Deformations of Lie algebras
The paper is devoted to the study of the deformations of nilpotent Lie algebras over an algebraically closed field of characteristic 0. After some remarks giving reasons for it the authors consider the deformations of the type \(a\circ b= [a,b]+ t\psi(a,b)\), where \(a\circ b\) is a new multiplication on Lie algebra \(L\) and \(\psi\in Z^ 2(L,L)\). The main result states that nilpotent Lie algebra \(L\) has a non-trivial deformation if some codimension 1 ideal \(K\) of \(L\) has a nonsingular derivation. The proof is based on Dixmier's construction of 2-cochains in \(C^ 2(L,L)\) from 1-cochains in \(C^ 1(K,K)\). After that the authors consider some classes of nilpotent Lie algebras in which such an ideal may be constructed. In the paper Dixmier's construction is applied to obtain non-trivial deformations for some other Lie algebras as well.
scientific article; zbMATH DE number 554205
scientific article