算子值概率论的一个shuffle代数观点
A shuffle algebra point of view on operator-valued probability theory
摘要：
我们扩展洗牌代数的角度标量值非交换概率论的算子值的情况下。给定一个算子值概率空间，一个代数$B$作用在它上面（在左边和右边），我们将$B$上的多线性映射的算子与一个随机变量的算子值分布和自由累积量联系起来。这些运算符定义了非交叉分区的PROS的表示。利用高阶范畴理论中的概念，特别是$2$ -幺半范畴，我们定义了一个unshuffle的霍普夫代数上的一个基本PROS的概念。我们介绍了一个PROS的话插入，并表明，后者和PROS的非交叉分区unshuffle的霍普夫代数（在$2$ -monoidal意义上）。这两个关系的手段的映射unshuffle双代数（在$2$ -monoidal意义上），我们称之为分裂映射。最后，我们得到了一个对应于双收集同态的shuffle代数中自由矩-累积量关系的左半shuffle不动点方程。右半洗牌和洗牌法律的布尔和单调的非交换概率论的框架内解释，分别。保留字：算子值非交换概率论，高阶范畴理论，对偶范畴，运算，固有运算，PROS，洗牌代数，半洗牌
英文摘要：
We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to the operator-valued case. Given an operator-valued probability space with an algebra $B$ acting on it (on the left and on the right), we associate operators in the operad of multilinear maps on $B$ to the operator-valued distribution and free cumulants of a random variable. These operators define a representation of a PROS of non-crossing partitions. Using concepts from higher category theory, specifically $2$-monoidal categories, we define a notion of unshuffle Hopf algebra on an underlying PROS. We introduce a PROS of words insertions and show that both the latter and the PROS of non-crossing partitions are unshuffle Hopf algebras (in a $2$-monoidal sense). The two relate by mean of a map of unshuffle bialgebras (in a $2$-monoidal sense) which we call the splitting map. Ultimately, we obtain a left half-shuffle fixed point equation corresponding to free moment-cumulant relations in a shuffle algebra of bicollection homomorphisms on the PROS of words insertions. Right half-shuffle and shuffle laws are interpreted in the framework of boolean and monotone non-commutative probability theory, respectively. Keywords: operator-valued non-commutative probability theory, higher category theory, duoidal categories, operads, properads, PROS, shuffle algebra, half-shuffles
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