In this paper, we obtain analogues of Jørgensen’s inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya and Parker [7]. As...

We characterize the quasianti-Hermitian quaternionic operators in QQM by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two altern...

We study a second example of the phenomenon studied in the article “The complex Lorentzian Leech lattice and the bimonster”. We find 14 roots in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram given by the incidence graph of projective plane over F2. We prove that the reflections in these 14 roots generate the automorphism group of L. The ...

We study a second example of the phenomenon studied in the article “The complex Lorentzian Leech lattice and the bimonster”. We find 14 reflections in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram given by the incidence graph of projective plane over F2. We prove that these 14 reflections generate the automorphism group of L. The investi...

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M,Q) of dimension 4n ≥ 8. In particular, we show that any self-dual quaternionic connection D of (M,Q) induces an holomorphic structure ∂̄ on Θ. We construct a Penrose transform which identifies solutions of the Penrose operator P on (M,Q) defined by D with the space of ∂̄-...

From the classical differential equation of Jacobi fields, one naturally defines the Jacobi operator of a Riemannian manifold with respect to any tangent vector. A straightforward computation shows that any real, complex and quaternionic space forms satisfy that any two Jacobi operators commute. In this way, we classify the real hypersurfaces in quaternionic projective spaces all of whose tange...

The main result is that any continuous real-linear operator A on a quaternionic Hubert space has a unique decomposition A=A0+iiAl + izAi+iiA3, where the A„ are continuous linear operators and (fi,f2,'3) is any right-handed orthonormal triad of vector quaternions. Other results concern the place of the colinear and complex-linear operators in this characterisation and the effect on the Av of a r...

Further results are reported for the one-component quaternionic wave equation recently introduced. A Lagrangian is found for the momentum-space version of the free equation; and another, nonlocal in time, is found for the complete equation. Further study of multiparticle systems has us looking into the mathematics of tensor products of Hilbert spaces. The principles of linearity and superpositi...

A manifold (M, I, J,K) is called hypercomplex if I, J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian hypercomplex manifold is called HKT (hyperkähler with torsion) if the (2,0)-form Ω associated with the corresponding Sp(n)-structure satisfies ∂Ω = 0. A Hermitian metric ω on a complex manifold is called balanced if d∗ω = 0. We show that balanced HKT metrics...