Centrosymmetric Lie algebras and boost-dilations (Q1910882)

Let the matrix \(J\in \text{gl}(n)\) be defined by the elements \(J_{ab}=\delta_{a,n+1-b}\) and be used further to introduce the mapping \(X\to X^c\equiv JXJ\Rightarrow X^c_{ab}=X_{n+1-a,n+1-b}\) for any \(X\in \text{gl}(n)\). Centrosymmetric matrices are defined then as those invariant under this mapping, or precisely, \(c(n)=\{X\in \text{gl}(n)\mid X^c = X\}\). They form the subalgebra \(c(n)\subset \text{gl}(n)\). It is proved that \(c(n)\) is a noncompact and nonsimple algebra that contains its exponential. It is also shown that \(c(2m)\) involves a boost-dilation subgroup of the conformal group of the Lorentzian metric with signature zero. The algebras \(c(2)\) and \(c(3)\) are examined in detail, and as is proved \(c(3)\) is a nonsolvable algebra whose derived algebra is isomorphic to \(\text{sl}(2,\mathbb{R})\). Finally it is shown that nonsolvability is inherent for all \(c(2m+1)\) algebras with \(m\geq 1\).