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Invariant Markov semigroups on quantum homogeneous spaces
Journal of Noncommutative Geometry ( IF 0.9 ) Pub Date : 2021-09-28 , DOI: 10.4171/jncg/404
Biswarup Das 1 , Uwe Franz 2 , Xumin Wang 2
Affiliation  

Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected co-ideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected co-ideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.

The classical sphere {$S^{N-1}$}, the free sphere {$S^{N-1}_+$}, and the half-liberated sphere {$S^{N-1}_*$} are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behaviour of the eigenvalues of their Laplace operator.



中文翻译:

量子齐次空间上的不变马尔可夫半群

研究了线性泛函的不变性和量子齐次空间上函数代数的线性映射,特别是对于预期共理想 *-子代数的特殊情况。建立了这种不变函数之间的几个一对一的对应关系。添加一个正条件,这会产生作用于预期共理想 *-子代数的不变量子马尔可夫半群和基础紧致量子群上的某些状态卷积半群的一对一对应关系。这提供了一种在这些量子齐次空间上对不变量子马尔可夫半群进行分类的方法。这些半群的生成元被视为这些空间上的拉普拉斯算子。

经典球体 {$S^{N-1}$}、自由球体 {$S^{N-1}_+$} 和半解放球体 {$S^{N-1}_*$被视为示例,并对这些球体上的马尔可夫半群的生成器进行分类。我们根据其拉普拉斯算子的特征值的渐近行为计算这三个球族的谱维数。

更新日期:2021-10-13
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