Solvable Lie algebras with naturally graded nilradicals and their invariants. (English) Zbl 1095.17003

For every natural number \(n\geq1\) let \(Q_{2n}\) be the (real or complex) nilpotent Lie algebra defined by a basis \(\{X_1,\dots,X_{2n}\}\) satisfying \([X_1,X_i]=X_{i+1}\) for \(i=2,\dots,2n-2\), and \([X_k,X_{2n+1-k}]=(-1)^kX_{2n}\) for \(k=2,\dots,n\). The paper under review presents several interesting properties of the solvable Lie algebras whose nilradical is isomorphic to \(Q_{2n}\). These solvable Lie algebras can be completely classified (Propositions 5 and 6 in the paper). In particular, their dimensions are either \(2n+1\) or \(2n+2\). The generalized Casimir invariants of the algebras of dimension \(2n+1\) are described in Theorem 1, and one then proves that the algebras of dimension \(2n+2\) have no invariants if \(n\geq3\).
Another interesting property concerns the existence of contact forms. Thus, for each \(n\geq3\) there exists only one \((2n+1)\)-dimensional solvable Lie algebra with the radical \(Q_{2n}\) which does not admit a contact form (Proposition 8). Possible applications of this fact in the contact geometry are then pointed out.
As the authors mention in the concluding section of their paper, the present results complete the investigation of the generalized Casimir invariants of indecomposable Lie algebras with a naturally graded nilradical of maximal nilindex. This investigation was initiated in the paper by L. Snobl and P. Winternitz [J. Phys. A, Math. Gen. 38, No. 12, 2687–2700 (2005; Zbl 1063.22023)].


17B30 Solvable, nilpotent (super)algebras
53D10 Contact manifolds (general theory)
22E70 Applications of Lie groups to the sciences; explicit representations


Zbl 1063.22023
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