论文标题
十倍的方式
The Tenfold Way
论文作者
论文摘要
十倍的方式在2010年左右的物理学中变得很重要：这意味着有十种根本不同的物质。但是它可以追溯到1964年，当时C. T. C. Wall分类了真正的超级分区代数。这些是有限维真实$ \ mathbb {z}/2 $ raded代数，每个非零均质元素都是可逆的。他发现，除了$ \ mathbb {r} $，$ \ mathbb {c} $和$ \ mathbb {h} $之外，这些$ \ \ \ m mathbb {h} $均给出了纯粹的超级分区代数，还有七个。他还表明，这十个代数都是真实或复杂的克利福德代数。八个真实的八个莫里塔等效类代表了真正的克利福德代数的所有莫里塔等效类别，而两个复杂的代数则对复杂的克利福德代数也是如此。因此，十倍的方式将真实且复杂的周期性结合在一起。在这篇说明性文章中，我们可以快速证明有十个真正的超级分区代数，并对十倍方式的应用说一些话。
The tenfold way became important in physics around 2010: it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when C. T. C. Wall classified real super division algebras. These are finite-dimensional real $\mathbb{Z}/2$-graded algebras where every nonzero homogeneous element is invertible. He found that besides $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$, which give purely even super division algebras, there are seven more. He also showed that these ten algebras are all real or complex Clifford algebras. The eight real ones represent all eight Morita equivalence classes of real Clifford algebras, and the two complex ones do the same for the complex Clifford algebras. The tenfold way thus unites real and complex Bott periodicity. In this expository article we give a quick proof that there are ten real super division algebras, and say a bit about applications of the tenfold way.
