2 $，我们给出一个完整的分类：$ \ mathsf {car} _ {θ_1} \ simeq \ mathsf {car} _ {θ_2} $ iff $θ_1= \pmpMθ_2\pmθ_2\ mod \ mod \ mathbb {z} $。
Given a skew-symmetric real $n\times n$ matrix $Θ$ we consider the universal enveloping $C^*$-algebra $\mathsf{CAR}_Θ$ of the $*$-algebra generated by $a_1, \ldots, a_n$ subject to the relations \[ a_i^* a_i + a_i a_i^* = 1, \ \] \[ a_i^* a_j = e^{2 πi Θ_{i,j}} a_j a_i^*, \] \[ a_i a_j = e^{-2 πi Θ_{i,j}} a_j a_i. \] We prove that $\mathsf{CAR}_Θ$ has a $C(K_n)$-structure, where $K_n = \left[ 0,\frac{1}{2} \right]^n$ is the hypercube and describe the fibers. We classify irreducible representations of $\mathsf{CAR}_Θ$ in terms of irreducible representations of a higher-dimensional noncommutative torus. We prove that for a given irrational skew-symmetric $Θ_1$ there are only finitely many $Θ_2$ such that $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$. Namely, $\mathsf{CAR}_{Θ_1} \simeq \mathsf{CAR}_{Θ_2}$ implies $(Θ_1)_{ij} = \pm (Θ_2)_{σ(i,j)} \mod \mathbb{Z}$ for a bijection $σ$ of the set $\{(i,j) : i < j, \ i, j = 1, \ldots, n\}$. For $n = 2$ we give a full classification: $\mathsf{CAR}_{θ_1} \simeq \mathsf{CAR}_{θ_2}$ iff $θ_1 = \pm θ_2 \mod \mathbb{Z}$.
