A quaternion algebra A over a field K is isomorphic to the matrix ring M2(K) if and only if there exists a zerodivisor ε in A. Given such an ε, we can exhibit an explicit isomorphism; otherwise a zerodivisor will be computed first by finding a point on a conic (see [Vig80, Cor. I.2.4]).
Given an associative algebra, we also have an algorithm to recognize if the algebra is a quaternion algebra, and, if so, return an isomorphism to a quaternion algebra in standard form.
Isomorphism: BoolElt Default: false
Returns true if and only if the quaternion algebra A with base field F is isomorphic to M2(F), or equivalently if A has no ramified places. The field F has to be Q, Fq(X) (with q odd) or a number field.If A is isomorphic to M2(F) and Isomorphism is set to true, then M2(F) and an isomorphism A to M2(F) are also returned.
Given a quaternion algebra A and a zerodivisor ε ∈A, the function returns the matrix algebra M2(F) and an isomorphism A to M2(F).
> A := QuaternionAlgebra<Rationals() | -1, 1>; > eps := A.3-1; > MinimalPolynomial(eps), Norm(eps); x^2 + 2*x 0
Thus, since ε has reduced norm 0, it is a zerodivisor: indeed, ε(ε + 2)=0.
> M2F, phi := MatrixRing(A,eps);
> [<MinimalPolynomial(A.i), MinimalPolynomial(phi(A.i))> : i in [1..3]];
[
<x^2 + 1, x^2 + 1>,
<x^2 - 1, x^2 - 1>,
<x^2 - 1, x^2 - 1>
]
Returns true if and only if the associative algebra B is a quaternion algebra; if true, it returns the associated quaternion algebra A in standard form and an algebra homomorphism from B to A. The algorithm used is [Voi05, Algorithm 4.2.9].
> A := AssociativeAlgebra(QuaternionAlgebra<Rationals() | -1,1>); > vecs := [&+[Random(10)*A.i : i in [1..4]] : j in [1..4]]; > Mchange := Matrix(Rationals(),4,4,&cat[Eltseq(vecs[i]) : i in [1..4]]); > Mchange := Mchange^(-1); > seq := [<i,j,k,((vecs[i]*vecs[j])*Mchange)[k]> : i,j,k in [1..4]]; > A := AssociativeAlgebra<Rationals(),4 | seq>; > bl, Aquat, phi := IsQuaternionAlgebra(A); > bl; true > Aquat; Quaternion Algebra with base ring Rational Field > Aquat.1^2, Aquat.2^2; 25 -3924/25 > phi; Mapping from: AlgAss: A to AlgQuat: Aquat given by a rule
We now verify the functionality when a zerodivisor is encountered.
> A := Algebra(MatrixAlgebra(Rationals(),2)); > IsQuaternionAlgebra(A); true Quaternion Algebra with base ring Rational Field Mapping from: AlgAss: A to Quaternion Algebra with base ring Rational Field given by a rule
The algebra k<x, y> with x2=y2=xy + yx=0 is not semisimple; the ideal generated by x, y is a nontrivial two-sided ideal. Similarly, a commutative algebra is not a quaternion algebra.
> A := Algebra(FPAlgebra<Rationals(), x,y | x^2, y^2, x*y+y*x>); > IsQuaternionAlgebra(A); false > A := Algebra(FPAlgebra<Rationals(), x | x^4+x^2+1>); > IsQuaternionAlgebra(A); false
In characteristic 2, the algorithm also performs correctly, both for an associative but non-quaternion algebra and for the "universal" example of a quaternion algebra.
> A := Algebra(FPAlgebra<GF(2), x,y | x^2, y^2, x*y+y*x+1>); > IsQuaternionAlgebra(A); false > F<a,b,x,y,z,w> := FieldOfFractions(PolynomialRing(GF(2),6)); > M := [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1], > [0,1,0,0],[a,1,0,0],[0,0,0,1],[0,0,a,1], > [0,0,1,0],[0,0,1,1],[b,0,0,0],[b,b,0,0], > [0,0,0,1],[0,0,a,0],[0,b,0,0],[a*b,0,0,0]]; > A<alpha,beta> := AssociativeAlgebra<F,4 | M>; > alpha^2+alpha+a; (0 0 0 0) > beta^2+b; (0 0 0 0) > > bl, Aquat, phi := IsQuaternionAlgebra(A); > bl; true > Aquat; Quaternion Algebra with base ring Multivariate rational function field of rank 6 over GF(2) > theta := phi(x+y*alpha+z*beta+w*alpha*beta); > Trace(theta); y > Norm(theta); a*b*w^2 + a*y^2 + b*z^2 + b*z*w + x^2 + x*y
Given a quaternion algebra A over Q or a quaternion order R over Z, this function returns a 2 x 2-matrix representation of A, defined over a quadratic extension.