论文标题
超对称单类别
Supersymmetric monoidal categories
论文作者
论文摘要
根据Kapranov的想法，我们提出了超对称性单体超类别的想法。粗略地，这是一个单型类别，其中对象和形态为$ {\ bf z}/2 $，配备了同构，$ x \ otimes y \ to y \ otimes x $ otimes x $ ot parity $ \ vert x \ vert x \ vert x \ vert y \ vert y \ vert y \ vert y \ vert $ on phyt on homogene oken homogeene okenogene okenogene对象。有两个基本的例子：旋转组的类固醇，以及配备半张量产品的酷儿矢量空间的类别；可以从这些示例中得出其他重要示例（例如线性自旋物种的类别）。还有两个通用结构。第一个是超类的外部代数（由于ganter-kapranov）。第二个是我们介绍的一个名为Clifford Eversion的结构。这定义了一定的2类超对称单类超类别与对称的对称单相关超类别之间的等效性。我们利用理论更好地理解了酷儿超级的某些方面，例如q-合作函数和schur-sergeev二元性理论中$ \ sqrt {2} $的某些因素。
We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are ${\bf Z}/2$-graded, equipped with isomorphisms $X \otimes Y \to Y \otimes X$ of parity $\vert X \vert \vert Y \vert$ on homogeneous objects. There are two fundamental examples: the groupoid of spin-sets, and the category of queer vector spaces equipped with the half tensor product; other important examples can be derived from these (such as the category of linear spin species). There are also two general constructions. The first is the exterior algebra of a supercategory (due to Ganter--Kapranov). The second is a construction we introduce called Clifford eversion. This defines an equivalence between a certain 2-category of supersymmetric monoidal supercategories and a corresponding 2-category of symmetric monoidal supercategories. We use our theory to better understand some aspects of the queer superalgebra, such as certain factors of $\sqrt{2}$ in the theory of Q-symmetric functions and Schur--Sergeev duality.
