Lie algebras of linear systems and their automorphisms

Abstract

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 $\mathfrak{gl}^M_\infty$.