Classification of 7-dimensional Einstein nilradicals

EDISON ALBERTO FERNÁNDEZ CULMA

Norman, Oklahoma

Workshop; Workshop On Ricci Solitons And Symmetry; 2012

Department of Mathematics, University of Oklahoma

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety $N_n(C)$ of $n$-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which $GL_n(CC)$-orbits in $N_n(CC)$ have a critical point of the squared norm of the moment map. We have proved in arXiv:1105.4489 a classification theorem of such distinguished orbits for $n=7$. The set $N_7(CC)/GL_7(CC)$ is formed by 148 nilpotent Lie algebras and 6 one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques (based on the earlier results of J. Lauret and Y. Nikolayevsky) to decide whether it has a distinguished orbit or not. In the talk, we will give many examples which cover all possible strategies that we have used to obtain the full classification; each Lie algebra in the work of Magnin (Adjoint and Trivial Cohomology Tables for Indecomposable Nilpotent Lie Algebras of Dimension less than or equal to 7 over CC, 2007) has been worked out in some of these ways.