Complex product structures on Lie algebras

Abstract

A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. Any Lie algebra g with such a structure is even-dimensional and its complexification has a hypercomplex structure. In addition, g splits into the direct sum of two Lie subalgebras of the same dimension, and each of these is shown to have a left-symmetric algebra (LSA) structure. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.

Double Products and Hypersymplectic Structures on ℝ4 n

“…In [3,4] (5) and (6) trivially hold, obtaining in this case a semidirect product which coincides with aff(A). This family of algebras and various geometric properties were considered in [3,4].…”

“…We note that JE = −EJ, and therefore {J, E} is a complex product structure on n ∇,∇ ′ (see [3]). The symplectic form ω 1 satisfies…”

“…In order to see that the compatibility condition (5) holds, we observe that , 1), (1, 3), (3, 1), (3,3) and ∇ e j ∇ ′ e k = 0 otherwise. From results in §2, we may construct the 8-dimensional nilpotent Lie group N ∇,∇ ′ (see Theorem 2.1), whose underlying manifold is R 4 × R 4 .…”

“…Take f ∈ A + , which is the annihilator of g − . It is known that df ∈ 2 A + ⊕ A + ⊗ A − (see [3]), so we only have to see that the component of (df ) in 2 A + is zero. For x, y ∈ g + , we have showing that df ∈ A + ⊗ A − .…”

Complex product structures on 6-dimensional nilpotent Lie algebras

“…The notion of a complex product structure on a (real) Lie algebra was introduced in [4]; it is given by two anticommuting endomorphisms of the Lie algebra, one of them a complex structure and the other one a product structure. Such a structure determines in a unique way a torsion-free connection on the Lie algebra such that the complex structure and the product structure are both parallel, and this connection restricts to flat torsion-free connections on two complementary totally geodesic subalgebras.…”