Holonomy algebras of pseudo-quaternionic-Kählerian manifolds of signature (4,4)

Abstract

Possible holonomy algebras of pseudo-quaternionic-Kählerian manifolds of signature (4, 4) are classified. Using this, a new proof of the classification of simply connected pseudo-quaternionic-Kählerian symmetric spaces of signature (4, 4) is obtained.

Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4

“…We show that if a pseudo-hyper-Kählerian manifold (M, h) of signature (4, 4n + 4), n ≥ 1 is locally symmetric, then n = 2 and we give explicitly the curvature tensor and holonomy algebra of the obtained space. For the case of signature (4,4) the analogous result is obtained in [8].…”

“…Denote by sp(1, 1) Hp the subalgebra of sp(1, n+1) Hp that annihilates H n ⊂ H 1,n+1 . The space R(sp(1, 1) Hp ) is found in [8,Proposition 1]. Note that any R given by elements C rs , B rs , D rs and such that all the rest elements are zero belongs to R(sp(1, 1) Hp ).…”

“…If m = n and h 0 = sp(1), then any R ∈ R(g) is given as in Proposition 4.1 with A 01 , A 02 , A 03 ∈ h. Since the elements C 01 ∈ H, A 01 ∈ h, S 01 ∈ H n , D 01 ∈ Im H can be chosen in arbitrary way, g is a Berger algebra. If m = n and h 0 = Ri, then from [8,Section 4] it follows that in addition to the above case C 01 = 0 and C 02 ∈ R ⊕ Ri is arbitrary, hence g is a Berger algebra. Suppose that m < n. Then h 0 = Ri.…”

“…Hence, θ 0 (X) ∈ Im H has a non-zero projection to Rj ⊂ Im H. Since R(I s q, X) = (0, 0, 0, θ s (X)) ∈ g, there exists α, β ∈ R such that α = 0 and (0, 0, 0, αj + βk) ∈ g. We may assume that α 2 + β 2 = 1. In [8,Lemma 6] it is shown that there exists x, y ∈ R such that x 2 + y 2 = 1 and with respect to the new basis with p ′ = (x + iy)p and q ′ = (x + iy)q the elements (0, 0, 0, i) ∈ g and (0, 0, 0, αj + βk) ∈ g have the form (0, 0, 0, i) and (0, 0, 0, j), respectively. Note that S 03 = −iS 02 = 0.…”

“…In the present paper we classify all possible holonomy algebras g ⊂ sp(1, n+1) of pseudo-hyper-Kählerian manifolds of signature (4, 4n + 4), n ≥ 1. For n = 0 this classification is obtained in [8]. The main results is stated in Section 3.…”