On the cohomology of a new class of Lie algebras generalizing Borel subalgebras
It is well known that the second cohomology groups of Lie algebras in an adjoint module describe its infinitesimal deformations. The condition \(H^ 2({\mathfrak g},{\mathfrak g})=0\) means that \({\mathfrak g}\) is rigid. In the paper under review cohomologies of semi-direct products \({\mathfrak g}= \tau+{\mathfrak n}\), where \({\mathfrak n}\) is a nilpotent Lie algebra and \(\tau\) is a torus of derivations of \({\mathfrak n}\) are considered. \textit{R. Carles} [Ann. Inst. Fourier 34, No. 3, 65-82 (1984; Zbl 0544.17004)] proved that any solvable rigid Lie algebra will have this form. For a Lie algebra \({\mathfrak n}\) defined by generators and defining relations a formula for calculating invariants over the maximal torus \(H({\mathfrak n},{\mathfrak n})^ T\) is proposed. A Lie algebra \({\mathfrak g}\) is called pseudo-nilpotent, if the supplement of \([{\mathfrak g},{\mathfrak g}]\) in \({\mathfrak g}\) generates \({\mathfrak g}\) and \(\bigcap_{p\geq 1} C^ p{\mathfrak g}=0\), where \(C^ p{\mathfrak g}\) is the descending central series of \({\mathfrak g}\). A pseudo-nilpotent Lie algebra has maximal rank if the minimal number of generators is equal to the dimension \(s\) of the maximal torus. Such a Lie algebra \({\mathfrak m}\) is isomorphic to the quotient of a free Lie algebra \({\mathfrak L}_ s\) with \(s\) generators over an ideal contained in \([{\mathfrak L}_ s,{\mathfrak L}_ s]\) and stable under the action of the maximal torus \(T_ s\). If \(P\) is the set of weights of \(T_ s\) over \({\mathfrak m}\), then \({\mathfrak m}\) is the quotient by a \(T_ S\)-stable ideal of a Lie algebra admitting \(P\) as a set of weights and is maximal according to this property. This Lie algebra corresponding to \(P\) is called a Lie model associated to \(P\). For any model \({\mathfrak m}\) the group \(H^ k({\mathfrak m},{\mathfrak m})\) is zero for \(k=0,2\) and equal to \(T\) for \(k=1\). In particular, \(H^ k({\mathfrak g},{\mathfrak g})=0\) for \(k\leq 2\), if \({\mathfrak g}= T\oplus {\mathfrak m}\).
scientific article; zbMATH DE number 168584
scientific article