Classification of minimal algebras over any field up to dimension $6$

Abstract

Abstract. We give a classification of minimal algebras generated in degree 1, defined over any field k of characteristic different from 2, up to dimension 6. This recovers the classification of nilpotent Lie algebras over k up to dimension 6. In the case of a field k of characteristic zero, we obtain the classification of nilmanifolds of dimension less than or equal to 6, up to k-homotopy type. Finally, we determine which rational homotopy types of such nilmanifolds carry a symplectic structure.

Symplectic resolution of orbifolds with homogeneous isotropy

“…This coincides with L 6,10 in Table 2 in [2]. The symplectic form of M is ω = e 1 e 6 + e 2 e 5 − e 3 e 4 (see Table 3 in [2]). Now we consider the map ϕ(x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = (x 1 , −x 2 , −x 3 , −x 4 , −x 5 , x 6 ).…”

“…To give an explicit example of a resolution, we shall take a symplectic 6-nilmanifold from [2] and perform a suitable quotient to get a symplectic 6-orbifold with homogeneous isotropy. For instance we take the nilmanifold corresponding to the Lie algebra L 6,10 of Table 2 in [2], which is symplectic since it appears in Table 3 of [2]. Take the group of (7 × 7)-matrices given by the matrices , where x i ∈ R, for any i = 1, .…”

Bi-Lagrangian structures on nilmanifolds

“…We now consider the 6-dimensional case. The symplectic nilpotent Lie algebras of dimension 6 have been determined (independently) by [19], [16] and [3]. The latter two references contain explicit symplectic forms.…”

“…Remark 4.1. The lists in [3] and [16] for symplectic forms on 6-dimensional nilpotent Lie algebras contain several errors:…”

“…We follow the notation for Lie algebras in [3]: A n denotes an n-dimensional abelian Lie algebra and L m or L m,k an m-dimensional nilpotent Lie algebra.…”