Complex product structures on 6-dimensional nilpotent Lie algebras

Abstract

Abstract. We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex product structure on a 6-dimensional nilpotent Lie algebra must be nilpotent in the sense of Cordero-Fernández-Gray-Ugarte. A study is made of the torsion-free connection associated to the complex product structure and we consider also the associated hypercomplex structures on the 12-dimensional nilpotent Lie algebras obtained by complexification.

Abelian Hermitian geometry

“…We point out that the double product (A, ·) ⊲⊳ (A, * ) is a nilpotent Lie algebra if and only if both (A, ·) and (A, * ) are nilpotent commutative associative algebras[2].…”

Interpolation of geometric structures compatible with a pseudo Riemannian metric

“…Remark 3.2 For further reference we verify that I k is symmetric for b defined in (1). It suffices to check that g ♭ (u) (v) = g ♭ (v) (u) for all vector fields u, v on M and τ g ♭ −1 (σ) = σ g ♭ −1 (τ ) for all one forms σ, τ on M. Both assertions follow from the symmetry of g (for the second one, use the fact that σ = g ♭ (x) and τ = g ♭ (y) for some vector fields x, y on M).…”

Construction of nice nilpotent Lie groups

“…Type: 321 631:1 0, 0, 0, e 12 , e 13 , e 14 246 N 6,3,4 631:2 0, 0, 0, e 12 , e 13 , e 24 246 N 6,3,3 631:3 0, 0, 0, e 12 , e 13 , e 23 + e 14 246 N 6,2,8 631:4 0, 0, 0, e 12 , e 13 , e 24 + e 15 136 N 6,2,6 631:5a 0, 0, 0, e 12 , e 13 , e 24 + e 35 136 N 6,3,1a 631:5b 0, 0, 0, −e 12 , e 13 , e 35 + e 24 136 N 6,3,1 631:6 0, 0, 0, e 12 , e 13 , e 25 + e 34 136 N 6,3,1 Type: 33 63:1 0, 0, 0, e 12 , e 13 , e 23 36 N 6,3,6 Type: 411 621:1 0, 0, 0, 0, e 12 , e 15 346 N 4,2 ⊕ R 2 621:2 0, 0, 0, 0, e 12 , e 25 + e 13 246 N 5,2,2 ⊕ R 621:3 0, 0, 0, 0, e 12 , e 34 + e 15 146 N 6,3,2 Type: 42 62:1 0, 0, 0, 0, e 12 , e 13 36 N 5,3,2 ⊕ R 62:2 0, 0, 0, 0, e 12 , e 34 26 N 3,2 ⊕ N 3,2 62:3 0, 0, 0, 0, e 12 , e 24 + e 13 26 N 6,3,5 62:4a 0, 0, 0, 0, e 13 + e 24 , e 12 + e 34 26 N 3,2 ⊕ N 3,2 62:4b 0, 0, 0, 0, e 24 − e 13 , e 34 + e 12 26 N 6,4,4a Table 7 -Continued to next page 1 There is a misprint in [18], where the lower descending series is incorrectly written as (6432). Table 7 -Continued from previous page Name g Gong [18] Type: 51 61:1 0, 0, 0, 0, 0, e 12 46 N 3,2 ⊕ R 3 61:2 0, 0, 0, 0, 0, e 34 + e 12 26 N 5,3,1 ⊕ R Type: 6 6:1 0, 0, 0, 0, 0, 0 6 R 6 0, 0, 0, 0, e 12 , e 15 + e 23 , e 25 + e 14 257J Type: 421 731:1 0, 0, 0, 0, e 12 , e 13 , e 15 357 N 6,3,4 ⊕ R 731:2 0, 0, 0, 0, e 12 , e 13 , e 25 357 N 6,3,3 ⊕ R 731:3 0, 0, 0, 0, e 12 , e 34 , e 15 257 N 4,2 ⊕ N 3,2 731:4 0, 0, 0, 0, e 12 , e 13 , e 23 + e 15 357 N 6,2,8 ⊕ R 731:5 0, 0, 0, 0, e 12 , e 13 , e 34 + e 15 257F 731:6 0, 0, 0, 0, e 12 , e 13 , e 25 + e 34 257E 731:7 0, 0, 0, 0, e 12 , e 24 + e 13 , e 15 257B 731:8 0, 0, 0, 0, e 12 , e 34 , e 25 + e 13 257H 731:9 0, 0, 0, 0, e 12 , e 13 , e 25 + e 14 257C 731:10 0, 0, 0, 0, e 12 , e 13 , e 15 + e 24 257A 731:11 0, 0, 0, 0, e 12 , e 24 + e 13 , e 34 + e 15 257G 731:12 0, 0, 0, 0, e 12 , e 23 + e 14 , e 13 + e 25 257D 731:13 0, 0, 0, 0, e 12 , e 13 , e 25 + e 16 247 N 6,2,6 ⊕ R 731:14a 0, 0, 0, 0, e 12 , e 13 , e 36 + e 25 247 N 6,3,1a ⊕ R 731:14b 0, 0, 0, 0, −e 12 , e 13 , e 36 + e 25 247 N 6,3,1 ⊕ R 731:15 0, 0, 0, 0, e 12 , e 13 , e 26 [18] 731:24a 0, 0, 0, 0, e 24 + e 13 , −e 12 + e 34 , e 46 + e 15 + e 23 137B 1 731:24b 0, 0, 0, 0, e 24 − e 13 , −e 12 + e 34 , e 23 + e 46 + e 15 137B Type: 43 73:1 0, 0, 0, 0, e 12 , e 13 , e 23 47 N 6,3,6 ⊕ R 73:2 0, 0, 0, 0, e 12 , e 13 , e 14 37A 73:3 0, 0, 0, 0, e 12 , e 13 , e 24 37B 73:4 0, 0, 0, 0, e 12 , e 13 , e 23 + e 14 37C 73:5 0, 0, 0, 0, e 12 , e 34 , e 13 + e 24 37D 73:6a 0, 0, 0, 0, e 12 , e 14 + e 23 , e 13 + e 24 37B 73:6b 0, 0, 0, 0, e 12 , −e 14 + e 23 , e 24 + e 13 37B 1 73:7a 0, 0, 0, 0, e 23 + e 14 , e 24 + e 13 , e 12 + e 34 37D 73:7b 0, 0, 0, 0, e 14 + e 23 , e 24 − e 13 , e 12 + e 34 37D 1 Type: 511 721:1 0, 0, 0, 0, 0, e 12 , e 16 457 N 4,2 ⊕ R 3 721:2 0, 0, 0, 0, 0, e 12 , e 26 + e 13 357 N 5,2,2 ⊕ R 2 721:3 0, 0, 0, 0, 0, e 12 , e 16 + e 34 257 N 6,3,2 ⊕ R 721:4 0, 0, 0, 0, 0, e 12 , e 45 + e 13 + e 26 157 Type: 52 72:1 0, 0, 0, 0, 0, e 12 , e 13 47 N 5,3,2 ⊕ R 2 72:2 0, 0, 0, 0, 0, e 12 , e 34 37 N 3,2 ⊕ N 3,2 ⊕ R 72:3 0, 0, 0, 0, 0, e 12 , e 13 + e 24 37 N 6,3,5 ⊕ R 72:4a 0, 0, 0, 0, 0, e 24 + e 13 , e 34 + e 12 37 N 3,2 ⊕ N 3,2 ⊕ R 72:4b 0, 0, 0, 0, 0, e 24 − e 13 , e 34 + e 12 37 N 6,4,4a ...…”