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This section introduces two general constructions for
*-algebras:
- (i)
- The algebra of adjoints of a system of
reflexive (alternating, reflexive, or Hermitian)
forms [φ1, ..., φe] defined
on a common vector space V.
- (ii)
- The group algebra K[G], where K is any
ring and G is a finite group.
We also provide a constructor function for simple *-algebras.
A reflexive form on a K-vector space V is a
bilinear function φ: V x V to K such that,
whenever φ(u, v)=0 for u, v∈V, we also have
φ(v, u)=0. Reflexive forms φ and ψ
on V are isometric if φ(u, v)=ψ(u, v)
for all u, v∈V; they are similar if there exists
a∈K such that φ and aψ are isometric.
The radical of a reflexive form φ is the
subspace (rad)φ={u∈V:φ(u, V)=0},
and φ is nondegenerate if (rad)φ=0.
A fundamental result of Birkhoff and von Neumann states that
there are three similarity classes of reflexive forms:
alternating, symmetric, and Hermitian.
Each such form φ is represented by a matrix and
an automorphism of K. Specifically, regarding V as the
space of row vectors, we specify F and α such that
φ(u, v)=uαFv^((tr)), where α
is the identity if φ is bilinear, or the automorphism
x |-> /line(x) of order 2 if φ is Hermitian.
Thus a matrix g∈(GL)(d, K) is an
isometry (respectively similarity)
of the reflexive form if
gαFg^((tr))=F (respectively
gαFg^((tr))=aF).
Auto: RngIntElt Default: 0
The group of isometries of the (possibly degenerate)
reflexive form represented by the matrix F with entries
in a finite field GF(pe). The
field automorphism associated with F is specified by
the parameter Auto, which represents the exponent
f in the map x |-> xpf. The default is that F is
bilinear on its base ring.
Auto: RngIntElt Default: 0
The group of similarities of the (possibly degenerate)
reflexive form represented by the matrix F with entries
in a finite field GF(pe). The
field automorphism associated with F is specified by
the parameter Auto, which represents the exponent
f in the map x |-> xpf. The default is that F is
bilinear on its base ring.
A system of forms is a sequence
[φ1, ..., φe], where each φi
is a reflexive form on a common K-vector space V.
The radical of a system of forms is the intersection
of the radicals of the individual forms in the system;
A system is nondegenerate if
its radical is zero.
A classical group on a K-vector space
V preserves a reflexive form on V that is unique up
to similarity. Hence systems of forms arise naturally from
sets of classical groups having V as their common defining
module.
Systems of forms also arise naturally from the study
of p-groups. Let G be a finite p-group.
Let G=γ1(G)>γ2(G)> ... >γm(G)=1 denote
the lower central series of G, and let
1=ζ1(G)<ζ2(G)< ... <ζn(G)=G denote its upper
central series. Let Φ(G) be the Frattini
subgroup of G, and put
N:=< Φ(G), ζn - 1(G) >.
Then V=G/N and W=γ2(G)/γ3(G) are
(GF)(p)-vector spaces and commutation in G induces
a bilinear map V x V to W. One now obtains a system
of forms associated to G by choosing bases for V and W.
Just as matrices are useful representations of bilinear
forms, so are systems of forms convenient
computational models for bilinear maps.
PGroupToForms(G,p) : GrpPC, RngIntElt -> SeqEnum
Return a system of forms associated to the p-group G,
which must be of type GrpPC.
PGroupToForms(G,p) : GrpMat, RngIntElt -> SeqEnum
Return a system of forms associated to the matrix group G,
which must be a class 2 p-group.
We construct a system of forms associated to
a Sylow 7-subgroup of (GL)(3, 7).
> S := ClassicalSylow(GL (3, 7), 7);
> G := PCGroup(S);
> Forms := PGroupToForms(G);
> Forms;
[
[0 1]
[6 0]
]
Now we apply the same construction working directly with the
matrix group as a p-group of class 2.
> Forms := PGroupToForms(S);
> Forms;
[
[0 1]
[6 0]
]
It is preferable to input a PC-group if such a
representation can readily be obtained. The option to
use a matrix group for p-groups of class 2 enables
the user to construct an associated system of forms
in situations where it requires considerably more time
to first construct a PC-representation.
Using the following functions one can ascertain whether
or not a given algebra has an assigned involution, and
also access the map if it has.
IsStarAlgebra(A) : AlgGrp -> BoolElt
Return true if and only if A has an assigned involution.
Star(A) : AlgGrp -> Map
Return the involution associated to
the *-algebra A.
Let S=[φ1, ..., φe] be a system of
reflexive forms on a K-vector space V. Let
R denote the algebra (End)K(V) and R^((op))
denote its opposite ring. Then the algebra of
adjoints of S is defined as follows:
(Adj)(S) =
{(x, y)∈R x R^((op)):
φi(ux, v)=φi(u, yv) forall
u, v∈V, forall i∈{1, ..., e}}.
As the φi are reflexive forms,
(x, y)∈(Adj)(S)
if and only if (y, x)∈(Adj)(S).
If S is nondegenerate, then y
is uniquely determined by x; thus we identify
(Adj)(S) with its projection onto R and
the assignment x * :=y equips (Adj)(S) with
an involution.
The function to compute (Adj)(S) is an implementation
of the algorithms in [BW12a, Proposition 5.1]
and [BW12b, Section 5].
Autos: SeqEnum Default: [0,..,0]
Given a sequence S containing a nondegenerate system of reflexive
forms, this function returns the *-algebra of adjoints of the forms
in S. The parameter Autos is used to specify a list of
Frobenius exponents associated with the given forms. By default all
forms are considered to be bilinear over their common base ring,
which must be a finite field. Note that the individual forms in
the system are allowed to be degenerate.
We construct the algebra of adjoints of a
particular pair of
forms on a vector space of dimension 3 over
(GF)(5 2). The matrices have entries in
(GF)(5) but we regard them as forms over
the larger field, and define the first (symmetric)
form to be Hermitian over the larger field.
> MA := MatrixAlgebra(GF (25), 3);
> F := MA![1,2,0,2,3,4,0,4,1];
> G := MA![0,1,0,4,0,0,0,0,0];
> A := AdjointAlgebra([F, G] : Autos := [1, 0]);
> IsStarAlgebra(A);
true
> Degree(A);
6
> BaseRing(A);
Finite field of size 5
Observe that the function has converted the given
system of forms to a new system over (GF)(5)
and returned an algebra of degree 6 over the subfield.
Now we access the involution on A and apply it to
a generator of A.
> star := Star(A);
> A.3;
[1 0 2 0 2 0]
[0 1 0 2 0 2]
[1 0 4 0 4 0]
[0 1 0 4 0 4]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
> A.3@star;
[4 0 3 0 3 0]
[0 4 0 3 0 3]
[4 0 1 0 1 0]
[0 4 0 1 0 1]
[0 0 0 0 0 0]
[0 0 0 0 0 0]
If G is a finite group and R is any ring, then the
group algebra A=R[G] possesses a natural involution.
Thus each group algebra may be regarded as a *-algebra.
The natural involution on the group algebra A
induced by inversion on the underlying group.
Specifically, if a=∑g∈Gαgg∈A, then
a * =∑g∈Gαgg - 1.
Construct the group algebra R[G] equipped with the
natural involution afforded by inversion in G.
We construct the group algebra Z[S 3] as a *-algebra.
> G := SymmetricGroup(3);
> K := Integers();
> A := GroupAlgebraAsStarAlgebra(K, G);
> IsStarAlgebra(A);
true
Now access the involution on A and apply it to
an element of A.
> star := Star(A);
> a := A![0,0,1,0,3,0];
> a;
(1, 3, 2) + 3*(1, 2)
> a@star;
(1, 2, 3) + 3*(1, 2)
Alternatively, Z[S 3] can be constructed using the
standard constructor and the involution can be
attached later.
> A := GroupAlgebra(K, G);
> IsStarAlgebra(A);
false
> StarOnGroupAlgebra(A);
Mapping from: AlgGrp: A to AlgGrp: A given by a rule [no inverse]
> IsStarAlgebra(A);
true
The Artinian simple *-algebras --
those having no proper *-invariant ideals -- were
classified by Albert [Alb61].
They come in two basic flavours: classical
and exchange. The classical types are simple as
algebras and arise as adjoints of nondegenerate reflexive
forms. A simple *-algebra of exchange type is a direct
sum of two isomorphic simple algebras where the involution
interchanges the two factors.
Naturally extending the Magma classification of reflexive
forms (see the manual entry for the function
ClassicalForms) a simple *-algebra S is
assigned a name as follows:
- "symplectic" if S is defined by an
alternating form;
- "orthogonalcircle" if S is defined by
a symmetric form in odd dimension;
- "orthogonalplus" if S is defined by a
symmetric form of maximal Witt index;
- "orthogonalminus" if S is defined by a
symmetric form of non-maximal Witt index;
- "unitary" if S is defined by an Hermitian
form; and
- "exchange" if S has exchange type.
Every simple *-algebra is isomorphic to a standard
simple *-algebra having one of these six names defined
naturally on a suitable vector space.
We note that involutions on simple *-algebras are often
classified as being "of the first kind" or
"of the second kind" according to whether or not they
induce the identity on the center of the algebra.
Thus, involutions of the second kind are unitary and
exchange, and the others are all involutions of the
first kind [KMRT98].
Given a name, a positive integer d, and a field K,
this function constructs the standard copy of the simple
*-algebra of type name defined naturally on a
K-vector space of dimension d.
We construct the standard copy of the simple *-algebra
of exchange type on the vector space of dimension 4 over
(GF) (16).
> K := GF(16);
> S := SimpleStarAlgebra("exchange", 4, K);
> Dimension(S);
8
> IsStarAlgebra(S);
true;
> w := K.1;
> s := S.1 * S.2 + w * S.1;
> s;
[ K.1 1 0 0]
[ 0 0 0 0]
[ 0 0 0 0]
[ 0 0 0 0]
> star := Star(S);
> s@star;
[ 0 0 0 0]
[ 0 0 0 0]
[ 0 0 K.1 0]
[ 0 0 1 0]
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