Algebra, Hyperalgebra and Lie-Santilli Theory

The theory of hyperstructures can offer to the Lie-Santilli Theory a variety of models to specify the mathematical representation of the related theory. In this paper we focus on the appropriate general hyperstructures, especially on hyperstructures with hyperunits. We define a Lie hyperalgebra over a hyperfieldas well as a Jordan hyperalgebra, and we obtain some results in this respect. Finally, by using the concept of fundamental relations we connect hyper algebras to Lie algebras and Lie-Santilli-addmissible algebras.

The structure of the laws in physics is largely based on symmetries. The objects in Lie theory are fundamental, interesting and innovating in both mathematics and physics. It has many applications to thespectroscopy of molecules, atoms, nuclei and hadrons. The central role of Lie algebra in particle physics is well known. A Lie-admissible algebra, introduced by Albert, is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket [a,b] = ab − ba. Examples include associative algebras, Lie algebras and Okubo algebras. Lie admissible algebras arise in various topics, including geometry of invariant affine connections on Lie groups and classical and quantum mechanics.