Studying of characteristics of One Hypercomplex Generalization of Quaternions

A.S. Turenko

Institute for Information Recording National Academy of Science of Ukraine,

Kyiv, Shpaka str. 2, 03113, Ukraine

Studying of characteristics of One Hypercomplex Generalization of Quaternions

The main properties of generalization of hypercomplex system of quaternion as antiquaternion are presented in this article. Definitions and studied antiquaternions conjugation are introduced, their rate and zero divisor, and how to perform operations on them.

Key words: antiquaternion, hypercomplex numerical system, zero divisor, pseudonorm, conjugate antiquaternion.

Introduction

There are many applications of hypercomplex numerical systems in the methods of processing and presenting information. The quaternion system has particular importance, applications which can solve many practical problems: the navigation and management of mobile objects, in mechanics, electrodynamics, cryptography, digital signal processing, and others.

Numerous application of quaternions is caused by their properties which allow to carry out various operations with vectors in three-dimensional the Cartesian system of coordinates. Therefore, it is advisable to consider the properties of other hypercomplex numerical systems, such as a generalization of hypercomplex quaternion as antiquaternion system. Such studies will allow to solve new practical problems or facilitate decision which has been previously discussed.

 

Definition of the problem

This work investigates the question of synthesis of hypercomplex numerical system of antiquaternion as a result of the application to the system of complex numbers doubling procedure of Grassmann - Clifford by system of double numbers. The main properties of antiquaternions, algorithms of performance of a set of the algebraic operations, necessary for use the system of antiquaternions in mathematical modeling, are studied.

 

Definitions and basic properties of antiquaternions

It is known [1, 2], the four-dimensional hypercomplex numerical system called a system of quaternions with basis the multiplication table of elements which is;

 

 

.

Quaternions are the result of anticommutative doubling of complex numbers by same system of numbers. Or, using a system of markings, introduced in [3], we can write:

. (1)

If we redouble the system of complex numbers by system of binary numbers with multiplication table

 

,

 

we obtain a system of antiquaternions , or through operator of doubling:

. (2)

Indeed, if we take the composition of bases and , we obtain a basis

.

Multiplication table of the obtained hypercomplex numerical system is constructed using multiplying of elements of this basis. However, we believe that the basic elements of the same name are multiplied by the rules of systems and . When multiplying them together remains commutative only, when at least one multiplier is or . Basic elements and are multiplied anticommutatative.

 

.

We will give some examples of multiplication of basic elements, taking into account these rules:

.

If to rename two-symbolical names of basic elements in one-symbolical:

, , , ,

то that we will receive the multiplication table of basic elements of system of antiquaternions:

 

 

 

(3)

 

 

The principle of multiplication of basic elements is represented in fig. 1, on which basic elements of antiquaternion’s system are represented by triangle tops. Product of any two elements from this three is equal to the third if movement from the first to the second multiplier coincides with the arrow direction, if the movement is opposite to the direction of the arrow - the third with a minus sign.

Fig. 1 Schematic image of the multiplication table of basic elements of antiquaternion’s system.

 

Thus, anti-quaternions are numbers of a look

, (4)

where: .

 

Conclusions

From the aforesaid follows that a set of arithmetic and algebraic operations in hypercomplex numerical system of anti-quaternions is defined, which allows to use this numerical system for creation of mathematical models in various areas of science and technology.

These operations allow to build various functions from antiquaternions, such as exponential, logarithmic, trigonometrical and hyperbolic functions, that will be a subject of further scientific researches.

 

 

1. Cantor I. L. Hypercomplex numbers. / Cantor I. L., Solodovnikov A. S. - M.: "Science", 1973. - 144.

2. Sinkov M. V. Finite-dimensional hypercomplex numerical systems. Theory bases. Applications. / M. V. Sinkov, Yu.E. Boyarinova, Ya.A. Kalinovsky. - K.: Infodruk 2010. – 388 р.

3. Kalinovsky Ya.A. High-dimensional isomorphic hypercomplex numerical systems and their use for efficiency increase calculations / Ya.A.Kalinovsky, Yu.E.Boyarinova. - К.: Infodruk 2012. – 183 р..

4. Catoni F. Two-dimensional hypercomplex Numbers and related Trigonometries and Geometries / Catoni F., Cannata R., Catoni V., Zampetti P. – Advances in Applied Clifford Algebras, 2004. — Vol. 14. — No. 1. — P.47-68.

5. Smirnov A.V. Commutative algebra of scalar quaternions / A.V. Smirnov. - Vladikavkaz mathematical magazine, 2004. – Vol. 6. — No. 2. — P.50-57.

6. Fuhrman Ya.A. Complex-valued and hypercomplex systems in problems of processing multidimensional Signals / Fuhrman Ya.A. Krevetsky A.V. Rozhentsov A.A. Hafizov R. G., Leukhin A.N. Egomin L.A. - M.: Fyzmatlіt, 2003. – 456 р.

7. Eliovich A. A. About norm of biquaternions and Other algebras with central interfaces [Electronic resource] / Eliovich A. A — Access mode: hypercomplex.xpsweb.com/page.php?lang= ru&id=176. 2004.

 

 

A.S. Turenko

Institute for Information Recording National Academy of Science of Ukraine,

Kyiv, Shpaka str. 2, 03113, Ukraine

Studying of characteristics of One Hypercomplex Generalization of Quaternions

The main properties of generalization of hypercomplex system of quaternion as antiquaternion are presented in this article. Definitions and studied antiquaternions conjugation are introduced, their rate and zero divisor, and how to perform operations on them.

Key words: antiquaternion, hypercomplex numerical system, zero divisor, pseudonorm, conjugate antiquaternion.

Introduction

There are many applications of hypercomplex numerical systems in the methods of processing and presenting information. The quaternion system has particular importance, applications which can solve many practical problems: the navigation and management of mobile objects, in mechanics, electrodynamics, cryptography, digital signal processing, and others.

Numerous application of quaternions is caused by their properties which allow to carry out various operations with vectors in three-dimensional the Cartesian system of coordinates. Therefore, it is advisable to consider the properties of other hypercomplex numerical systems, such as a generalization of hypercomplex quaternion as antiquaternion system. Such studies will allow to solve new practical problems or facilitate decision which has been previously discussed.

 

Definition of the problem

This work investigates the question of synthesis of hypercomplex numerical system of antiquaternion as a result of the application to the system of complex numbers doubling procedure of Grassmann - Clifford by system of double numbers. The main properties of antiquaternions, algorithms of performance of a set of the algebraic operations, necessary for use the system of antiquaternions in mathematical modeling, are studied.