Generators of SO(8) group have been described by using direct product of the Gamma matrices and the Pauli Sigma matrices. We have obtained these generators in terms of generalized split octonion also. These generators have been used to describe the rotational transformation of vectors for SO(7) symmetry group.
The 8 $times$ 8 matrix representation of SO(8) Symmetry has been defined by using the direct product of Pauli matrices and Gamma matrices. These 8 $times$ 8 matrices are being used to describe the rotations in SO(8) symmetry. The comparison of 8$time
s$8 matrices with octonions has also been shown. The transformations of SO(8) symmetry are represented with the help of Octonions and split Octonions spinors.
We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $Bbbk$-split
in the sense that they factor (inside the tensor category of bimodules) over $Bbbk$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $Bbbk$-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $Bbbk[x,y]/(x^2,y^2,xy)$.
We present the complete family of space-times with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves) that are of algebraic type III and for which the cosmological constant ($Lambda_c$) is non-zero. The possible
presence of an aligned pure radiation field is also assumed. These space-times generalise the known vacuum solutions of type N with arbitrary $Lambda_c$ and type III with $Lambda_c=0$. It is shown that there are two, one and three distinct classes of solutions when $Lambda_c$ is respectively zero, positive and negative. The wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively, and the structure of the family of wave surfaces in the background space-time is described. The weak singularities which occur in these space-times are interpreted in terms of envelopes of the wave surfaces.
In this paper we establish Springer correspondence for the symmetric pair $(mathrm{SL}(N),mathrm{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaves construction due to Grinberg. As applications, we obtain result
s on cohomology of Hessenberg varieties and geometric constructions of irreducible representations of Hecke algebras of symmetric groups at $q=-1$.
The three-point amplitude is the key building block in the on-shell approach to scattering amplitudes. We show that the classical objects computed by massive three-point amplitudes in gauge theory and gravity are Newman-Penrose scalars in a split-sig
nature spacetime, where three-point amplitudes can be defined for real kinematics. In fact, the quantum state set up by the particle is a coherent state fully determined by the three-point amplitude due to an eikonal-type exponentiation. Having identified this simplest classical solution from the perspective of scattering amplitudes, we explore the double copy of the Newman-Penrose scalars induced by the traditional double copy of amplitudes, and find that it coincides with the Weyl version of the classical double copy. We also exploit the Kerr-Schild version of the classical double copy to determine the exact spacetime metric in the gravitational case. Finally, we discuss the direct implication of these results for Lorentzian signature via analytic continuation.