Para-Hypercomplex Structures on a Four-Dimensional Lie Group

Compact complex surfaces with geometric structures related to split quaternions

“…Also, every Inoue surface S + defined by means of a real parameter t is the quotient of the group Sol 4 1 by a discrete subgroup [32, Proposition 9.1]. Left-invariant para-hypercomplex structures descending to these types of elliptic and Inoue surfaces have been constructed in [3,8]. Compatible metrics have been given in [20] where it is shown that the respective para-hyperhermitian structures are locally, but not globally, conformally para-hyperkähler, the metric on the elliptic surfaces being locally conformally flat.…”

“…The structures obtained in this way define a holonomy reduction to the group SU (1, 1) ∼ = SL(2, R) and are an indefinite analog of hyperkähler structures which have holonomy SU (2) ∼ = Sp (1). Mathematically, these structures are described by quadruples (g, I, S, T ) where g is a signature (2, 2) metric and I, S, T are parallel endomorphisms of the tangent bundle such that: I 2 = −S 2 = −1, T = IS = −SI, g(IX, IY ) = −g(SX, SY ) = g(X, Y ) (1) In the literature such structures are called hypersymplectic [18], neutral hyperkähler [22], para-hyperkähler [8,12], pseudo-hyperkähler [13], etc. They are not preserved by a conformal change of the metric and a natural conformally invariant generalization is to relax the condition for covariant constancy of I, S, T to their integrability (see Section 2).…”

Однородные Пара-Кэлеровы Многообразия Эйнштейна

“…В частности, в работе [24] получена классификация левоинвариантных пара-гиперкомплексных структур на четырехмерных груп-пах Ли. При некоторых дополнительных предположениях такие структуры были описаны в [27]. Конструкция левоинвариантных пара-гиперкомплексных структур на группе Ли G сводится к разложению соответствующей алгебры Ли g в прямую сумму подалгебр g + , g − и конструкции комплексной структу-ры J, переставляющей g + и g − .…”

“…Вычисления дают b 2 = 1, a 2 = 3. Из формулы (27) и леммы 7.9 следует, что ψ = 6π 2 = 6(3α 1 + 2α 2 ) и ρ = 6 ω α2 ∧ ω −α2 + 3ω α1+α2 ∧ ω −(α1+α2) + 3ω 2α1+α2 ∧ ω…”

Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces

“…These structures happen to be Hermitian, but not Kählerian (the first case from Table 1). In fact, it was shown in [5] that ch 2 is one of three non-abelian 4-dimensional Lie algebras admitting both the hypercomplex and the para-hypercomplex structure. Snow [18] investigated 4-dimensional solvable simply-connected real Lie groups with commutator subalgebra of dimension less than three.…”