论文标题
双对产生的通用张量类别
Universal tensor categories generated by dual pairs
论文作者
论文摘要
令$ v _*\ otimes v \ rightArrow \ mathbb {c} $是可计数尺寸的复杂矢量空间$ v $和$ v _*$的非降级配对。 Mackey Lie代数$ \ Mathfrak {g} = \ Mathfrak {gl}^m（v，v，v _*）$与此相对应的对应于所有$ v $的所有内词$ v $，$ v _*$在双niualismorphism $ v _*c稳定下是稳定的。我们研究了由$ \ Mathfrak {g} $ - 模块$ v $，$ v _*$及其代数duals $ v^*$ v^*$和$ v^*_*$生成的Tensor Grothendieck类别$ \ MATHBB {T} $。这是先前文献中考虑的类别的类似物，主要区别在于琐碎的模块$ \ mathbb {c} $不再是$ \ mathbb {t} $中的注入。我们描述了$ \ mathbb {c} $ in $ \ mathbb {t} $中的注入性船体$ i $ i $ i $ i $ i $ i $ i $ \ mathbb {c} $，并表明类别$ \ mathbb {t} $是koszul。此外，我们证明了$ i $具有换向代数的自然结构。然后，我们在$ \ mathbb {t} $中定义了另一个类别$ _i \ Mathbb {t} $，该$是$ i $ -modules。我们的主要结果是类别$ {} _ i \ mathbb {t} $也是koszul，此外，$ {} _ i \ mathbb {t} $在Abelian $ \ Mathbb {c} $ - 线性Tensor类别中是由两个对象$ x $ x $ x $ x $ x'中的$ x $ x'在Abelian $ \ Mathbb {C} $ - lineart tensor中。 $ y'\ hookrightarrow y $和一个配对$ x \ otimes y \ rightarrow \ text {\ textbf {1}} $其中\ textbf {1}是单个单位。我们通过讨论类别的正交和符号类似物$ \ mathbb {t} $和$ {} _ i \ Mathbb {t} $来结束论文。
Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring consists of all endomorphisms $φ$ of $V$ for which the space $V_*$ is stable under the dual endomorphism $φ^*: V^*\rightarrow V^*$. We study the tensor Grothendieck category $\mathbb{T}$ generated by the $\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$. This is an analogue of categories considered in prior literature, the main difference being that the trivial module $\mathbb{C}$ is no longer injective in $\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in $\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition, we prove that $I$ is endowed with a natural structure of commutative algebra. We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$ is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian $\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing $X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.
