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 $⟨\cdot ,\cdot ⟩:U\otimes W\to \mathbb{C}$, to which we attach three Lie algebras ${\mathfrak{sl}}_{U,W}\subset {\mathfrak{gl}}_{U,W}\subset {\mathfrak{gl}}_{U,W}^M$. If both $U$ and $W$ are countable dimensional, then, up to isomorphism, there is a unique linear system $(V,V_{\ast })$. In this case ${\mathfrak{sl}}_{V,V_{\ast }}$ and ${\mathfrak{gl}}_{V,V_{\ast }}$ are the well-known Lie algebras ${\mathfrak{sl}}_{\infty }$ and ${\mathfrak{gl}}_{\infty }$, while the Lie algebra ${\mathfrak{gl}}_{V,V_{\ast }}^M$ is the Mackey Lie algebra introduced in \cite{PSer}. We review results about the monoidal categories ${\mathbb{T}}_{{\mathfrak{sl}}_{U,W}}$ and ${\mathbb{T}}_{{\mathfrak{gl}}_{U,W}^M}$ 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 ${\mathbb{T}}_{{\mathfrak{gl}}_{\infty }^M}$, we compute the automorphism group of ${\mathfrak{gl}}_{\infty }^M$.