STEFAN IVANOV, IVAN MINCHEV, and DIMITER VASSILEV 9 Besides the non-uniqueness due to the action of SO(3), the 1-form η can be changed by a conformal factor, in the sense that if η is a form for which we can find associated almost complex structures and metric g as above, then for any Ψ SO(3) and a positive function μ, the form μ Ψ η also has an associated complex structures and metric. In particular, when μ = 1 we obtain a whole unit sphere of contact forms, and we shall denote, as already mentioned, by Q the corresponding sphere bundle of associated triples of almost complex structures. With the above consideration in mind we introduce the following notation. Notation 2.3. We shall denote with (M, η) a QC-manifold with a fixed globally defined contact form. (M, g, Q) will denote a QC-manifold with a fixed metric g and a sphere bundle of almost complex structures Q. In this case we have in fact a Sp(n)Sp(1) structure, i.e., we are working with a fixed metric on the horizontal space. Correspondingly, we shall denote with η any (locally defined) associated contact form. We recall the definition of the Lie groups Sp(n), Sp(1) and Sp(n)Sp(1). Let us identify Hn = R4n and let H acts on Hn by right multiplications, λ(q)(W ) = W · q−1. This defines a homomorphism λ : {unit quaternions} −→ SO(4n) with the convention that SO(4n) acts on R4n on the left. The image is the Lie group Sp(1). Let λ(i) = I0,λ(j) = J0,λ(k) = K0. The Lie algebra of Sp(1) is sp(1) = span{I0,J0,K0}. The group Sp(n) is Sp(n) = {O SO(4n) : OB = BO, B Sp(1)} or Sp(n) = {O Mn(H) : O ¯t = I}, and O Sp(n) acts by (q1,q2,...,qn)t O (q1,q2,...,qn)t. Denote by Sp(n)Sp(1) the product of the two groups in SO(4n). Abstractly, Sp(n)Sp(1) = (Sp(n) × Sp(1))/Z2. The Lie algebra of the group Sp(n)Sp(1) is sp(n) sp(1). Any endomorphism Ψ of H can be decomposed with respect to the quaternionic structure (Q,g) uniquely into four Sp(n)-invariant parts Ψ = Ψ+++ + Ψ+−− + Ψ−+− + Ψ−−+, where Ψ+++ commutes with all three Ii, Ψ+−− commutes with I1 and anti-commutes with the others two and etc. Explicitly, 4Ψ+++ = Ψ I1ΨI1 I2ΨI2 I3ΨI3, 4Ψ+−− = Ψ I1ΨI1 + I2ΨI2 + I3ΨI3, 4Ψ−+− = Ψ + I1ΨI1 I2ΨI2 + I3ΨI3, 4Ψ−−+ = Ψ + I1ΨI1 + I2ΨI2 I3ΨI3. The two Sp(n)Sp(1)-invariant components are given by (2.7) Ψ[3] = Ψ+++, Ψ[−1] = Ψ+−− + Ψ−+− + Ψ−−+. Denoting the corresponding (0,2) tensor via g by the same letter one sees that the Sp(n)Sp(1)-invariant components are the projections on the eigenspaces of the Casimir operator (2.8) = I1 I1 + I2 I2 + I3 I3 corresponding, respectively, to the eigenvalues 3 and −1, see [CSal]. If n = 1 then the space of symmetric endomorphisms commuting with all Ii,i = 1, 2, 3 is
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