On porcupine varieties in Lie algebras
If \({\mathfrak g}\) is a real or complex Lie algebra then each element \(x\) is contained in its centralizer \({\mathfrak z} (x,{\mathfrak g})\). If \({\mathfrak e}\) is a subset of \({\mathfrak g}\), such as a vector subspace or, better still, a subalgebra, then the set \[ {\mathfrak Z} ({\mathfrak e}) {\buildrel {\text{def}} \over =} \bigcup \biggl\{ {\mathfrak z} (y,{\mathfrak g}) \mid 0 \neq y \in {\mathfrak e} \biggr\}=\biggl\{ x \in {\mathfrak g} \mid (\exists y \in {\mathfrak e})\;y \neq 0,[x,y]=0 \biggr\} \] is shown to be a Zariski closed subset of \({\mathfrak g}\). Suppose that a subalgebra \({\mathfrak e}\) of a real or complex Lie algebra \({\mathfrak g}\) satisfies certain embedding properties and let \({\mathfrak h}\) be a Cartan subalgebra. Then the set of all subalgebras \({\mathfrak h} \cap \varphi ({\mathfrak e})\) is finite as \(\varphi\) ranges through the group of all inner automorphisms of \({\mathfrak g}\). If, in addition, \({\mathfrak e}\) does not contain any nonzero ideals of \({\mathfrak g}\), then every conjugacy class of Cartan subalgebras of \({\mathfrak g}\) contains a member \({\mathfrak h}\) such that \({\mathfrak h} \cap {\mathfrak e}=\{0\}\). This implies, in particular, that under these circumstances, the codimension \(\dim {\mathfrak g}-\dim {\mathfrak e}\) of \({\mathfrak e}\) in \({\mathfrak g}\) is at least the rank of \({\mathfrak g}\) \((=\dim {\mathfrak h})\) and that the ``porcupine variety \({\mathfrak Z} ({\mathfrak e})\)'' is a proper subset of \({\mathfrak g}\). Sharper information is provided on the dimension of this variety and sufficient conditions are given under which it does not exceed \(\dim {\mathfrak g}-2\). These results are applied to the solution of a problem in geometric control theory on Lie groups.
scientific article; zbMATH DE number 562488