Too restrictive IsSingleValued method selected for algebra map

[hulpke]

(This was reported by Heiko Dietrich):

sc1:=
[ [ [ [  ], [  ] ], [ [ 3, 4, 5 ], [ Z(7)^3, Z(7)^4, Z(7)^3 ] ], [ [ 5
], [ Z(7)^3 ] ], [ [ 6 ], [ Z(7)^3 ] ], [ [  ], [  ] ], [ [  ], [  ] ]
],
 [ [ [ 3, 4, 5 ], [ Z(7)^0, Z(7), Z(7)^0 ] ], [ [  ], [  ] ], [ [ 4,
5, 6 ], [ Z(7)^3, Z(7)^4, Z(7)^2 ] ], [ [ 5, 6 ], [ Z(7)^3, Z(7)^4 ]
], [ [ 6 ], [ Z(7)^3 ] ], [ [  ], [  ] ] ],
 [ [ [ 5 ], [ Z(7)^0 ] ], [ [ 4, 5, 6 ], [ Z(7)^0, Z(7), Z(7)^5 ] ],
[ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ],
 [ [ [ 6 ], [ Z(7)^0 ] ], [ [ 5, 6 ], [ Z(7)^0, Z(7) ] ], [ [  ], [
] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ],
 [ [ [  ], [  ] ], [ [ 6 ], [ Z(7)^0 ] ], [ [  ], [  ] ], [ [  ], [
] ], [ [  ], [  ] ], [ [  ], [  ] ] ], [ [ [  ], [  ] ], [ [  ], [  ]
], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ],
     [ [  ], [  ] ] ], -1, 0*Z(7) ];;

sc2 :=
[ [ [ [  ], [  ] ], [ [ 3, 4, 5, 6 ], [ Z(7)^3, Z(7)^4, Z(7)^2, Z(7)^3
] ], [ [ 6 ], [ Z(7)^3 ] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [
] ] ],
 [ [ [ 3, 4, 5, 6 ], [ Z(7)^0, Z(7), Z(7)^5, Z(7)^0 ] ], [ [  ], [  ]
], [ [ 4, 5, 6 ], [ Z(7)^3, Z(7)^4, Z(7)^2 ] ], [ [ 5, 6 ], [ Z(7)^3,
Z(7)^4 ] ], [ [ 6 ], [ Z(7)^3 ] ], [ [  ], [  ] ] ],
 [ [ [ 6 ], [ Z(7)^0 ] ], [ [ 4, 5, 6 ], [ Z(7)^0, Z(7), Z(7)^5 ] ],
[ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ],
 [ [ [  ], [  ] ], [ [ 5, 6 ], [ Z(7)^0, Z(7) ] ], [ [  ], [  ] ], [
[  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ], [ [ [  ], [  ] ], [ [
6 ], [ Z(7)^0 ] ], [ [  ], [  ] ], [ [  ], [  ] ],
     [ [  ], [  ] ], [ [  ], [  ] ] ], [ [ [  ], [  ] ], [ [  ], [  ]
], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ],
-1, 0*Z(7) ];;

L1:=LieAlgebraByStructureConstants(GF(7),sc1);;
L2:=LieAlgebraByStructureConstants(GF(7),sc2);;

iso := AlgebraHomomorphismByImages(L1,L2,Basis(L1),Basis(L2));

does not return fail, though the map is not an algebra homomorphism:

g:=Basis(L1)[1];; h:= Basis(L1)[2];;
Image(iso,g*h) = Image(iso,g) * Image(iso,h);

If we only map element lists:

iso := AlgebraHomomorphismByImages(L1,L2,List(Basis(L1),x->x),List(Basis(L2),x->x));

all is OK (i.e. fail is returned).

The issue is that the IsSingleValued method chosen is (line 406 in vspchom.gi) the one "for left module g.m.b.i." that only checks for vector space mapping.

[fingolfin]

That looks like a fundamental flaw with allowing a morphism to be a "morphism of algebras" and a "morphism of modules" at the same time ?

[hulpke]

Yes, that is my worry (and why I did not try to fix it). The only rescue would be a convention that IsSingleValued needs to test for all categories an object claims to be part of, but that is also a kludge.