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Basic Algebras
New Features:
- A homomorphism of basic algebras A
and B
can be constructed using
the hom-constructor:
where S
ia an appropriate
matrix. The intrinsic IsAlgebraHomomorphism
(A, B,
)
can be
used to verify the homomorphsim property while the intrinsic Image
can be used to compute the image of a homomorphism.
- The intrinsic Restriction
(M, B,
)
constructs the restriction
map for module M
along the algebra homomorphism
.
- The intrinsic AutomorphismGroup(A)
will compute the automorphism
group of the basic algebra A
. Isomorphism of basic algebras A
and B
can be tested using the intrinsic IsIsomorphic(A, B)
.
The implementation generalizes an algorithm of Bettina Eick for nilpotent
structure constant algebras. Related intrinsics are
GradedAutomorphismGroup and IsGradedIsomorphic.
- Ideals, subalgebras and quotient algebras may now be constructed using
the standard constructors ideal, sub and quo. The subalgebra
is returned
as a basic algebra together with the inclusion homomorphism. The returned
subalgebra includes the minimal idempotent that act as an identity on the
given elements. The intrinsic Annihilator(A, S)
constructs the
annihilator of the ideal generated by Elements S
in the basic algebra A
.
- The intrinsics Centre and Centralizer may now be used to
construct the centre of a basic algebra and the centralizer of a
collection of elements in a basic algebra.
- The intrinsic AssociatedGradedAlgebra(A)
constructs the associated
graded algebra of a basic algebra.
- Other new functions allow the user to create the basic algebra of the
action algebra on the direct sum of a sequence of modules over a matrix
algebra or group algebra. An example is the basic algebra of a block
of a group algebra.
Next: Matrix Algebras
Up: Linear Associative Algebras
Previous: Associative Algebras