Invariant Tensor Product

Various forms of invariant tensor products appeared in the literature implicitly, for example, in Schur’s orthogonality for finite groups. In many cases, they are employed to study the space HomG(π1, π2) where one of the representations π1 and π2 is irreducible. In this paper, we formulate the concept of invariant tensor product uniformly. We also study the invariant tensor functor associated with discrete series representations for classical groups. For motivations and applications.

Download PDF

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.

Download PDF

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.

Download PDF

How to Prove the Riemann Hypothesis

The aim of this paper is to prove the celebrated Riemann Hypothesis. I have already discovered a simple proof of the Riemann Hypothesis. The hypothesis states that the nontrivial zeros of the Riemann zeta function have real part equal to 0.5. I assume that any such zero is s=a+bi. I use integral calculus in the first part of the proof. In the second part I employ variational calculus. Through equations (50) to (59) I consider (a) as a fixed exponent, and verify that a=0.5. From equation (60) onward I view (a) as a parameter (a <0.5) and arrive at a contradiction. At the end of the proof (from equation (73)) and through the assumption that (a) is a parameter, I verify again that a=0.5.

Download PDF

Exploring Rack-bi algebra

Basic Lie theory relies heavily on the fundamental links between associative algebras, Lie algebras and groups. The study focuses on self-distributive multiplication on co algebras reveals that it leads to rack bi algebra, which is universal enveloping of the Lie algebra.

Get PDF Link Here