线性系的李代数及其自同构
Lie algebras of linear systems and their automorphisms
摘要：
本论文的目的是研究附着在线性系上的李代数的自同构群。一个线性系统是一对向量空间$(U,W)$，其中有一个非退化配对$\langle\cdot,\cdot\rangle\colon U\otimes W\to \mathbb{C}$，我们给它附加了三个李代数$\mathfrak{sl}_{U,W}\subset \mathfrak{gl}_{U,W}\subset\mathfrak{gl}^M_{U,W}$。如果$U$和$W$都是可数维的，那么直到同构，存在唯一的线性系统$(V,V_*)$。在这种情况下，$\mathfrak{sl}_{V,V_*}$和$\mathfrak{gl}_{V,V_*}$是众所周知的李代数$\mathfrak{sl}_\infty$和$\mathfrak{gl}_\infty$，而李代数$\mathfrak{gl}^M_{V,V_*}$是在\cite{PSer}中引入的麦基李代数。我们回顾了张量模的monoidal范畴$\mathbb{T}_{\mathfrak{sl}_{U,W}}$和$\mathbb{T}_{\mathfrak{gl}^M_{U,W}}$的结果，这两个范畴都等价于前面在\cite{DPS}中引入的范畴$\mathbb{T}_{\mathfrak{sl}_\infty}$。利用范畴$\mathbb{T}_{\mathfrak{sl}_\infty}$和$\mathbb{T}_{\mathfrak{gl}^M_\infty}$之间的关系，计算了$\mathfrak{gl}^M_\infty$的自同构群.
英文摘要：
The objective of this thesis is to study the automorphism groups of the Lie algebras attached to linear systems. A linear system is a pair of vector spaces $(U,W)$ with a nondegenerate pairing $\langle\cdot,\cdot\rangle\colon U\otimes W\to \mathbb{C}$, to which we attach three Lie algebras $\mathfrak{sl}_{U,W}\subset \mathfrak{gl}_{U,W}\subset\mathfrak{gl}^M_{U,W}$. If both $U$ and $W$ are countable dimensional, then, up to isomorphism, there is a unique linear system $(V,V_*)$. In this case $\mathfrak{sl}_{V,V_*}$ and $\mathfrak{gl}_{V,V_*}$ are the well-known Lie algebras $\mathfrak{sl}_\infty$ and $\mathfrak{gl}_\infty$, while the Lie algebra $\mathfrak{gl}^M_{V,V_*}$ is the Mackey Lie algebra introduced in \cite{PSer}. We review results about the monoidal categories $\mathbb{T}_{\mathfrak{sl}_{U,W}}$ and $\mathbb{T}_{\mathfrak{gl}^M_{U,W}}$ of tensor modules, both of which turn out to be equivalent as monoidal categories to the category $\mathbb{T}_{\mathfrak{sl}_\infty}$ introduced earlier in \cite{DPS}. Using the relations between the categories $\mathbb{T}_{\mathfrak{sl}_\infty}$ and $\mathbb{T}_{\mathfrak{gl}^M_\infty}$, we compute the automorphism group of $\mathfrak{gl}^M_\infty$.
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