Representation categories of Mackey Lie algebras as universal monoidal categories
Representation categories of Mackey Lie algebras as universal monoidal categories
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DOI：
10.4310/pamq.2017.v13.n1.a3
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
A. Chirvasitu;I. Penkov
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文献类型：
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作者：
A. Chirvasitu;I. Penkov
设$\mathbb{K}$是特征为$0$的代数闭域。我们研究了在所有对称的$\mathbb{K}$-由两个对象$A$和$B$生成的线性么半群范畴中通用的$\mathbb{T}_\α$，使得$A$有一个可能是超限的滤子。我们将$\mathbb{T}_\α$构造为李代数$\mathfrak{gl}^M(V_*，V)$的表示范畴，它由维为$\α$的向量空间$V_*和$V$的固定可对角化对$V_*和$V$的自同态组成。这里的$\α$是一个任意基数。我们明确地刻画了$\mathbb{T}_\α$的简单对象和内射对象，并证明了范畴$\mathbb{T}_\α$是Koszul。我们特别注意在$A$上的过滤是有限的情况。在这种情况下，$\Alpha=\Aleph_t$表示\mathbb{Z}_{\geq 0}$中的$t\。
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We study a monoidal category $\mathbb{T}_\alpha$ which is universal among all symmetric $\mathbb{K}$-linear monoidal categories generated by two objects $A$ and $B$ such that $A$ has a, possibly transfinite, filtration. We construct $\mathbb{T}_\alpha$ as a category of representations of the Lie algebra $\mathfrak{gl}^M(V_*,V)$ consisting of endomorphisms of a fixed diagonalizable pairing $V_*\otimes V\to \mathbb{K}$ of vector spaces $V_*$ and $V$ of dimension $\alpha$. Here $\alpha$ is an arbitrary cardinal number. We describe explicitly the simple and the injective objects of $\mathbb{T}_\alpha$ and prove that the category $\mathbb{T}_\alpha$ is Koszul. We pay special attention to the case where the filtration on $A$ is finite. In this case $\alpha=\aleph_t$ for $t\in\mathbb{Z}_{\geq 0}$.