论文标题
分区代数有价值的简单复合物
Division algebra valued energized simplicial complexes
论文作者
论文摘要
我们查看由h：g到k定义的连接laplacians l，g，其中g是有限的集合，k是一个规范的划分环，不需要交换，也不需要相关性，但具有共轭导致标准作为h^* h的平方根。目标空间k可以是标准的真实分区代数，例如四元组或代数数字字段，例如二次场。对于结果的一部分，我们甚至可以假设K是希尔伯特空间上的操作员代数的Banach代数。然后，G上的k值函数h然后定义了连接矩阵l，g，其中条目在k中。我们表明，l和g的dieudonne决定因素都等于g上所有字段值的乘积的abelianization。 g（x，y）。如果k是复数的场C，则可以研究h（g，h）的依赖性h（g，h）的光谱。具有简单频谱的一组矩阵定义了a | g | - 二维非紧密的kaehler歧管，该歧管通常是断开连接的，我们可以为此计算每个连接的组件的基本组。
We look at connection Laplacians L,g defined by a field h:G to K, where G is a finite set of sets and K is a normed division ring which does not need to be commutative, nor associative but has a conjugation leading to the norm as the square root of h^* h. The target space K can be a normed real division algebra like the quaternions or an algebraic number field like a quadratic field. For parts of the results we can even assume K to be a Banach algebra like an operator algebra on a Hilbert space. The K-valued function h on G then defines connection matrices L,g in which the entries are in K. We show that the Dieudonne determinants of L and g are both equal to the abelianization of the product of all the field values on G. If G is a simplicial complex and h takes values in the units U of K, then g^* is the inverse of L and the sum of the energy values is equal to the sum of the Green function entries g(x,y). If K is the field C of complex numbers, we can study the spectrum of L(G,h) in dependence of the field h. The set of matrices with simple spectrum defines a |G|-dimensional non-compact Kaehler manifold that is disconnected in general and for which we can compute the fundamental group of each connected component.
