The Composition Algebra of a Cyclic Quiver

Abstract

Note that A is also called the quiver of type An_x with cyclic orientation. We denote by Ao the set of vertices of A (and often we will identify Ao with Z/nZ, or also with the set {xx, x2, •-, xn) or just with {1, 2, ..., n}, with arrows xt—>xi+l or i->i + l). There are n one-dimensional representations, corresponding to the vertices of A; these are (up to isomorphism) all the simple objects of T. The simple representation corresponding to the vertex a of A will be denoted by 5(a), and if S' is isomorphic to S(a), we will write [5'] = a. Given a simple representation 5, and I eNu there is (up to isomorphism) a unique indecomposable representation S[l] of length / with top 5, and we obtain in this way all indecomposable representations (again up to isomorphism). It follows that we can index the isomorphism classes in T by the set IT of n-tuples of partitions; the representation of A corresponding to n e U. will be denoted by M{JZ); see § 3.3.