Modules over Basic Algebras#

A module \(M\) over a basic algebra \(B\) is presented as a sequence of matrices, one for each generator of the algebra.

Indecomposable Projective Modules#

The indecomposable projective modules are defined from the structure of the algebra and have associated path trees that solve the homomorphism lifting problem.

ProjectiveModule(B, i): AlgBas, RngIntElt -> ModRng#: The \(i^{th}\) projective module of the basic algebra \(B\).

PathTree(B, i): AlgBas, RngIntElt -> ModRng#: The path tree of the \(i^{th}\) projective module of the basic algebra \(B\).

ActionGenerator(B, i): AlgBas, RngIntElt -> SeqEnum#: The sequence of matrices for the generators of the basic algebra \(B\) acting on the \(i^{th}\) projective module of \(B\).

IdempotentActionGenerators(B, i): AlgBas, RngIntElt -> SeqEnum#: The sequence of matrices for the idempotent generators of the basic algebra \(B\) acting on the \(i^{th}\) projective module of \(B\).

NonIdempotentActionGenerators(B, i): AlgBas, RngIntElt -> SeqEnum#: The sequence of matrices for the nonidempotent generators of the basic algebra \(B\) acting on the \(i^{th}\) projective module of \(B\).

Injection(B, i, v): AlgBas, RngIntElt, ModRngElt -> AlgBasElt#: Given a vector \(v\) in in the \(i^{th}\) projective module of the basic algebra \(B\), the function returns the image of inclusion of \(v\) into \(B\).

Creation#

AModule(B, Q): AlgBas, SeqEnum[AlgMatElt] -> ModRng#: Given a basic algebra \(B\) and a sequence \(Q\) of elements in a matrix algebra the function returns the \(B\)-module \(M\) on which the generators of \(B\) act by multiplication by the corresponding elements of \(Q\).

ProjectiveModule(B, S): AlgBas, SeqEnum[RngIntElt] -> ModAlg, SeqEnum, SeqEnum#: Given a sequence \(S = [s_1,s_2, ... ]\), the function returns a projective module which is the direct sum of \(s_1\) copies of the first projective of the algebra \(B\), \(s_2\) copies of the second, etc. It also returns the sequence of inclusions and projections from and to the indecomposable projective modules.

IrreducibleModule(B, i): AlgBas, RngIntElt -> ModAlg#
SimpleModule(B, i): AlgBas, RngIntElt -> ModAlg#: The \(i^{th}\) irreducible module of the algebra \(B\). The module is the quotient of the \(i^{th}\) projective module by its radical.

ZeroModule(B): AlgBas -> ModAlg#: The zero \(B\)-module.

RightRegularModule(B): AlgBas -> ModAlg#: The algebra \(B\) as a right module over itself. The module is the direct sum of the projectives modules of \(B\).

RegularRepresentation(v): AlgBasElt -> AlgMatElt#: If \(v\) is an element of a basic algebra given as a vector in the underlying space, then the function computes the matrix of the action by right multiplication of the element on the algebra.

Restriction(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt -> ModAlgBas#: If \(B\) is a subalgebra of the basic algebra \(A\), \(\xi\) is the embedding of \(B\) into \(A\), and \(M\) is an \(A\)-module, then the function returns the restriction of M to a B-module.

ChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, Map -> ModAlgBas#
ChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt -> ModAlgBas#: Given a module \(M\) over an algebra \(A\) and an algebra homomorphism \(\xi\) from \(B\) to \(A\), the function returns the module \(M\) as a \(B\)-module.

JacobsonRadical(M): ModAlg -> ModAlg#: The Jacobson radical of the module \(M\).

Socle(M): ModAlg -> ModAlg#: The socle of the module \(M\). The sum of the simple submodules of \(M\).

Access Functions#

Algebra(M): ModAlg -> AlgBas#: Given a module \(M\) over a basic algebra \(B\), the function returns \(B\).

Dimension(M): ModAlg -> RngIntElt#: The dimension of the module \(M\) over its base ring.

Action(M): ModAlg -> AlgMat#: The matrix algebra of the action of the algebra of \(M\) on \(M\).

IsomorphismTypesOfRadicalLayers(M): ModAlgBas -> SeqEnum#: Given a module \(M\) over a basic algebra, returns the sequence of isomorphism types of simple composition factors in each layer of the radical filtration of \(M\).

IsomorphismTypesOfSocleLayers(M): ModAlgBas -> SeqEnum#: Given a module \(M\) over a basic algebra, returns a sequence of isomorphism types of simple composition factors in each socle layer with reversed order, i. e. isomorphism types of the socle of \(M\) will appear last.

IsomorphismTypesOfBasicAlgebraSequence(S): SeqEnum -> SeqEnum#: Given a sequence of irreducible modules \(S\) for a basic algebra \(A\), return a sequence of isomorphism types comparing with the simple modules of \(A\).

Example: Restriction To Center (ex-6be5c5)#

We show the restriction of a module over an algebra \(A\) to a subalgebra of \(A\).

> G := SmallGroup(32,7);
> A := BasicAlgebra(G);
> C, mu := Center(A);
> X := RightRegularModule(A);
> Z := JacobsonRadical(X);
> L := Restriction(Z,C,mu);
> L;
AModule L of dimension 31 over GF(2)
> A eq Algebra(L);
True
> IndecomposableSummands(L);
[
    AModule of dimension 1 over GF(2),
    AModule of dimension 30 over GF(2)
]
> Dimension(Socle(L));
16

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Next we show how to pull back modules along a quotient map. We use the same algebra \(A\).

> U := ideal<A|[A.13 +A.17]>;
> Q, theta := quo<A|U>;
> X := ProjectiveModule(Q,1);
> Y := ChangeAlgebras(X,A,theta);
> Y;
AModule Y of dimension 16 over GF(2)

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Example: ChangeAlgebras 2 (ex-5bc3d0)#

Here is another example of pulling back a module along a quotient map. This one involves algebras with more than one idempotent.

> load m11;
Loading "/usr/local/dmagma/libs/pergps/m11"
M11 - Mathieu group on 11 letters - degree 11
Order 7 920 = 2^4 * 3^2 * 5 * 11;  Base 1,2,3,4
Group: G
> A:= BasicAlgebraOfPrincipalBlock(G,GF(2));
> A;
Basic algebra of dimension 22 over GF(2)
Number of projective modules: 3
Number of generators: 9
> DimensionsOfProjectiveModules(A);
[ 8, 8, 6 ]
> I := ideal<A|[A.9]>;
> B, mu := quo<A|I>;
> B;
Basic algebra of dimension 6 over GF(2)
Number of projective modules: 2
Number of generators: 5
> P := ProjectiveModule(B,1);
> P;
AModule P of dimension 3 over GF(2)
> Q := ChangeAlgebras(P,A,mu);
> Algebra(Q) eq A;
true

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Example: Radical Layers (ex-e824ff)#

In this example, we investigate the structure of the projective modules of a basic algebra.

> G := PSL(3,3);
> N := Normalizer(G,Sylow(G,2));
> A := BasicAlgebraOfHeckeAlgebra(G,N,GF(2));
> DimensionsOfProjectiveModules(A);
[ 1, 2, 3, 9, 9, 1, 1, 1, 1 ]
> IsomorphismTypesOfRadicalLayers(ProjectiveModule(A,4));
[
    [ 4 ],
    [ 2, 3, 4, 5 ],
    [ 4, 4, 5 ],
    [ 5 ]
]
> IsomorphismTypesOfSocleLayers(ProjectiveModule(A,4));
[
    [ 4 ],
    [ 3, 4, 5 ],
    [ 2, 4, 5 ],
    [ 4, 5 ]
]

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So we see that, unlike a group algebra, a Hecke algebra can have indecomposable projective modules whose socles are not simple.

Predicates#

The following functions return a boolean value.

IsSemisimple(M): ModAlg -> BoolElt, SeqEnum#: Returns true if the module \(M\) is a semisimple module and false otherwise. If true, then the function also returns a list of the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of the simple modules of the algebra as composition factors in a composition series for the module.

IsProjective(M): ModAlg -> BoolElt, SeqEnum#: Returns true if the module \(M\) is projective. The function also returns a sequence of multiplicities of the standard projective modules as direct summands of the projective cover of \(M\).

IsInjective(M): ModAlg -> BoolElt, SeqEnum#: Returns true if the module \(M\) is injective. The function also returns a sequence of multiplicities of the standard injective modules as direct summands of the injective hull of \(M\).

Elementary Operations#

m * b: ModAlgElt, AlgBasElt -> ModAlgElt#: Given an element \(b\) in a basic algebra \(B\) and an element \(m\) in a module \(M\) over \(B\), \(m*b\) is the product.

Example: A Modules (ex-bb4604)#

We obtain the dimensions of the radical layers of the group algebra of an extra special group of order 243 over a field of characteristic 3.

> G := ExtraSpecialGroup(3,2);
> G;
Permutation group G acting on a set of cardinality 243
> ff := GF(3);
> A := BasicAlgebra(G,ff);
> A;
Basic algebra of dimension 243 over GF(3)
Number of projective modules: 1
Number of generators: 6
> P := ProjectiveModule(A,1);
> P;
AModule P of dimension 243 over GF(3)
> R := JacobsonRadical(P);
> R;
AModule R of dimension 242 over GF(3)
> while Dimension(R) ne 0 do
>     T := JacobsonRadical(R);
>     print Dimension(R) - Dimension(T);
>     R := T;
> end while;
4
11
20
30
36
39
36
30
20
11
4
1

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Example: AModules 2 (ex-2f301a)#

We consider the mod-2 group algebra of an extraspecial group of order 128, and construct the module induced from the trivial module on the subgroups of order 4 generated by the first generator of the group.

> G := ExtraSpecialGroup(2,3);
> G;
Permutation group G acting on a set of cardinality 128
> F := GF(2);
> A := BasicAlgebra(G,F);
> A;
Basic algebra of dimension 128 over GF(2)
Number of projective modules: 1
Number of generators: 8
> 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)

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Note that A.1 is the unique idempotent (identity element) in the group algebra, whereas A.2 is G.1 - 1 where G.1 is the first generator of the group.

> A.2;
(0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g := A.1+A.2;

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Now we check the order of g.

> g^2;
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 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 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g^4;
(1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    0 0 0 0 0 0 0 0 0 0 0 0)
> g^4 eq A!1;
true

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So g has order 4.

> P := ProjectiveModule(A,1);
> P;
AModule P of dimension 128 over GF(2)

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Note that P is generated by P.1 which corresponds to the identity element of A if we think of P as the algebra A as a module over itself. Now we create the induced module as the submodule generated by \((\hbox{\tt g}-1)^3\), since \((\hbox{\tt g}-1)^4 = 0\).

> U := sub<P|P.1*A.6>;
> U;
AModule U of dimension 32 over GF(2)

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Because the dimension is a quarter of the order of the group we can be sure that we have the right thing by just checking that U is generated by a g fixed point.

> U.1*g eq U.1;
true

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