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 ...

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley– Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N + 1) and sp(p, q) in the Cartan series CN+1, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relati...

It has long been known that differential forms on a complex manifold M2n can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for a quaternionic manifold M4n using the fact that its cotangent space T ∗ mM is isomorphic to the quaternionic vector space H. This defines an action of the group Sp(1) of unit quater...

An operator $I$ on a real Lie algebra $A$ is called complex structure if $I^2=-Id$ and the $\sqrt{-1}$-eigenspace $A^{1,0}$ subalgebra in complexification of $A$. A hypercomplex triple structures $I,J$ $K$ satisfying quaternionic relations. We call nilpotent quaternionic-solvable there exists finite filtration by quaternionic-invariant subalgebras with commutative subquotients which converges t...

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 ∂̄-...

Taking the complex nature of quantum mechanics which we observe today as a low energy effect of a broken quaternionic theory we explore the possibility that dark matter arises as a consequence of this underlying quaternionic structure to our universe. We introduce a low energy, effective, Lagrangian which incorporates the remnants of a local quaternionic algebra, investigate the stellar product...

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S1 can be constructed from another hyperKähler manifold (of lower dimention) with an action of S1 which fixes one complex structure and rotates the other two and vice versa. On the other hand, the...

We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts existence of a 2-design every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this terms entanglement breaking rank certain quantum channel. show that quantity is equal size smallest weighted 2-design. Next, finite ...