离散极小曲面代数
Discrete Minimal Surface Algebras
摘要：
本文将离散极小曲面代数（DMSA）看作是高维球面上极小曲面的广义非交换模拟.这些代数自然出现在膜理论中，它们的表示序列被用作正则化。在证明了代数的定义关系是一致的，并且可以计算包络代数的基之后，我们给出了关于sl（n）的子集的DMSA的几个明确的例子（任何半单李代数本身提供了一个平凡的例子）。一类特殊的DMSA是Yang-Mills代数。引入表示图来研究维数d ≤ 4的DMSA的表示，表示的性质与图的性质有关.张量积的表示图（一般）是对应图的笛卡儿积。我们提供了明确的例子，不可约表示和重合的特征值，分类所有的酉表示相应的代数。
英文摘要：
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
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