Cartan Subalgebras of $\mathfrak{gl}_\infty$

http://dx.doi.org/10.4153/CMB-2003-056-1
Canad. Math. Bull. 46(2003), 597-616

  Published:2003-12-01
 Printed: Dec 2003

Let $V$ be a vector space over a field $\mathbb{K}$ of characteristic zero and $V_*$ be a space of linear functionals on $V$ which separate the points of $V$. We consider $V\otimes V_*$ as a Lie algebra of finite rank operators on $V$, and set $\mathfrak{gl} (V,V_*) := V\otimes V_*$. We define a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of $\mathfrak{gl} (V,V_*)$ under the assumption that $\mathbb{K}$ is algebraically closed. A subalgebra of $\mathfrak{gl} (V,V_*)$ is a Cartan subalgebra if and only if it equals $\bigoplus_j \bigl( V_j \otimes (V_j)_* \bigr) \oplus (V^0 \otimes V_*^0)$ for some one-dimensional subspaces $V_j \subseteq V$ and $(V_j)_* \subseteq V_*$ with $(V_i)_* (V_j) = \delta_{ij} \mathbb{K}$ and such that the spaces $V_*^0 = \bigcap_j (V_j)^\bot \subseteq V_*$ and $V^0 = \bigcap_j \bigl( (V_j)_* \bigr)^\bot \subseteq V$ satisfy $V_*^0 (V^0) = \{0\}$. We then discuss explicit constructions of subspaces $V_j$ and $(V_j)_*$ as above. Our second main result claims that a Cartan subalgebra of $\mathfrak{gl} (V,V_*)$ can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra $\mathfrak{h}$ which coincides with the maximal locally nilpotent $\mathfrak{h}$-submodule of $\mathfrak{gl} (V,V_*)$, and such that the adjoint representation of $\mathfrak{h}$ is locally finite.