坐标代数与量子空间的对合表示
Involutive representations of coordinate algebras and quantum spaces
摘要：
证明了与具有$\frak{su}(2)$非交换性的量子空间相关的坐标算子的$\frak{su}(2)$李代数可以方便地用$SO(3)$-协变多微分对合表示表示.我们证明了由量子化映射作用于通常的指数函数所得到的量子化平面波是由算子的极分解结合源于$SU(2)$的Wigner定理的约束所确定的。选择$^*$ -表示的一个子族，我们证明了所得到的星积等价于有限维李代数$\mathfrak{su}(2)$对偶的Poisson流形的Kontsevich积。我们讨论的结果，表明一种方法来扩展的建设，任何半简单的非单连通李群和目前的非交换标量场理论是免费的微扰UV/IR混合。
英文摘要：
We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently represented by $SO(3)$-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for $SU(2)$. Selecting a subfamily of $^*$-representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra $\mathfrak{su}(2)$. We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.
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