Construction of nice nilpotent Lie groups

Abstract

We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension n up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for n ≤ 9. On every nilpotent Lie algebra of dimension ≤ 7, we determine the number of inequivalent nice bases, which can be 0, 1, or 2.We show that any nilpotent Lie algebra of dimension n has at most countably many inequivalent nice bases.

Ricci-flat and Einstein pseudoriemannian nilmanifolds

“…Extending these results to higher dimensions is made difficult by the fact that nilpotent Lie algebras are not classified, although with the same methods we have been able to classify nice nilpotent Lie algebras of dimension 8 and 9 (see [6]). Nevertheless, it is natural to ask whether the low-dimensional behaviour generalizes.…”

“…In this section we survey our work on the classification of nice Lie algebras and state some open questions; for details, we refer to [6].…”

“…. , e 6 } such that de 1 = 0 = · · · = de 4 , de 5 = e 13 + e 24 , de 6 = e 12 + e 34 (where, as usual, we have written e ij in lieu of e i ∧ e j ); the label 62:4a is the name of this Lie algebra in the classification of [6]. A nice Lie algebra (g, B) is reducible if there exist nice Lie algebras (g 1 , B 1 ), (g 2 , B 2 ) such that g = g 1 ⊕ g 2 and B = B 1 ∪ B 2 , and irreducible otherwise.…”

“…Thus, when e M∆ (g) = e M∆ (h), we obtain a Lie algebra automorphism f = gh −1 : g → g. If g and h lie in the same connected component of D n (in particular, they have the same signature), then we can write f = exp t so that exp t/2 defines an isometry between the metric Lie algebras (g, g i (e i ) 2 ) and (g, h i (e i ) 2 ). Thus, X determines the metric in an essentially unique way, at least for fixed signature; we refer to [6] for more details.…”

“…In [6], we defined a nice Lie algebra as a pair (g, B), with B a nice basis on the Lie algebra g, and an equivalence of nice Lie algebras as an isomorphism that maps basis elements to multiples of basis elements. This is the natural definition for classification purposes, since the nice condition is unaffected by rescaling any element of the basis.…”