Quaternion Algebras 

Miriam Ciavarella
Universit? degli Studi di Torino
Italy
miriam.ciavarella@unito.it 

Marina Marchisio
Universit? degli Studi di Torino
Italy
marina.marchisio@unito.it  

Introduction 

 

The aim of this  worksheet is to define some procedures in order to make computations in a quaternion algebra  over the field of rational number . 

Quaternion algebra means something more general than the algebra of Hamilton's quaternions (for which there exists already a Maple package) .  

Let we recall some definitions. A quaternion algebra  over   is a central simple algebra of dimension 4 over . To give a quaternion algebra  is equivalent to give a pair  of non-zero rational numbers so that is defined as the -algebra of basis  where the elements verify the relations and .  

The conjugation is the -endomorphism  of  which extends the non-trivial -automorphism  of , defined by ConjQ. This is  an involutive anti-automorphism of  and we denote by ConjQ the quaternion conjugate of the element . 

Let ; let we recall the definitions of two rational numbers associated to h:  the reduced trace of  is TrQConjQ and the reduced norm of  is  NormQConjQ.  The invertible elements of  are those elements  with non-zero reduced norm.  

Let v be a place of  with completition  (so it is either the p-adic numbers field  for some prime p or the real number field R) .  We say that B is unramified at v if  ( defined as the tensor product over ) is isomorphic to the  matrices over . We say that B is ramified at v if  is the quaternion division algebra over . The set of  places of  where a quaternion algebra over   ramifies  is always finite and even,  and their product  is called  the reduced discriminant of B. By the Classification Theorem,  the places where B is ramified determine B up to isomorphism as an algebra; thus it can be very useful to give a procedure which computes the reduced discriminant of a given quaternion algebra B=(a,b). 

We recall that if  B is ramified at infinity then B is a definite quaternion algebra over Q;  in this case the reduced discriminant is the product of an odd number of prime numbers and if we write B=(a,b) then both a and b are negative numbers. 

We define the following  procedures for computing some operations over a quaternion algebra B=(a,b): 

1)The procedure ProdQuat returns the product of two elements of B;   

 

2)The procedure  InvQ returns the inverse of a non-zero-element of B; 

 

3)The procedure TrQ  returns the reduced trace of an element of B; 

 

4)The procedure  NormQ returns the reduced norm of an element of B; 

 

5)The procedure  Discriminant returns the reduced discriminant of B. 

 

Product of two Quaternion Numbers 

Initialization 

>

restart:

 

>	with(LinearAlgebra):

 

Procedure Definition 

 

Let  be a quaternion algebra over Q, where the elements i, j verify and .  The new procedure ProdQuat  in Maple computes the product of two quaternions in B. The algorithm is based on an isomorphism between B and GL(2,Q). 

Input data are the pair (a,b)  of non zero rational numbers and two elements . 

The parameters a and b can be rational numbers or symbols. The output data is the quaternion product of h and g in B. 

 

>

ProdQuat:=proc(a,b,h,g)                                                                      local terminenoto1,terminenoto2, m1,mi,mj,mk,hm1,hm2,prodquatm,prodquat,coef;                                           terminenoto1:=h-coeff(h,i)*i-coeff(h,j)*j-coeff(h,k)*k: terminenoto2:=g-coeff(g,i)*i-coeff(g,j)*j-coeff(g,k)*k: m1:=Matrix([[1,0],[0,1]]):                             mi:=Matrix([[sqrt(a),0],[0,-sqrt(a)]]): mj:=Matrix([[0,1],[b,0]]): mk:=mi.mj: hm1:=MatrixScalarMultiply(m1,terminenoto1)+MatrixScalarMultiply(mi,coeff(h,i))+MatrixScalarMultiply(mj,coeff(h,j))+MatrixScalarMultiply(mk,coeff(h,k)): hm2:=MatrixScalarMultiply(m1,terminenoto2)+MatrixScalarMultiply(mi,coeff(g,i))+MatrixScalarMultiply(mj,coeff(g,j))+MatrixScalarMultiply(mk,coeff(g,k)): prodquatm:=MatrixMatrixMultiply(hm1,hm2): coef:=solve({x+y*sqrt(a)=prodquatm[1,1], z+w*sqrt(a)=prodquatm[1,2], z*b-b*w*sqrt(a)=prodquatm[2,1], x-y*sqrt(a)=prodquatm[2,2]}, {x,y,z,w}); assign(coef); prodquat:=x+y*i+z*j+k*w; end:

 

Examples 

Example 1. 

>	h[1]:=i+k;

 

(2.3.1)

 

>	g[1]:=-1/12*i-1/12*k;

 

(2.3.2)

 

>	ProdQuat(3,5,h[1],g[1]);

 

(2.3.3)

 

Example 2. 

>	h[2]:=2*i-3*j+k;

 

(2.3.4)

 

>	g[2]:=i+k;

 

(2.3.5)

 

>	ProdQuat(3,1,h[2],g[2]);

 

(2.3.6)

 

Inverse of a Quaternion Number 

Initialization 

>

restart:

 

>	with(LinearAlgebra):

 

>	with(linalg):

 

Procedure Definition 

 

We give a new procedure InvQ in Maple for computing the inverse of a non-zero element h of B=(a,b). 

Input data are are the pair (a,b)  of non zero rational numbers which define B and a quaternion number h. The parameters a and b can be rational numbers or symbols. 

The output data is an element InvQ(h) of B such that h*InvQ(h)=1. 

 

>

InvQ:=proc(a,b,h) local InvQ,terminenoto,m1,mi,mj,mk,hm,coef,mh; terminenoto:=h-coeff(h,i)*i-coeff(h,j)*j-coeff(h,k)*k: m1:=Matrix([[1,0],[0,1]]): mi:=Matrix([[sqrt(a),0],[0,-sqrt(a)]]): mj:=Matrix([[0,1],[b,0]]): mk:=mi.mj: hm:=MatrixScalarMultiply(m1,terminenoto)+MatrixScalarMultiply(mi,coeff(h,i))+MatrixScalarMultiply(mj,coeff(h,j))+MatrixScalarMultiply(mk,coeff(h,k)): mh:=inverse(hm):   coef:=solve({x+y*sqrt(a)=mh[1,1], z+w*sqrt(a)=mh[1,2], z*b-b*w*sqrt(a)=mh[2,1], x-y*sqrt(a)=mh[2,2]}, {x,y,z,w}); assign(coef); InvQ:=x+y*i+z*j+k*w; end:

 

Examples 

Example 1. 

>	h[1]:=i+k;

 

(3.3.1)

 

>	InvQ(3,5,h[1]);

 

(3.3.2)

 

Example 2. 

>	h[2]:=1-i+2*j+1/2*k;

 

(3.3.3)

 

>	InvQ(2,3,h[2]);

 

(3.3.4)

 

Reduced Trace of a Quaternion 

Initialization 

>

restart:

 

>	with(LineaeAlgebra):

 

Procedure Definition 

 

The new procedure TrQ in Maple computes the reduced trace of a quaternion number.  

Input data is quaternion number h. The output data is the reduced trace of h, which is a rational number. 

 

>	TrQ:=proc(h) local TrQ,terminenoto; terminenoto:=h-coeff(h,i)*i-coeff(h,j)*j-coeff(h,k)*k: TrQ:=2*terminenoto: end:

 

Examples 

Example 1. 

>	h[1]:=2+4*i;

 

(4.3.1)

 

>	TrQ(h[1]);

 

(4.3.2)

 

Example 2. 

>	h[2]:=1/2+4*i-11*j+k;

 

(4.3.3)

 

>	TrQ(h[2]);

 

(4.3.4)

 

Reduced Norme of a Quaternion 

Initialization 

>

restart:

 

>	with(LinearAlgebra):

 

Procedure Definition 

 

We give a new procedure NormQ in Maple for computing the reduced norm of a quaternion number.  

Input data are are the pair (a,b)  of non zero rational numbers which define B and a quaternion number h. The parameters a and b can be rational numbers or symbols. 

The output data is the reduced norm NormQ(h) of h, which is a rational number.  

 

>	NormQ:=proc(a,b,h) local NormQ,terminenoto;                                           terminenoto:=h-coeff(h,i)*i-coeff(h,j)*j-coeff(h,k)*k: NormQ:=terminenoto^2-a*coeff(h,i)^2-b*coeff(h,j)^2+a*b*coeff(h,k)^2: end:

 

Examples 

Example 1. 

>

h[1]:=i;

 

(5.3.1)

 

>	NormQ(3,5,h[1]);

 

(5.3.2)

 

Example 2. 

>	h[2]:=-2+3*i-1/3*k;

 

(5.3.3)

 

>	NormQ(3,4,h[2]);

 

(5.3.4)

 

Discriminant of a Quaternion Algebra B 

Initialization 

>

restart:

 

>	with(padic):

 

Procedure Definition 

 

We  give a new procedure (Discriminant) in Maple for computing the discriminant of a quaternion algebra B=(a,b). 

Input data are the rational numbers a, b defining the quaternion algebra   where  and . The output data is the product of the finite places of Q which ramify in B.  

 

>

Discriminant:=proc(a,b) local a2,b2,u2,v2,eu,ev,wu,wv,alpha, beta, u, v, d ,e, p,Lu,Lv,Discriminant; a2:=ordp(a,2): b2:=ordp(b,2): u2:=a/(2^a2): v2:=b/(2^b2): eu:=modp((u2-1)/2,2): ev:=modp((v2-1)/2,2): wu:=modp((u2^2-1)/8,2): wv:=modp((v2^2-1)/8,2): if (-1)^(eu*ev+a2*wv+b2*wu)=-1 then d:=2 else d:=1 end if: for p from 3 to max(abs(a),abs(b)) do if type(p,prime) and gcd(a*b,p)<>1 then alpha:=ordp(a,p): beta:=ordp(b,p): u:=a/(p^alpha): v:=b/(p^beta): e:=modp((p-1)/2,2): Lu:=modp(u^((p-1)/2),p): Lv:=modp(v^((p-1)/2),p) : if Lv=p-1 then Lv:=-1 end if: if Lu=p-1 then Lu:=-1 end if: if (-1)^(alpha*beta *e)*(Lu)^beta*(Lv)^alpha=-1 then d:=d*p end if: end if: end do: Discriminant:=d; end:

 

Examples 

Example 1. 

>	Discriminant(-2,-5);

 

(6.3.1)

 

Example 2. 

>	Discriminant(3,5);

 

(6.3.2)

 

References 

 

Roger C. Alperin, Free Subgroups of Quaternion Algebras, Proceedings of the American Mathematical Society, Volume 118, Number 1, May 1993. 

 

Marie-France Vigneras, Arithm?tique des Alg?bres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, 1980. 

 

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