Studies of the Regular and Irregular Iso representations of the Lie-Santilli Isotheory

As it is well known, the Lie theory is solely applicable to dynamical systems consisting of point-like particles moving in vacuum under linear and Hamiltonian interactions (systems known as exterior dynamical systems). One of the authors (R.M. Santilli) has proposed an axiom-preserving broadening of the Lie theory, known as the Lie-Santilli iso theory, that is applicable to dynamical; systems of extended, nonspherical and deformable particles moving within a physical medium under Hamiltonian as well as non-linear and non-Hamiltonian interactions (broader systems known as interior dynamical systems).

In this paper, we study apparently for the first time regular and irregular iso representations of Lie-Santilli iso algebras occurring when the structure quantities are constants or functions, respectively. A number of applications to particle and nuclear physics are indicated. It should be indicated that this paper is specifically devoted to the study of iso representations under the assumption of a knowledge of the Lie-Santilli iso theory, as well as of the isotopies of the various branches of 20th century applied mathematics, collectively known as iso mathematics, which is crucial for the consistent formulation and elaboration of iso theories.

Non-associative slave-boson decomposition

Everybody knows that the algebra of non-perturbative operators in quantum theory exists but nobody knows its exact form. In this paper the idea is discussed that the constraint (2.2) in t-J model of high-temperature superconductivity is a new generating relation for an algebra of operators the product of which gives us the electron operator. On the perturbative level the algebra of quantum fields is defined by canonical (anti) commutative relations. The algebra of non-perturbative operators should be more complicated and should be generated not only by canonical (anti) commutative relations but should exist other generating relations as well. In this paper we discuss the idea that the constraint (2.2) is an anti associator in a non-associative algebra of quantum non-perturbative operators.

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A canonical semi-classical star product

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controlling the deformation theory of associative algebras. We prove that there is a canonical solution (choice independent) within the class of graphs without circuits, i.e. at the level of the free operad, without imposing the Jacobi identity.

The proof is a consequence of the unique factorization property of the pre-Lie algebra of graphs (tree operad), where composition is the insertion of graphs. The restriction to graphs without circuits, i.e. at “tree level”, accounts for the interpretation as a semi-classical solution. The fact that this solution is canonical should not be surprising, in view of the Hausdorff series, which lies at the core of almost all quantization prescriptions.

Dilatation structures I. Fundamentals

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is apart of a series of papers which show that such a structure allows to do non-commutative analysis, in the sense of differential calculus, on a large class of metric spaces, some of them fractals. We also describe a formal, universal calculus with binary decorated planar trees, which underlies any dilatation structure.

The purpose of this paper is to introduce dilatation structures on metric spaces. A dilatation structure is a concept in between a group and a differential structure. Any metric space (X,d) endowed with a dilatation structure has an associated tangent bundle. The tangent space at a point is a conical group that is the tangent space has a group structure together with a one-parameter group of auto morphisms. Conical groups generalize Carnot groups, i.e nilpotent groups endowed with a graduation. Each dilatation structure leads to a non-commutative differential calculus on the metric space (X, d).

Deformations of ternary algebras

The aim of this paper is to extend to ternary algebras the classical theory of formal deformations of algebras introduced by Gerstenhaber. The associativity of ternary algebras is available in two forms, totally associative case or partially associative case. To any partially associative algebra corresponds by anti-commutation a ternary Lie algebra. In this work, we summarize the principal definitions and properties as well as classification in dimension 2 of these algebras. Then we focuss ourselves on the partially associative ternary algebras, we construct the first groups of a cohomolgy adapted to formal deformations and then we work out a theory of formal deformation in a way similar to the binary algebras.

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On compact realifications of exceptional simple Kantor triple systems

Let A be the realification of the matrix algebra determined by Jordan algebra of hermitian matrices of order three over complex composition algebra. We define an in volutive auto morphism on A with a certain action on the triple system obtained from A which give models of simple compact Kantor triple systems. In addition, we give an explicit formula for the canonical trace form and the classification for these triples and their corresponding exceptional real simple Lie algebras. Moreover, we present all realifications of complex exceptional simple Lie algebras as Kantor algebras for a compact simple Kantor triple system defined on a structurable algebra of skew-dimension one.

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Lie-admissible co-algebras

After introducing the concept of Lie-admissible co-algebras, we study a remarkable class corresponding to co-algebras whose co-associator satisfies invariance conditions with respect to the symmetric group 3. We then study the convolution and tensor products. An interesting class of Lie-admissible co-algebras is obtained by dualizing the Gi-associative algebras. These Lie-admissible algebras has been introduced and developed. Let us point out these initially notations.

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Classification of Canonical Bases for (n−1)−dimensional Subspaces of n− Dimensional Vector Space

Canonical bases for (n-1)-dimensional sub spaces of n-dimensional vector space are introduced and classified in the article. This result is very prospective to utilize canonical bases at all applications. For example, maximal sub algebras of Lie algebras can be found using them.

The canonical bases for (n-1)-dimensional sub spaces of n− dimensional vector space are introduced in the article, and all non equivalent of them are classified (Theorem 2). This result generalizes a particular result for 5-dimensional sub spaces of 6-dimensional vector space obtained in the previous article of the same author.To analyze the general case, reduced row echelon forms of matrices are utilized; about reduced row echelon forms. In addition to the principal result, all non-equivalent reduced row echelon forms for (n−1)×n matrices of the rank (n−1) are found.

Lie Group Methods for Eigenvalue Function

By considering a C∞ structure on the ordered non-increasing of elements of Rn, we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results are applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.

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A Meeting of Great Minds, Sophus Lie and John Nash throughout their Works

It is well known that Marius Sophus Lie (1842-1899) and John Forbes Nash (1928-2015) are great mathematicians. Sophus Lie comes from Norway and John Nash from United States of America. Their stories have certain resemblances and remarkable relations. This editorial would emphasize some of them. When they have started their university studies, their respective first interests were not mathematics.

That is to say, Lie has been in Astronomy and Nash in Chemical Engineering. Whereas,when they worked on mathematics, the first had Lobatchevski award in 1897 and the second, Nobel prize 1994 and Abel award 2015 (Niels Abel is the uncle of the wife of Sophus Lie: Anna Birch). In addition, their contributions in geometry are considerable, particularly in differential equations. Lie worked on transformation groups relative to partial differential equations, in other words, on Lie groups and on special non-associative algebras named Lie algebras.