Algebra: A Text-Book of Determinants, Matrices, and by W. L. Ferrar

By W. L. Ferrar

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5). A path w of positive length l > 0 is called a cycle if s(w) = t(w). Cycles are often also called oriented cycles in the literature. A cycle of length 1 is called loop. 4 The path category The composition of two paths v = (i|αl , . . , α1 |h) and w = (j|βm , . . , β1 |i) is defined by wv = (j|βm , . . , β1 , αl , . . , α1 |h). Notice that we defined the composition of paths in the same order as functions, which is not at all standard in the literature, but rather up to the taste of the author.

A contravariant functor F : C → D is very similar to a covariant functor except that it inverts the direction of the morphisms, that is F g ∈ D(F y, F x) for any g ∈ C (x, y) and consequently F (h ◦ g) = F g ◦ F h for any g and h. We shall meet many functors during the course of this book and limit ourselves here to two simple examples of covariant functors. 10 (a) Let Q be a finite quiver. Define a functor F : rep Q → vec by F V = i∈Q0 Vi for any representation V of Q and F g = i∈Q0 gi for any morphism of representations g.

Ri → 0 → . . → 0)⊕ (V1 → . . → Vj−1 → Mj → . . Mi → 0 → . . → 0). The indecomposability of V implies now that the latter one is zero, since Rj = 0. This shows that Vh = 0 for h < j and that Vαh is bijective for h = j, . . , i−1. We observe that in case j = 1 all these statements are trivially true or void. Thus V is isomorphic to 1 1 d d 0 → . . → 0 → K d −→ . . −→ K d → 0 → . . → 0, where d denotes the dimension of the spaces Vj , . . , Vi . But this representation is isomorphic to the direct sum of d copies of [j, i].

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