[GAP Forum] question concerning the GAP package qpa (how to get an inclusion mprphism?)

[Øyvind Solberg]

Dear GAP Forum and Bernhard,
Suppose we are given a finite dimensional quotient  A = kQ/I  of a path
algebra and two modules  M  and  N  over  A, where we know that  N is
a submodule of  M.  If  M  and  N  are given as two representations/
modules over  A and you have not told QPA how  N  is a submodule of
M, then there is in general no way QPA can find the inclusion you are
thinking about.  In general a module N can be a submodule of a given
module  M  in infinitely many ways.  However, if you know by which
elements  N  is generated by inside  M, say  m_1, m_2,..., m_t, then the
command

gap> g := SubRepresentationInclusion(M, [m1,m2,...,mt]);

will produce an inclusion from a module  N'  isomorphic to N  into M, but
where the images inside  M  are the same.  Then to get an inclusion from
N you could find an isomorphism between  N  and  N' by

gap> alpha := IsomorphismOfModules( N, N' );

and then find the composition  alpha*g.

If you are just abstractly knowing that  N  is a submodule of  M, then I 
don't
know an algorithm to find an inclusion of  N  into M.

Best regards, Oeyvind Solberg.