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 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.

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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...

Matrix Lie Groups: An Introduction

Lie theory, the theory of Lie groups, Lie algebras, and their applications is a fundamental part of mathematics that touches on a broad spectrum of mathematics, including geometry (classical, differential, and algebraic), ordinary and partial differential equations, group, ring, and algebra theory, complex and harmonic analysis, number theory, and physics (classical, quantum, and relativistic).

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It typically relies upon an array of substantial tools such as topology, differentiable manifolds and differential geometry, covering spaces, advanced linear algebra, measure theory, and group theory to name a few. However, we will considerably simplify the approach to Lie theory by restricting our attention to the most important class of examples, namely those Lie groups that can be concretely realized as (multiplicative) groups of matrices.

Lie theory began in the late nineteenth century, primarily through the work of the Norwegian mathematician Sophus Lie, who called them “continuous groups,” in contrast to the usually finite permutation groups that had been principally studied up to that point. An early major success of the theory was to provide a viewpoint for a systematic understanding of the newer geometries such as hyperbolic, elliptic, and projective, that had arisen earlier in the century. 

Verification of Some Properties of the C-nilpotent Multiplier in Lie Algebras

The purpose of this paper is to obtain some inequalities and certain bounds for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras. Also, we give an inequality for the dimension of the c-nilpotent multiplier of L connected with dimension of the Lie algebras γd (L) and L / Zd−1 (L) . Finally, we compare our results with the previously known result.

All Lie algebras referred to in this article are (of finiteor infinite dimension) over a fixed field F and the square brackets [ , ] denotes the Lie product. Let 0→R→F→L→0 be a free presentation of a Lie algebra L, where F is a free Lie algebra.

This is analogous to the definition of the Baer-invariant of a group with respect to the variety of nilpotent groups of class at most c given by Baer [1-3] (for more information on the Baer invariant of groups).

The purpose of this paper is to obtain some inequalities for the dimension of the c-nilpotent multiplier of finite dimensional nilpotent Lie algebras and their factor Lie algebras (Corollary 2.3 and Corollary 2.5). Finally, we compare our results to upper bound given.

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