2.8. Quivers 21 Definition 2.8.8. A subrepresentation of a representation (Vi,xh) of a quiver Q is a representation (Wi,xh) where Wi Vi for all i I and where xh(Wh ) Wh and x h = xh|W h : Wh −→ Wh for all h E. Definition 2.8.9. The direct sum of two representations (Vi,xh) and (Wi,yh) is the representation (Vi Wi,xh yh). As with representations of algebras, a nonzero representation (Vi) of a quiver Q is said to be irreducible if its only subrepresentations are (0) and (Vi) itself, and it is said to be indecomposable if it is not isomorphic to a direct sum of two nonzero representations. Definition 2.8.10. Let (Vi,xh) and (Wi,yh) be representations of the quiver Q. A homomorphism ϕ : (Vi) −→ (Wi) of quiver representations is a collection of maps ϕi : Vi −→ Wi such that yh ϕh = ϕh xh for all h E. Problem 2.8.11. Let A be a Z+-graded algebra, i.e., A = n≥0 A[n], and A[n] · A[m] A[n + m]. If A[n] is finite dimensional, it is useful to consider the Hilbert series hA(t) = dim A[n]tn (the generating function of dimensions of A[n]). Often this series converges to a ratio- nal function, and the answer is written in the form of such a function. For example, if A = k[x] and deg(xn) = n, then hA(t) = 1 + t + t2 + · · · + tn + · · · = 1 1 t . Find the Hilbert series of the following graded algebras: (a) A = k[x1,...,xm] (where the grading is by degree of polyno- mials). (b) A = k x1,...,xm (the grading is by length of words). (c) A is the exterior (= Grassmann) algebra ∧k[x1,...,xm] gen- erated over some field k by x1,...,xm with the defining relations xixj + xjxi = 0 and x2 i = 0 for all i, j (the grading is by degree). (d) A is the path algebra PQ of a quiver Q (the grading is defined by deg(pi) = 0, deg(ah) = 1). Hint: The closed answer is written in terms of the adjacency matrix MQ of Q.
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