We define the affine VW supercategory , which arises from studying the action of the periplectic Lie superalgebra \(\mathfrak {p}(n)\) on the tensor product \(M\otimes V^{\otimes a}\) of an arbitrary representation M with several copies of the vector representation V of \(\mathfrak {p}(n)\). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in \(\mathfrak {p}(n)\otimes \mathfrak {p}(n)\). When M is the trivial representation, the action factors through the Brauer supercategory \(\textit{s}\mathcal {B}{} \textit{r}\). Our main result is an explicit basis theorem for the morphism spaces of and, as a consequence, of \(\textit{s}\mathcal {B}{} \textit{r}\). The proof utilises the close connection with the representation theory of \(\mathfrak {p}(n)\). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.

中文翻译：

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仿射大众超级类别

我们定义了仿射VW超类，它是通过研究李维超代数\（\ mathfrak {p}（n）\）对一个张量积\（M \ otimes V ^ {\ otimes a} \）的作用而产生的任意表示中号与所述矢量表示的几个拷贝V的（\ mathfrak {p}（N）\）\。在一般线性群表示的背景下，它的作用类似于简并仿射Hecke代数的作用；主要障碍是\（\ mathfrak {p}（n）\ otimes \ mathfrak {p}（n）\）中缺少二次Casimir元素。当M是平凡的表示形式时，通过Brauer超类别的作用因子\（\ textit {s} \ mathcal {B} {} \ textit {r} \）。我们的主要结果是\（\ textit {s} \ mathcal {B} {} \ textit {r} \）和的态射空间的显式基本定理。证明利用了与\（\ mathfrak {p}（n）\）的表示理论的紧密联系。作为应用程序，我们明确描述了所有同构代数的中心，并表明它在传递给相关的渐变和变形的情况下表现良好。