论文标题
Brauer类别的表示理论II：咖喱代数
The representation theory of Brauer categories II: curried algebra
论文作者
论文摘要
$ \ mathfrak {gl}（v）= v \ otimes v^*$的表示形式是线性地图$μ\ colon \ colon \ mathfrak {gl}（v）（v）\ otimes m \ to m $ to M $满足某种身份。通过咖喱，给出线性地图$μ$等同于给出线性地图$ a \ colon v \ otimes m \ otimes m \ to v \ otimes m $，并且可以将条件转换为$μ$的条件，以表示$ a $ a $的条件。此替代配方不使用$ v $的双重，并且对于张量的类别中的任何对象$ v $ $ \ mathcal {c} $都是有意义的。我们将咖喱通用线性代数的此类对象表示为$ v $。可以为从矢量空间及其双重构建的许多代数进行咖喱过程，我们详细检查了几个案例。我们表明，许多众所周知的组合类别等同于线性物种张量类别中熟悉的代数的咖喱形式。例如，名义brauer类别“”是symbletic lie代数的咖喱形式。该观点将这些类别置于新的角度，具有一些技术应用程序，并提出了新的探索方向。
A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $μ\colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $μ$ is equivalent to giving a linear map $a \colon V \otimes M \to V \otimes M$, and one can translate the condition for $μ$ to be a representation to a condition on $a$. This alternate formulation does not use the dual of $V$, and makes sense for any object $V$ in a tensor category $\mathcal{C}$. We call such objects representations of the curried general linear algebra on $V$. The currying process can be carried out for many algebras built out of a vector space and its dual, and we examine several cases in detail. We show that many well-known combinatorial categories are equivalent to the curried forms of familiar Lie algebras in the tensor category of linear species; for example, the titular Brauer category "is" the curried form of the symplectic Lie algebra. This perspective puts these categories in a new light, has some technical applications, and suggests new directions to explore.
