ExactG2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$hbox {G}_{2}$$end{document}-structures on unimodular Lie algebras

摘要

We consider seven-dimensional unimodular Lie algebras g admitting exact G2structures, focusing our attention on those with vanishing third Betti number b3(g). We discuss some examples, both in the case when b2(g) similar to = 0, and in the case when the Lie algebra g is (2,3)-trivial, i.e., when both b2(g) and b3(g) vanish. These examples are solvable, as b3(g) = 0, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to g. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact G2-structure. From this, it follows that there are no compact examples of the form (similar toG,.), where G is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, similar to. G is a co-compact discrete subgroup, and. is an exact G2-structure on similar toG induced by a left-invariant one on G.