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A short review about nonassociative algebraic systems (mainly nonassociative algebras) and their physical applications is presented. We begin with some motivations, then we give a brief historical overview about the formation and development of the concept of hypercomplex number system and about...
Foundations of the theory of binary analytic composition laws with two-sided neutral element are discussed.
A survey of the methods of the theory of quasigroups and loops in algebra and geometry is presented in order to attract the attention of mathematicians and physicists to promising applications of this new branch of mathematics in applied sciences.
Algebraic systems called local geodesic loops and their tangent Akivis algebras are considered. Their possible role in the theory of gravity is discussed. Quantum conditions for infinitesimal quantum events are proposed.
Continuous Moufang transformations are introduced and discussed. Commutation relations for infinitesimal Moufang transformations are established. The resulting Lie algebra has quite impressive structure equations, well known from the theory of alternative algebras.
We show explicitly the connection between the infinitesimal form of the birepresentation of the Moufang loop and the octonionic representation of SO(8). Possible types of Sugawara construction for the birepresentations are considered.
In this paper the geometric interpretation of the exceptional Lie groups F4, E6, E7, and E8 is given. These groups are groups of motions of elliptic hyperbolic planes over nonassociative algebras of octaves and split octaves and their tensor products with algebras of usual and split complex...
Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.
The relativistic first-order wave equations for massive particles with spin 0,1,1/2 are formulated in terms of a factorization of the Klein–Fock equation by means of the algebra of octonions. An analogous method applied to Hamiltonian of the quantum isotropic oscillator leads to the natural...
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