Complexification of the Exceptional Jordan Algebra and Its Application to Particle Physics

Recent papers contributed revitalizing the study of the exceptional Jordan algebra $\mathfrak{h}_{3}(\mathbb{O})$ in its relations with the true Standard Model gauge group $\mathrm{G}_{SM}$. The absence of complex representations of $\mathrm{F}_{4}$ does not allow $\Aut\left(\mathfrak{h}_{3}(\mathbb{O})\right)$ to be a candidate for any Grand Unified Theories, but the automorphisms of the complexification of this algebra, i.e., $\mathfrak{h}_{3}^{\mathbb{C}}(\mathbb{O})$, are isomorphic to the compact form of $\mathrm{E}_{6}$ and similar constructions lead to the gauge group of the minimal left-right symmetric extension of the Standard Model.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications

In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable \emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.

Static Perfect Fluid Space-Time on Almost Kenmotsu Manifolds

In this work, we intend to investigate the characteristics of static perfect fluid space-time metrics on almost Kenmotsu manifolds. At first we prove that if a Kenmotsu manifold $M$ is the spatial factor of static perfect fluid space-time then it is $\eta$-Einstein. Moreover, if the Reeb vector field $\xi$ leaves the scalar curvature invariant, then $M$ is Einstein. Next we consider static perfect fluid space-time on almost Kenmotsu $(\kappa,\mu)'$-manifolds and give some characteristics under certain conditions.

A Note on the Representation of Clifford Algebras

In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $\Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $\mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

New Parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and Some Explicit Formulas for Its Logarithm

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\bbr)\cong\mathrm{SL}(2,\bbr)$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\bbr)\to\mathrm{SL}(2,\bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups.\\ Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

Symbol Correspondence for Euclidean Systems

The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.