Von Neumann algebras, subfactors, topology and quantum physics

给定一个因子M的有限指数的子因子N，标准不变量是n的第n次上同调的第零上同调的0到无穷远的直和。M除以N的张量幂，可视为N-N-双模。这个分次向量空间有很多结构，特别是来自简单张量积的环的结构。这与代数几何中的“正则环”有着惊人的相似之处。事实上，已经证明了这个正则环的完备性给出了一个标准不变量与原始不变量相同的正则子因子！我们将详细探讨这种结构，特别是用于描述和分析正则环的标准不变量上的平面代数结构。各种组合结构，如Hadamard矩阵和拉丁方，产生了这些正则环，但存在严重的计算复杂性问题。随机n×n矩阵的大n极限也给出了这样的结构。从技术上讲，上面提到的N上的张量积是Connes张量积，我们正在研究这个张量积如何与量子物理直接相关。福克空间是二次量子化的数学引擎，二次量子化是使量子力学与狭义相对论兼容的必要过程。有费米子和玻色子两种版本。在我们的研究中，我们追求一个基于子代数和代数的更一般的Fock空间，小的代数对应于二次量子化中的标量，大的代数对应于第一个量子化的Hilbert空间，人们不能再像在费米子和玻色子的情况下那样自由地交换粒子。但是湮灭算符和创造算符仍然存在，并且形成了一个类似于射影簇的正则环的环。在我们的情况下，标准环/Fock空间具有更多的结构，即平面代数的结构。粗略地说，这为每一张地图提供了操作。想象一下像美国这样的国家的地图。当把Fock空间的一个元素放入地图上的每个州时，就会得到整个国家的输出，这是Fock空间的另一个元素。要说我们的结构是一个平面代数，意味着对于每一个可以在一张纸上画出的可想到的地图，都存在这样一个关于Fock空间的运算。

Given a subfactor N of finite index of a factor M, the standard invariant is the direct sum for n from 0 to infinity of the zeroth cohomology of the nth. tensor power of M over N, viewed as an N-N-bimodule. This graded vector space has a lot of structure, in particular that of a ring coming from the simple tensor product. This bears a striking resemblance to the "canonical ring" of algebraic geometry.Indeed it has been shown that the completion of this canonical ring gives a canonical subfactor whose standard invariant is the same as the original one! This structure will be explored in all its detail, especially the planar algbra structure on the standard invariant which is used to describe and analyse the canonical ring. Various combinatorial structures such as Hadamard matrices and Latin squares give rise to these canonical rings but there are serious problems of computational complexity. The large n limit of random nxn matrices also gives such structures. Technically the tensor product over N referred to above is the Connes tensor product and we are looking at how this tensor product may be directly relevant to quantum physics. Fock space is the mathematical engine of second quantization, a process that, among other things, is necessary to make quantum mechanics compatible with special relativity. There are fermionic and bosonic versions. In our research we are pursuing a more general Fock space based on a subalgebra and an algebra.The small algebra corresponds to the scalars in second quantization and the large one corresponds to the first quantized Hilbert space.One can no longer freely exchange particles as in the fermionic and bosonic situations. But annihilation and creation operators still exist and form a ring analogous to the canonical ring of a projective variety. In our situation the canonical ring/Fock space has much more structure, namely that of a planar algebra. Roughly speaking this gives operations for every map. Imagine the map of a country like the United States. Upon putting an element of the Fock space into each State on the map one would get an output for the whole country, which is another element of the Fock space. To say that our structure is a planar algebra means that such an operation on Fock space exists for every conceivable map that can be drawn on a piece of paper.