Ol'shanskiĭ wedges in symmetric Lie algebras
Let \({\mathfrak g}= {\mathfrak h}\oplus {\mathfrak q}\) be a finite dimensional real symmetric Lie algebra. The author studies the wedge \(W\subset {\mathfrak g}\) such that \({\mathfrak h}\) is contained in the edge \(H(W)= W\cap-W\) of \(W\). A wedge \(W\) in \({\mathfrak g}\) is an Ol'shanskij wedge, if there is a wedge \(C\subseteq {\mathfrak q}\) such that \(W={\mathfrak h}\oplus C\) and the following holds: (1) \(e^{\text{ad} {\mathfrak h}} C=C\). (2) There is a Campbell- Hausdorff-neighborhood \(B\) in \({\mathfrak g}\) such that \(g(\text{ad }x)L_ x(W)= L_ x(W)\) for all \(x\in C\cap B\). Here \(L_ x(W)= \overline {W- \mathbb{R}^ -\cdot x}\) is the subtangent wedge at \(x\in W\). Theorem 1.5 (Characterization theorem for Ol'shanskij wedges). For a wedge \(W={\mathfrak h}\oplus C\) in a symmetric Lie algebra \({\mathfrak g}={\mathfrak h}\oplus{\mathfrak q}\) the following conditions are equivalent: (1) \(W\) is an Ol'shanskij wedge. (2) \(W\) is a Lie wedge. (3) \(e^{\text{ad }{\mathfrak h}} C=C\). (4) \([{\mathfrak h},c]\subseteq T_ c(C)\) for all \(c\in C\) (resp., \(c\in C^ 1(C)\), if \(C\) is generating), \(T_ c(C)= L_ c(C)\cap -L_ c(C)\). Further in Section 2 the author considers invariant Ol'shanskij wedges and Ol'shanskij semialgebras, and gives a complete classification of symmetric Lie algebras supporting invariant Ol'shanskij wedges, resp., Ol'shanskij semialgebras.
scientific article; zbMATH DE number 469227