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 hyperfield
as 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 the
spectroscopy 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.Read More...
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