论文标题
对操作员值概率理论的避免代数的观点
A shuffle algebra point of view on operator-valued probability theory
论文作者
论文摘要
我们将标量价值的非共同概率理论的随机代数视角扩展到了操作员价值案例。给定具有代数$ b $作用的操作员值概率空间（在左侧和右侧），我们将$ b $上的多连线图的操作员与操作员可价值分布和随机变量的自由累积物相关。这些操作员定义了非交叉分区的优点的表示。使用来自高级类别理论的概念，特别是$ 2 $ - 单型类别，我们在基础优点上定义了一个未解决的Hopf代数的概念。我们介绍了单词插入的专业人士，并表明后者和非交叉分区的专业人士都是未解决的Hopf代数（以$ 2 $ - 单型的感觉）。这两个依靠地图（以$ 2 $ - monoidal Sense）为平均值，我们称为分裂地图。最终，我们获得了与单词插入的专利相同的同型同态同态的代数中的自由力矩肿瘤关系相对应的左半垫圈固定点方程。右半剃须和洗牌定律分别在布尔和单调非共同概率理论的框架中解释。 关键字：运营商价值的非交通概率理论，高等类别理论，uluoidal类别，Operads，Poserads，Pros，Pros，Shuffle Algebra，Half Shuffles
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
