Dear Hanifa,

I suggest you to take a look some  applications at the following book:

Ward J.P., Quaternions and Cayley numbers: algebra and applications, Kluwer Academic, Springer Science+Business Media Dordrecht, 1997.

Regards,

Wiwat Wanicharpichat

not really: The quaternion algebra is a THREE dimensional Lie algebra over a field K generated f.i. by your i,j,k with anticommutativ multiplication and Jacobi identity (+/-1 is not in the algebra). The quaternion group is SU(2) , the group of unitary 2x2 matrices with determinant 1 (sometimes: U(1)). If ai+bj+ck is a general element of the now REAL Lie algebra then exp[I(ai+bj+ck)], with I²  = -1, is an element of the quaternion group in the neighborhood of the identity.

it is 4 dimensional 

https://en.wikipedia.org/wiki/Quaternion_algebra

The Quaternion group is the non Abelian group of order 8 defined as above in the question, 

https://en.wikipedia.org/wiki/Quaternion_group

o.k., sorry, I was concentrating on Lie algebras and groups.

However, 1st, define a composition algebra C over a field K, nonassociative and a map called the norm N(X) of C, X element of C: 3 nonstandard properties (!), but see below for an example.

Next, define pairs by the Cayley-Dixon formulas, (X,Y)(Z,W) = …  = (A,B), and N’((X,Y)) = N(X) – aN(Y), and call this algebra the quadratic algebra C, a is element of K

Next, C is called generalized quaternion algebra and the C are called generalized Cayley algebra.

Further, R are “the” quaternion” numbers, and the algebra of 2x2 matrices over K is a split quaternion algebra, the norm is the determinant.

Your quaternion group is a discrete subset of this quaternion algebra, just take the standard Pauli spin matrices and the unit matrix.

Dear Hanifa,

some additional information you can find here:  http://www.stahlke.org/dan/publications/quaternion-paper.pdf

Dear Hanifa,

some additional information you can find here:  http://www.stahlke.org/dan/publications/quaternion-paper.pdf

Dear Hanifa,

some additional information you can find here:  http://www.stahlke.org/dan/publications/quaternion-paper.pdf

Roughly saying: 

1) quaternion algebra  (dim4) - a generalization of quaternions to a field

quaternion group (order 8) - quaternions under multiplication

2) number theory, quadratic forms, computer graphic in 3D, elecctrodynamics

Greetings, Taadeusz

 The quaternion is not field, it is a division ring.

Thank you dear Tadeusz.

I will be grateful if you have some references in what you gave in your answer 

Thank you dear Tadeusz.

I will be grateful if you have some references in what you gave in your answer.

Dear Wiwat, yes it is a skew field (division ring), I think Tandeusz used the historical term for skew fields (which considered differs from fields by commutativity )

An application is in a inertial navigation, e.g. Group-theoretic methods in mechanics and applied mathematics / D. M. Klimov and V. Ph. Zhuravlev

Dear Hanifa,

I suggest you to take a look some  applications at the following book:

Ward J.P., Quaternions and Cayley numbers: algebra and applications, Kluwer Academic, Springer Science+Business Media Dordrecht, 1997.

Regards,

Wiwat Wanicharpichat

Dear Hanifa,

instead of references i have sent  you some papers on QA and QG.

Greetings, Tadeusz

Thank you dear Tadeusz, dear Wiwat, dear Dimetry, dear Yuri, dear Anton, all dear colleagues for your help

Dear Hnifa,

I have also found in my files two books.

I hope they will be helpful.

Greetings, Tadeusz

quaternion algebra is group algebra of the groups genertors i,j,k and -1 with relations ij=-ji=k, jk=-ki=j, ki=-ik=j

Thank you dear Janab so much, this is an other substantial relation between them. in addition to the  Quaternion group is a discrete subset of this quaternion algebra as showed Anton in his answer. 

Dear Cenap,

quaternion algebra H is not a group algebra for the classical quaternion group Q_8 since it is NOT 8-dimensional (Q_8 contains 8 elements). There is only a representation of Q_8 in H. 

Dear Cenap,

quaternion algebra H is not the group algebra for the classical quaternion group Q_8 since it is NOT 8-dimensional (Q_8 contains 8 elements). There is only a representation of Q_8 in H. 

yes u r right thankssss

Yes, I searched about that point in time. I found the notion: "Algebra of a finite group, in French ( l'algèbre d'un groupe fini ), this one requires that the dimension of the algebra is the order of the group.   The second notion is the group algebra of a group as said Janab, it meant taking all the linear combinations of the elements of the group through a field, just short information doesn't talk about the dimension. My question: are both notions the same?

See these links please:

fisrt notion:

 https://fr.wikipedia.org/wiki/Alg%C3%A8bre_d%27un_groupe_fini

second notion:

http://mathworld.wolfram.com/GroupAlgebra.html

Now, I think if we take the second notion, we get an algebra a 4- dimensional algebra, because the genrators of Q8  are the basis of H. 

Now, I think if we take the second notion, we get an algebra a 4- dimensional algebra, because the genrators of Q8  are the basis of H. 

For he first notion, I think we have 8-dimensional, because we work on the real field then on the complex field, so the algebra of a group of order n becomes 2n-dimensional. 

What do you think?

Dear Hanifa,

in general, if you have an algebra A over a field F and some group G of invertible elements in A, you encounter two DIFFERENT but closely related objects.

1. The group algebra FG consisting of formal finite linear combinations of g in G with coefficients in F. Always dim FG= card G. Usually, FG\ne A.

2. The algebra A itself as a natural FG-module via inclusion G \subset A. It is equivalent to say that either G is represented in A, or that G acts on A by (left) shifts  a->ga, since there is a multiplication in A.

For quaternion algebra H=A and quaternion group Q=G we have respectively

1. The 8-dimensional (over the field R of real numbers)  group algebra RQ for which the basic elements '1' and '-1'  in Q are linearly independent.

2. The 4-dimensional (over the field R of real numbers)  quaternion algebra H  for which the quaternions 1 and -1  are linearly dependent: 1+(-1)=0.

Please pay attention  to the same difference for cases A=R and G={-1;1} or A= Mat(n,R), G=GL(n,R).

Dear Hanifa,

in general, if you have an algebra A over a field F and some group G of invertible elements in A, you encounter two DIFFERENT but closely related objects.

1. The group algebra FG consisting of formal finite linear combinations of g in G with coefficients in F. Always dim FG= card G. Usually, FG\ne A.

2. The algebra A itself as a natural FG-module via inclusion G \subset A. It is equivalent to say that either G is represented in A, or that G acts on A by (left) shifts  a->ga, since there is a multiplication in A.

For quaternion algebra H=A and quaternion group Q=G we have respectively

1. The 8-dimensional (over the field R of real numbers)  group algebra RQ for which the basic elements '1' and '-1'  in Q are linearly independent.

2. The 4-dimensional (over the field R of real numbers)  quaternion algebra H  for which the quaternions 1 and -1  are linearly dependent: 1+(-1)=0.

Please pay attention  to the same difference for cases A=R and G={-1;1} or A= Mat(n,R), G=GL(n,R).

Dear Hanifa,

in general, if you have an algebra A over a field F and some group G of invertible elements in A, you encounter two DIFFERENT but closely related objects.

1. The group algebra FG consisting of formal finite linear combinations of g in G with coefficients in F. Always dim FG= card G. Usually, FG\ne A.

2. The algebra A itself as a natural FG-module via inclusion G \subset A. It is equivalent to say that either G is represented in A, or that G acts on A by (left) shifts  a->ga, since there is a multiplication in A.

For quaternion algebra H=A and quaternion group Q=G we have respectively

1. The 8-dimensional (over the field R of real numbers)  group algebra RQ for which the basic elements '1' and '-1'  in Q are linearly independent.

2. The 4-dimensional (over the field R of real numbers)  quaternion algebra H  for which the quaternions 1 and -1  are linearly dependent: 1+(-1)=0.

Please pay attention  to the same difference for cases A=R and G={-1;1} or A= Mat(n,R), G=GL(n,R).

dears hanifa and yuri

u r right actually this is a quotient of RQ by killing i^2 = j^2= k^2 =-1 it is a quotient algbra of quaternion group algebra

1 and -1 are not linearly independent this is a problem

quaternion algbra is 4 dimensional whereas quaternion group algebra is 8 dimensional because 1 and -1 are linearli independent in group algebra

Dear Cenap,

I think that the relations [1]+[-1]=0, [i]+[-i]=0, [j]+[-j]=0, [k]+[-k]=0 are needed, brackets standing for copies of elements of Q in RQ (in order to not confound them with quaternions).

If you prefer the relations [i]^2=-[1],  ... should be written since the relation, say, [i]^2=[-1] holds in G or RG and "kills" nothing.

u r right the reason is 1 and -1

in RQ they are independent but in H algebra they are not independent so dimH = 4 and dim RQ = 8 which is reason im wrong not killing by group relations

Dear Yuri, I coulden"t understand this point:

1. The 8-dimensional (over the field R of real numbers) group algebra RQ for which the basic elements '1' and '-1' in Q are linearly independent.

Hello

I suggest you the book On Quaternions and  Octonions from J.H Conway and Derek A. Smith

Quaternion Group is a group  by 1,i, j, k with i² =j² =k² =-1,  ij=-ji=k

Quaternion Algebra is an Algebra allowing all the Algebraic operations (+ *  - division  ..)according to (1,i, j, k with i² =j² =k² =-1, ij=-ji=k)

Regards

Dear Hanifa Zekraouti,

As you said Quaternion group is a set with 8 elements {+1, _1, i,-i, j, -j, k, -k}, where the operations defined as above. It is non abelian

But Quaternion algebra is a special case of a general algebra.

Let F be a field of characteristic not equal to 2. A be a 4 dimensional (right)vector space over F with basis {1, i, j, k} Let a, b be two non zero elements of F. Now we define multiplication in A, so that is is bilinear and associative as below. 

i2 = a, j2= b, ij = -ji = k, k2 =ab, ik = -ki = ja, jk = -kj = -ib. The element a of F is identified with 1a of A and similar identification for other elements of F Then A is a (right) algebra over F. 

Now take F to be the field of real numbers and a=1 and b=-1.  The algebra now we get is Quaternion Algebra. Its dimension is 4

K.Premakumari.

Dear Hanifa Zekraouti,

As you said Quaternion group is a set with 8 elements {+1, _1, i,-i, j, -j, k, -k}, where the operations defined as above. It is non abelian

But Quaternion algebra is a special case of a general algebra.

Let F be a field of characteristic not equal to 2. A be a 4 dimensional (right)vector space over F with basis {1, i, j, k} Let a, b be two non zero elements of F. Now we define multiplication in A, so that is is bilinear and associative as below. 

i2 = a, j2= b, ij = -ji = k, k2 =ab, ik = -ki = ja, jk = -kj = -ib. The element a of F is identified with 1a of A and similar identification for other elements of F Then A is a (right) algebra over F. 

Now take F to be the field of real numbers and a=1 and b=-1.  The algebra now we get is Quaternion Algebra. Its dimension is 4

K.Premakumari.