Lie groups whose coadjoint orbits are of dimension smaller or equal to two
This note gives a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller than or equal to two. Let \(G\) be such a Lie group with Lie algebra \({\mathfrak g}\). The authors show that, up to a direct central factor, \({\mathfrak g}\) belongs to the following list of algebras: (i) \({\mathfrak su} (2)\) or \({\mathfrak sl} (2, \mathbb{R})\); (ii) \(\mathbb{R} T \oplus {\mathfrak a}\) where \({\mathfrak a}\) is an abelian ideal; (iii) \(\mathbb{R} T \oplus {\mathfrak h}\) where \({\mathfrak h}\) is the three-dimensional Heisenberg algebra spanned by \(X,Y,Z\), with \([X,Y] = Z\) and either \([T,X] = X\), \([T,Y] = - Y\), \([T,Z] = 0\), or \([T,X] = Y\), \([T,Y] = - X\), \([T,Z] = 0\); (iv) \({\mathfrak g}\) is six-dimensional with basis \(X_i\), \(Y_i\) \((1 \leq i \leq 3)\) and the nonvanishing brackets are \([X_1, X_2] = Y_3\), \([X_2, X_3] = Y_1\), \([X_3, X_1] = Y_2\); (v) \({\mathfrak g}\) is five-dimensional with basis \(X_i\) \((1 \leq i \leq 3)\), \(Y_j\) \((1 \leq j \leq 2)\) and the multiplicative law reads \([X_1, X_2] = X_3\), \([X_1, X_3] = Y_1\), \([X_2, X_3] = Y_2\). -- The proof is given by a case-by-case analysis.
scientific article; zbMATH DE number 807829