NEW CONSTRUCTION OF ALGEBRAS AS QUOTIENTS
Publicación Cuatrimestral. Vol. 5, No 2, Mayo/Agosto, Año 2020, Ecuador (p. 53-58) 53
Publicación Cuatrimestral. Vol. 5, No 2, Mayo/Agosto, 2020, Ecuador (p. 53-58). Edición continua
NEW CONSTRUCTION OF ALGEBRAS AS QUOTIENTS
José Játem
Research group Partial Differential Equations and Clifford analysis, Universidad Simón Bolívar, Caracas, Venezuela.
E-mail: jrjatem@gmail.com
Autor para la correspondencia: jrjatem@gmail.com
Recibido: 05-12-2019 / Aceptado: 17-08-2020 / Publicación: 31-08-2020
Editor Académico: Eusebio Ariza García
ABSTRACT
In this article we have presented a new approach to define algebras using for a natural number , the set of natural
numbers in base , none of their digits equal to zero. The study was developed in the context of vector -spaces and the
vector space definitions of the formal multiples of any element of the field , of the direct sum of vector spaces and
binary operations on vector spaces were used. The results obtained were the construction of a vector space denoted by ,
on the basis of the particular set of natural numbers in base mentioned, which allowed novel ways of defining the well-
known and very important algebras of complex numbers and that of quaternions on as quotients of ideals of , for
suitably chosen ideals . With this new approach and with the help of the vector spaces , known algebras can be presented
in a different way than those found up to now, by using certain ideals of those spaces in their quotient form. The spaces
can be over any field and other algebras such as Clifford algebras can be constructed using this procedure.
Keywords: Algebras, Quotients in algebras, Complex numbers and quaternions as quotients of algebras.
NUEVA CONSTRUCCIÓN DE ÁLGEBRAS COMO COCIENTES
RESUMEN
En este artículo se ha presentado un nuevo enfoque para definir álgebras usando para un número natural , el
conjunto de números naturales en base , ninguno de sus dígitos iguales a cero. El estudio se desarrolló en el contexto de
los -espacios vectoriales y se usaron las definiciones de espacio vectorial de los múltiplos formales de un elemento
cualquiera del cuerpo , de la suma directa de espacios vectoriales y operaciones binarias sobre espacios vectoriales.
Los resultados obtenidos fueron la construcción de un espacio vectorial denotado por , sobre la base del particular
conjunto de números naturales en base mencionado, que permitió novedosas formas de definir las conocidas y muy
importantes álgebras de los números complejos y la de los cuaterniones sobre como cocientes de ideales de , para
ideales convenientemente elegidos. Con este nuevo enfoque y con la ayuda de los espacios vectoriales se pueden
presentar álgebras conocidas de manera distinta a las encontradas hasta ahora, al usar en su forma de cociente ciertos
ideales de esos espacios . Los espacios pueden ser sobre cualquier cuerpo y otras álgebras como las álgebras de
Clifford se pueden construir usando este procedimiento.
Palabras clave: Algebras, cocientes en álgebras, Números complejos y quaterniones como cocientes en álgebras.
Artículo de
Investigación
Artículo de Investigación
Ciencias Matemáticas
José Játem
54
NOVA CONSTRUÇÃO DE ÁLGEBRAS COMO QUOCIENTES
RESUMO
Neste artigo apresentamos uma nova abordagem para definir as álgebras usando para um número natural , o
conjunto de números naturais na base, nenhum de seus dígitos igual a zero. O estudo foi desenvolvido no contexto de
espaços vetoriais e foram utilizadas as definições de espaço vetorial dos múltiplos formais de um elemento qualquer
do campo , da soma direta de espaços vetoriais e operações binárias em espaços vetoriais. Os resultados obtidos foram
a construção de um espaço vetorial denotado por , com base no conjunto particular de números naturais na base
mencionados, o que permitiu novas formas de definir as conhecidas e muito importantes álgebras de números complexos
e dos quatérnions em como quocientes de ideais de , para ideais adequadamente selecionados. Com esta nova
abordagem e com a ajuda dos espaços vetoriais, as álgebras conhecidas podem ser apresentadas de uma forma diferente
das encontradas até agora, usando certos ideais desses espaços na sua forma de quociente. Os espaços podem estar em
qualquer campo e outras álgebras, tais como álgebras de Clifford, podem ser construídas usando este processo.
Palavras-chave: Álgebras, quocientes nas álgebras, Números complexos e quatérnions como quocientes nas álgebras
Citación sugerida: Jatem, J. (2020). NEW CONSTRUCTION OF ALGEBRAS AS QUOTIENTS. Revista Bases de la
Ciencia, 5(2), 53-58. DOI: 10.33936/rev_bas_de_la_ciencia.v%vi%i.2107 Recuperado de:
https://revistas.utm.edu.ec/index.php/Basedelaciencia/article/view/2107
Orcid IDs:
Dr. José Játem: https://orcid.org/0000-0001-6153-6720
Dr. Eusebio Ariza García: https://orcid.org/0000-0001-7754-2666
NEW CONSTRUCTION OF ALGEBRAS AS QUOTIENTS
Publicación Cuatrimestral. Vol. 5, No 2, Mayo/Agosto, Año 2020, Ecuador (p. 53-58) 55
1. INTRODUCTION
Given any symbol x, we define the vector space over denoted by <x>, consisting of all formal real
multiples of x as the set
, together with the sum +:
→
, defined
for all rx,
by and the product : <x> → <x>, defined a
and
.
We will also deal with the (external) “direct sum” of vector spaces:
Let be a collection of vector spaces, the direct sum of all vector spaces V of is the set denoted by
V
V, whose elements are the tuples (u
V
)
V
, where u
V
, for each and
for almost
all (that means
for all but a finite number of vector spaces V of ).
This set turns into a vector space over with the operations:
Sum: +:
→
,
,
,
and the product :[
→
, defined for all
and all
r by r
= (rx
V
)
V
.
Notation: If and , with we will understand the tuple
, where
if
and
.
Given basis
of each vector space
, we consider the subset
of
. The union
V
is a basis of
.
Next we consider for all natural number k ≥ 2, the set
of all natural numbers written in basis k,
none of its digits null. The digits used to express the numbers of
are the elements of
(
when needed).
The next step is to consider the direct sum 1
x
k
*
<x> or
x
k
*
<x>, which is a vector
space denoted by (
when needed).
Vectors of are the tuples
r
x
x
x
k
, where r,
and
for almost all
with
, will be denoted by 1
and, for
The tuple
r
x
x
x
k
, with
, will be denoted by 1
and for all real number r, with
the tuple 1
r
x
x
x
k
with
,
We recall for all
we denote the
José Játem
56
tuple
r
x
x
x
k
, where
,
and
. Once these has been said, any tuple
r
x
x
x
k
, where
for all digit x different from
, can be written as
.
In order to provide with an algebra over structure, we define the operation : as follows:
i) , 1
1
ii) ,
,
,
, where
,
, is the ordered concatenation of the α-digits with the β-digits,
meaning
.
iii) Lastly, , , where
and
with
,
and then
It can be easily proven that this operation is associative and bilinear, which makes an algebra over
with 1
the multiplicative identity, only commutative when k = 2, because for all k > 2,
.
In order to keep the usual exponential notation for the digits
, we write
(n-times x) and
1
.
For themes related with general algebra, like vector spaces and direct sums see (Atiyah & Macdonald,
1969; Hartley & Hawkes, 1983).
2. ALGEBRAS AS QUOTIENTS
Our aim here is to build algebras over as quotients of conveniently chosen two sided ideals of .
For instance, lets choose the ideal I of
generated by 1
and the quotient
.
The set of digits
of is the singleton
and a basis of
is
1
1
1
, therefore the set
1
generates .
We claim that one basis of the quotient over I is the set 1
, which we will proceed to
prove right now.
i) generates
It can be easily check that
NEW CONSTRUCTION OF ALGEBRAS AS QUOTIENTS
Publicación Cuatrimestral. Vol. 5, No 2, Mayo/Agosto, Año 2020, Ecuador (p. 53-58) 57
Which implies actually generates
.
ii) is linearly independent:
Consider the null linear combination of vectors of
, where
a
, . With the product of
inherited from
multiply by
at both sides of the equality to obtain
, from which
and
It turns out, that
, and denoting with we have the following table for the
product on the basis of
.
I
I
i
i
-
Which means
and, algebraically speaking, both fields:
and , are the same object.
(Yaglom, 1968).
Another interesting construction is the one of the quaternions, usually denoted by (Gürlebeck 1997;
Hamilton, 1866).
On that purpose consider the two-sided ideal I of
generated by all elements of the form:
i)
, for all
ii) , for all x, , such that
iii) , and
In this case the set
is a basis of
fact that will be proven right
now.
i) generates
A basis of is
is
4
*
therefore the set
x
1
1
x
2
2
i
i
generates
.
Check that for all and
(1
R
+ I) if k is even
(x + I) if k is odd
, which implies that generates
ii) is linearly independent:
José Játem
58
Consider the null linear combination
,
where a, b, c, and right-multiply at both sides of the equality by
, to obtain
, which implies
and .
To conclude
just rename the elements of as
and consider the following table of the product restricted to :
1
i
J
k
1
1
i
J
K
i
i
-1
K
-j
j
j
-k
-1
I
k
k
j
-i
-1
3. CONCLUDING REMARKS
With this new approach and with the help of the vector spaces , known algebras can be presented
in a different way than those found up to now, by using certain ideals of those spaces in their quotient
form. The spaces can be over any field and other algebras can be constructed using this procedure.
In particular, as quotients of
the Clifford Algebras (Brackx, Delanghe & Sommen, 1982; Játem &
Vanegas, 2018), may also be built, which will appear in a second article now in preparation.
4. REFERENCES
Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley Publishing Co.,
Reading, Mass.-London-Don Mills, Ont.
Brackx, F., Delanghe, R., & Sommen, F. (1982). Clifford analysis. Research Notes in Mathematics, 76. Boston, London,
Melbourne: Pitman Advanced Publishing Company.
Gürlebeck, K., & Sprössig, W. (1997). Quaternionic and Clifford calculus for engineers and physicists. John Wiley &
Sons, Chichester.
Hamilton, W. R. (1866). Elements of quaternions. Longmans, Green, & Company.
Hartley, B., & Hawkes, T. O. (1983). Rings, Modules and Linear Algebra. Chapman and Hall, Edition 4.
Játem, J., & Vanegas, J. (2018). Caracterización de Álgebras de Clifford como anillos cocientes, MATEMATICA, 16(1),
57-60. Retrieved from http://www.revistas.espol.edu.ec/index.php/matematica/article/view/460/327
Yaglom, I. M. (1968). Complex Numbers in Geometry. Academic Press, New York.
