The centre of the associative algebra A.
Centraliser(A, S) : AlgAss, AlgAss -> AlgAss
The centralizer of the subalgebra S of the associative algebra A, that is,
the subalgebra of A commuting elementwise with S.
Idealiser(A, B: parameters) : AlgAss, AlgAss -> AlgAss
Side: MonStgElt Default: "Both"
Given an associative algebra A and a subalgebra B of A,
compute the idealizer of B in A, that is, the largest subalgebra of A
in which B is an ideal.
By default the two-sided idealizer, that is, the largest subalgebra in which B
is a two-sided ideal, is found;
the left- or right-idealizer can be found by setting the parameter
Side to "Left" or "Right" respectively.
For an associative structure constant algebra A, return the structure
constant algebra L with product given by the Lie bracket
(a, b) |-> a * b - b * a.
As a second value the map identifying the elements of A and L is returned.
Let A and B be subalgebras of an associative algebra with underlying
module M.
This function returns the submodule of M which is spanned by the elements
[a, b] = a * b - b * a, a ∈A, b ∈B.
For two subalgebras A and B of an associative algebra, return the ideal
generated by all [a, b] = a * b - b * a, a ∈A, b ∈B.
For two subalgebras A and B of an associative algebra, return the
left annihilator of B in A; that is, the subalgebra of A
consisting of all elements a such that a * b = 0 for all b ∈B.
For two subalgebras A and B of an associative algebra, return the
right annihilator of B in A; that is, the subalgebra of A
consisting of all elements a such that b * a = 0 for all b ∈B.
We create the Lie algebra sl3(Q) as a structure constant algebra.
First, we construct gl3(Q) from the full matrix algebra M3(Q) and
get sl3(Q) as the derived algebra of gl3(Q).
> gl3 := LieAlgebra(Algebra(MatrixRing(Rationals(), 3)));
> sl3 := gl3 * gl3;
> sl3;
Lie Algebra of dimension 8 with base ring Rational Field
Let's see how the first basis element acts.
> for i in [1..8] do
> print sl3.i * sl3.1;
> end for;
(0 0 0 0 0 0 0 0)
( 0 -1 0 0 0 0 0 0)
( 0 0 -2 0 0 0 0 0)
(0 0 0 1 0 0 0 0)
(0 0 0 0 0 0 0 0)
( 0 0 0 0 0 -1 0 0)
(0 0 0 0 0 0 2 0)
(0 0 0 0 0 0 0 1)
Since it acts diagonally, this element lies in a Cartan subalgebra. The next
candidate seems to be the fifth basis element.
> for i in [1..8] do
> print sl3.i * sl3.5;
> end for;
(0 0 0 0 0 0 0 0)
(0 1 0 0 0 0 0 0)
( 0 0 -1 0 0 0 0 0)
( 0 0 0 -1 0 0 0 0)
(0 0 0 0 0 0 0 0)
( 0 0 0 0 0 -2 0 0)
(0 0 0 0 0 0 1 0)
(0 0 0 0 0 0 0 2)
This also acts diagonally and commutes with sl3.1, hence we have luckily
found a full Cartan algebra in sl 3(Q).
We can now easily work out the root system. Obviously the root spaces
correspond to the pairs (sl3.2, sl3.4), (sl3.3, sl3.7) and
(sl3.6, sl3.8). The product of a positive root with its negative should
lie in the Cartan algebra.
> sl3.2*sl3.4;
( 1 0 0 0 -1 0 0 0)
> sl3.3*sl3.7;
(1 0 0 0 0 0 0 0)
> sl3.6*sl3.8;
(0 0 0 0 1 0 0 0)
Clearly some choices have to be made and we fix sl3.3 as the element
e α corresponding to the first fundamental root α, sl3.7
as e - α and get sl3.1 as h α = e α * e - α.
For the other fundamental root β we have to find an element
e β such that e α * e β is non-zero.
> sl3.3*sl3.2;
(0 0 0 0 0 0 0 0)
> sl3.3*sl3.4;
( 0 0 0 0 0 -1 0 0)
> sl3.3*sl3.6;
(0 0 0 0 0 0 0 0)
> sl3.3*sl3.8;
(0 1 0 0 0 0 0 0)
We choose sl3.8 as e β, sl3.6 as e - β and
consequently -sl3.5 as h β. This now determines
e α + β to be sl3.2 and e - α - β to be
sl3.4.
RestrictionOfScalars(A) : AlgAss[FldFun] -> AlgAss, Map
RestrictionOfScalars(A, F) : AlgAss[FldAlg], Fld -> AlgAss, Map
RestrictionOfScalars(A, F) : AlgAss[FldFun], FldFunG -> AlgAss, Map
Given an associative algebra A over an algebraic number or function
field K and an optional
coefficient field F of K, return the algebra isomorphic to A whose
coefficient field is F if given otherwise the coefficient field of K.
We illustrate some computations of restricting scalars, first over the
coefficient field.
> P<x> := PolynomialRing(Rationals());
> F<b> := NumberField(x^3-3*x-1);
> Z_F := MaximalOrder(F);
> A<alpha,beta,alphabeta> := QuaternionAlgebra<F | -3,b>;
> A := AssociativeAlgebra(A);
> A;
Associative Algebra of dimension 4 with base ring F
> RA, m := RestrictionOfScalars(A);
> RA;
Associative Algebra of dimension 12 with base ring Rational Field
> m;
Mapping from: AlgAss: A to AlgAss: RA given by a rule
> m(A.1);
(1 0 0 0 0 0 0 0 0 0 0 0)
> $1 @@ m;
(1 0 0 0)
> RA.6 @@ m;
( 0 b^2 0 0)
> m($1);
(0 0 0 0 0 1 0 0 0 0 0 0)
We now show an example where the scalars we are restricting to are not
necessarily the
coefficient field but a coefficient field of the coefficient field.
> P<x> := PolynomialRing(QuadraticField(23));
> F<b> := NumberField(x^3-3*x-1);
> Z_F := MaximalOrder(F);
> A<alpha,beta,alphabeta> := QuaternionAlgebra<F | -3,b>;
> A := AssociativeAlgebra(A);
> RA, m := RestrictionOfScalars(A);
> RA;
Associative Algebra of dimension 12 with base ring Quadratic Field with defining
polynomial x^2 - 23 over the Rational Field
> m;
Mapping from: AlgAss: A to AlgAss: RA given by a rule
> RA.6 @@ m;
( 0 b^2 0 0)
> m($1);
(0 0 0 0 0 1 0 0 0 0 0 0)
> RAQ, mQ := RestrictionOfScalars(A, Rationals());
> RAQ, mQ;
Associative Algebra of dimension 24 with base ring Rational Field
Mapping from: AlgAss: A to AlgAss: RAQ given by a rule
> mQ(A.1);
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> $1 @@ mQ;
(1 0 0 0)
> RAQ.15 @@ mQ;
( 0 0 b^2 + 2*$.1*b + 23 0)
> mQ($1);
(0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0)
Centraliser(A, s) : AlgAss, AlgAssElt -> AlgAss
The centralizer of the element s of the associative algebra A, that is,
the subalgebra of A commuting with s.
(a, b) : AlgAssElt, AlgAssElt -> AlgAssElt
The Lie bracket a * b - b * a of a and b, where a and b
are elements of an associative algebra A.
Returns true (and a coerced to F) iff a belongs to the base ring F
of its parent algebra.
Side: MonStgElt Default: "Right"
Returns the matrix representation of Side-multiplication by
the element a in the associative algebra A (which must have 1) on
the A-module M.
For an associative algebra A of dimension n return an isomorphic
matrix algebra. If A contains the identity-element, the matrix algebra will be
of degree n, otherwise it will be of degree n + 1.
Side: MonStgElt Default: "Right"
Given a finite-dimensional R-algebra A and a Side A-module
M (both free as R-modules), return the matrix algebra of
A-endomorphisms of M, and the R-algebra homomorphism from A
into this endomorphism ring.
Side: MonStgElt Default: "Right"
For an associative algebra A of dimension n over R return its regular
representation.
If B = (e1, e2, ..., en) is the stored basis for A, an element
a ∈A is mapped to the matrix in Rn x n which has as its i-th
row the coordinates of ei * a with respect to B.
As a second map, the homomorphism of A onto the regular representation is
returned.
By default, the right-regular representation is computed. This can be changed
to the left-regular representation (in which the i-th row of the image of a
contains the coordinates of a * ei) by setting the parameter Side to
"Left".
This section describes a few functions that can be used to obtain
information on the structure of a finite-dimensional associative algebra.
Al: MonStgElt Default: em "Default"
This returns the largest nilpotent ideal of A. This function works
for finite-dimensional associative algebras defined over a field of
characteristic 0, or over a finite field.
The algorithm used by default is taken from [CIW97].
The meataxe algorithm can be used by setting Al := "Meataxe".
We compute the Jacobson radical of the group algebra over the field
of three elements of a 3-group. In that case it is equal to the
augmentation ideal.
> G:= SmallGroup( 27, 5 );
> A:= GroupAlgebra( GF(3), G );
> JacobsonRadical( A );
Ideal of dimension 26 of the group algebra A
IndecomposableSummands(A) : AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]
Given an associative algebra A, return the direct sum decomposition of L as
a sequence of ideals of L whose sum is L and each of which cannot be
further decomposed into a direct sum of ideals. The second sequence return
contains the corresponding primitive central idempotents.
For a description of the algorithm we refer to [EG96].
Let Z be the centre of the associative algebra A, and let
J(Z) denote its Jacobson radical. This function returns a
sequence of primitive orthogonal idempotents in Z such that
their images in Z/J(Z) span J(Z). Each such idempotent
generates a two-sided ideal in A. The second return value
is the sequence of these ideals.
In particular, if A is a semisimple algebra, then this function returns
a sequence of primitive orthogonal idempotents spanning Z. Furthermore,
the ideals in the second sequence returned are simple algebras, and
their direct sum equals A.
For a description of the algorithm we refer to [EG96].
We compute the direct sum decomposition of a group algebra.
> G:= SmallGroup( 10, 2 );
> A:= GroupAlgebra( Rationals(), G );
> ee, II:= CentralIdempotents( A );
> ee[1];
1/10*Id(G) + 1/10*G.2 + 1/10*G.2^2 + 1/10*G.2^3 + 1/10*G.2^4 + 1/10*G.1 +
1/10*G.1 * G.2 + 1/10*G.1 * G.2^2 + 1/10*G.1 * G.2^3 + 1/10*G.1 * G.2^4
> II;
[
Ideal of dimension 1 of the group algebra A
Basis:
Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 + G.1 + G.1 * G.2 + G.1 * G.2^2 +
G.1 * G.2^3 + G.1 * G.2^4,
Ideal of dimension 1 of the group algebra A
Basis:
Id(G) + G.2 + G.2^2 + G.2^3 + G.2^4 - G.1 - G.1 * G.2 - G.1 * G.2^2 -
G.1 * G.2^3 - G.1 * G.2^4,
Ideal of dimension 4 of the group algebra A
Basis:
Id(G) - G.2^4 + G.1 - G.1 * G.2^4
G.2 - G.2^4 + G.1 * G.2 - G.1 * G.2^4
G.2^2 - G.2^4 + G.1 * G.2^2 - G.1 * G.2^4
G.2^3 - G.2^4 + G.1 * G.2^3 - G.1 * G.2^4,
Ideal of dimension 4 of the group algebra A
Basis:
Id(G) - G.2^4 - G.1 + G.1 * G.2^4
G.2 - G.2^4 - G.1 * G.2 + G.1 * G.2^4
G.2^2 - G.2^4 - G.1 * G.2^2 + G.1 * G.2^4
G.2^3 - G.2^4 - G.1 * G.2^3 + G.1 * G.2^4
]
We see that here the group algebra is the direct sum of two 1-dimensional
and two 4-dimensional ideals. The first idempotent is the sum over all group
elements divided by the group order.
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