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In this section we describe methods that facilitate
structural examinations of *-algebras.
All of the functions in this section require that
the base ring of the given algebra is a finite field
of odd order. The functions are implementations of
the methods described in [BW12a, Sections 4.2 and 4.3].
If A is a simple *-algebra, then we
constructively recognise A by finding
an explicit inverse isomorphisms between A and the
standard copy of the simple *-algebra which is
isomorphic to A. The latter is the output of the
function SimpleStarAlgebra with the
appropriate input parameters.
Given a matrix *-algebra A, this function first decides
whether or not A is a simple *-algebra of classical type.
If it is, the standard *-algebra, T, corresponding
to A, a *-isomorphism from A to T, and its
inverse from T to A are returned.
Given a matrix *-algebra A, this function first decides
whether or not A is a simple *-algebra of exchange type.
If it is, the standard *-algebra, T, corresponding
to A, a *-isomorphism from A to T, and its
inverse from T to A are returned.
We build a particular simple *-algebra of symplectic
type and recognise it constructively.
> MA := MatrixAlgebra(GF(7), 4);
> F := MA![0,1,3,4,6,0,0,1,4,0,0,2,3,6,5,0];
> F;
[0 1 3 4]
[6 0 0 1]
[4 0 0 2]
[3 6 5 0]
> A := AdjointAlgebra([F]);
> isit, T, f, g := RecogniseClassicalSSA(A);
> isit;
true;
A quick check that f is, as claimed,
a *-isomorphism.
> (A.1 + A.2)@f eq (A.1@f) + (A.2@f);
true
> (A.1 * A.2)@f eq (A.1@f) * (A.2@f);
true
> (A.2@Star(A))@f eq (A.2@f)@Star(T);
true
If A is an arbitrary *-algebra, then we
constructively recognise A as follows:
- (i)
- Find a decomposition A=J direct-sum T,
where J is the Jacobson radical of A and T is
a *-invariant semisimple complement to J in A;
- (ii)
- Find a decomposition
T=I1 direct-sum ... direct-sum It
of T into minimal *-ideals; and
- (iii)
- For each j∈{1, ..., t}
constructively recognise the simple *-algebra Ij.
RecogniseStarAlgebra(A) : AlgGrp -> BoolElt
Constructively recognise the *-algebra A given as
a matrix *-algebra or a group algebra.
There are several functions available that permit easy
access to structural information about
a *-algebra that has been constructively
recognised. (In fact all of these functions also
initiate a constructive recognition of the input
*-algebra if the recognition has not already been
carried out.) For all of the access functions A can
be either a matrix *-algebra or a group algebra.
IsSimpleStarAlgebra(A) : AlgGrp -> BoolElt
Return true if and only if A is a simple *-algebra.
SimpleParameters(A) : AlgGrp -> SeqEnum
Given a *-algebra A, this function returns the parameters that
determine (up to *-isomorphism) the minimal *-ideals of the
semisimple quotient A/J, where J is the Jacobson radical of A.
The parameters are returned in the form of a sequence.
Given a *-algebra A, this function returns the group of unitary
elements of A, namely the group consisting of all units in A
satisfying the condition x * =x - 1. The function is based on methods
described in [BW12a, Section 5].
Our first example illustrates how the *-algebra machinery
may be used to distinguish between group algebras over
(GF)(5) for the dihedral and quaternion groups of
order 8. Those group algebras are isomorphic
as algebras, but the example shows that they are
nonisomorphic as *-algebras.
> K := GF(5);
> G1 := SmallGroup(8, 3);
> G2 := SmallGroup(8, 4);
> A1 := GroupAlgebraAsStarAlgebra(K, G1);
> A2 := GroupAlgebraAsStarAlgebra(K, G2);
> J1, T1 := TaftDecomposition(A1);
> J2, T2 := TaftDecomposition(A2);
> Dimension(J1); Dimension(J2);
0
0
Thus (as we know from Maschke's theorem)
both (GF)(5)[D 8] and
(GF)(5)[Q 8] are semisimple. We now
recognise them as *-algebras and examine their
minimal *-ideals.
> RecogniseStarAlgebra(A1);
true
> RecogniseStarAlgebra(A2);
true
> SimpleParameters(A1);
[ <"orthogonalcircle", 1, 5>, <"orthogonalcircle", 1, 5>,
<"orthogonalcircle", 1, 5>, <"orthogonalcircle", 1, 5>,
<"orthogonalplus", 2, 5> ]
> SimpleParameters(A2);
[ <"orthogonalcircle", 1, 5>, <"orthogonalcircle", 1, 5>,
<"orthogonalcircle", 1, 5>, <"orthogonalcircle", 1, 5>,
<"symplectic", 2, 5>
]
Both group algebras decompose into four 1-dimensional
*-ideals, and one 4-dimensional *-ideal. However,
the latter has type "orthogonalplus" for
(GF)(5)[D 8], but type "symplectic" for
(GF)(5)[Q 8].
Our second example shows how to use *-algebra functions
to distinguish between two
p-groups of class 2 and order 43 6.
The first group is a Sylow 43-subgroup of
(GL)(3, 43 2).
> P1 := ClassicalSylow(GL(3, 43^2), 43);
> Forms1 := PGroupToForms(P1);
> A1 := AdjointAlgebra(Forms1);
> RecogniseStarAlgebra(A1);
true
> SimpleParameters(A1);
[ <"symplectic", 2, 1849> ]
The second group is constructed as a subgroup of
(GL)(3, (GF)(43)[x]/(x 2)).
> R<x> := PolynomialRing(GF(43));
> S, f := quo< R | x^2 >;
> G := GL(3, S);
> Ua := G![1,1,0,0,1,0,0,0,1];
> Wa := G![1,0,0,0,1,1,0,0,1];
> Ub := G![1,x@f,0,0,1,0,0,0,1];
> Wb := G![1,0,0,0,1,x@f,0,0,1];
> P2 := sub< G | [ Ua, Wa, Ub, Wb ] >;
> Forms2 := PGroupToForms(P2);
> A2 := AdjointAlgebra(Forms2);
> RecogniseStarAlgebra(A2);
true
> SimpleParameters(A2);
[ <"symplectic", 2, 43> ]
Since A 1 and A 2 are non-isomorphic *-algebras,
it follows that P 1 and P 2 are non-isomorphic groups.
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