so consider the indecomposable objects in the derived category of reps of the a2 quiver...

??then is there some obvious involution corresponding to reversing the direction of the arrow?? hmmm...

can we think of an abelian category of some restricted homological dimension (or something) as enriched over some category of (doubly) bounded chain complexes? or something? ...

vague feeling that various things that simon told me about might be relevant here...

a := 0->1

b := 1->1

c := 1->0

free resolution of c = c<-b<-a ...??... [l(c,0),l(a,1)] = 1 [l( ... -> c0 -> a1 -> b1 -> c1 -> ...

... <- a1 <- ??sa "triangulated category" ... ???.... ??peculiar multiple-level interpretations of "cartan matrix" ...???... ??perhaps this example is somewhat claustrophobic ... ??... try a->b->c vs a<-b->c ... ???....

d:=1->0->0 e:=0->1->0 f:=0->0->1 g:=1->1->0 h:=0->1->1 j:=1->1->1

d':=1->1->1 e':=0->1->1 f':=f g':=d' ?? h':=0->1->1 j':=d' ????.....

?? g':=1->2->2 h':=0->1->2 j':=1->2->3

d<-d'<-h'<-f' e<-e'<-f' f<-f' g<-g'<-h' ????? not very systematic ... ??... h<-h'<-f' j<-j'<-h' ?? h<-h' j<-j' ??maybe we should try some hopefully obvious way of matching up distinguished triangles in the .->.->. case with such in the .<-.->. case? ??suggesting in particular that we _are_ trying to get an equivalence of categories but of triangulated categories... ?? ... or something...

??also... consider the complex of reps of .->.->. as follows:

0->1->1

1->1->0

with vertical maps hopefully pretty obvious... ??is this an indecomposable complex or something?? ... ??are there a whole lot more indecomposable complexes to worry about here, or is this the only "unexpected" one?

??apparently it's supposed to be decomposable, since the representation algebroid is supposed to be homology dimension one or "hereditary" ... i should think about this...