With contributions derived from presentations at an international conference, Non-Associative Algebra and Its Applications explores a wide range of topics focusing on Lie algebras, nonassociative rings and algebras, quasigroups, loops, and related systems as well as applications of nonassociative algebra to geometry, physics, and natural sciences. This book covers material such as Jordan superalgebras, nonassociative deformations, nonassociative generalization of Hopf algebras, the structure of free algebras, derivations of Lie algebras, and the identities of Albert algebra. It also includes applications of smooth quasigroups and loops to differential geometry and relativity.
Table of Contents
On Non-unitary Representations of the Rotation Group and Magnetic Monopoles. Generalized Derivations of Quantum Polynomials. Abelian Group Gradings on Simple Algebras. Groebner-Shirshov Basis for Lie Algebras. Classification of Solvable Three-Dimensional Lip-Triple Systems. Nonassociative Algebra Structures on Irreducible Representations of the Simple Lie Algebra. On Locally Finite Split Lie Triple Systems. On a Special Kind of Malcev Algebras. New Realizations of Hadronic Supersymmetry Based on Octonions. Application of Octonionic Algebras in Hadronic Physics. Application of Octonionic Algebras in Hardonic Physics. On Flexible Right-Nilalgebras Satisfying x(zy) = y(zx). Lie Algebras: Applications to the Classical Electromagnetic Fields. Helicity Basis and Parity. A New Look at the Freudenthal. On the Theory of Left Loops. Approximation of Locally Compact Groups by Finite Quasigroups. Some Classes of Nonassociative Algebras. Some Results on the Theory of Smooth Bol-Bruck Loops. A Nonzero Element of Degree 7 in the Center of the Free Alternative Algebra. The Identities of the Simple Non-Special Jordan Algebra. Ternary Derivations of Finite Dimensional Real Division Algebras. The Transformation Algebras of Bernstein Graph Algebras. On Composition, Quadratic, and some Triple Systems. Combinatorial Rank of a Frobenius-Lusztig Kernel. On Representations on Right Nilalgebras of Right Nilindex Four and Dimension Four. An Introduction to Associator Quantization. Prosymmetric Spaces. The Exponential Function and the Fundamental Theorem of Algebra for the Cayley Dickson Algebras. Dimension Filtration on Binary Systems. Right Alternative Bimodules. Some Applications of Quasigroups and Loops in Physics. Gravity within the Framework of Nonassociative Geometry. One-to-One Correspondence between Bi-Linear and Triple Product Algebras. Algebras satisfying Local Symmetric Triality Principle. Operads and Nonassociative Deformations. Representations of Quantum Algebras at Roots of 1. Algebras, Hyperalgebras, Nonassociative Bialgebras, and Loops. Algebraic Structures on Lie Algebras, Vinberg Algebras. Algebraic and Differential Structures in Renormalized Perturbation Quantum Field Theory. Survey on Smooth Quasigroups Development. On Kikkawa Spaces. Bol and Bruck Identities in Recent Research. New Example of a Simple Jordan Superalgebra with Associative Even Part. Unital Irreducible Representations of Small Simple Jordan Superalgebras. The Lie Product on the Lie Bialgebra Duals of the Witt and Virasoro Algebras. Derivations and Automorphisms of Free Algebras. On Derivations and Automorphisms of a Lie Algebra. Subrings of Finite Division Rings. Realizaiton of Finite Groups by Nets in Complex Projective Plane.