Why Can’t There Be Infinite Descending Chain Of Quotient Representations? – Part 3

 

For a quiver Q, the category Rep(Q) of finite-dimensional representations of Q is abelian. A morphism f : V → W in the category Rep(Q) defined by a collection of morphisms f_i : V_i → W_i is injective (respectively surjective, an isomorphism) precisely if each of the linear maps f_i is.

There is a collection of simple objects in Rep(Q). Indeed, each vertex i ∈ Q₀ determines a simple object S_i of Rep(Q), the unique representation of Q up to isomorphism for which dim(V_j) = δ_ij. If Q has no directed cycles, then these so-called vertex simples are the only simple objects of Rep(Q), but this is not the case in general.

If Q is a quiver, then the category Rep(Q) has finite length.

Given a representation E of a quiver Q, then either E is simple, or there is a nontrivial short exact sequence

0 → A → E → B → 0

Now if B is not simple, then we can break it up into pieces. This process must halt, as every representation of Q consists of finite-dimensional vector spaces. In the end, we will have found a simple object S and a surjection f : E → S. Take E₁ ⊂ E to be the kernel of f and repeat the argument with E₁. In this way we get a filtration

… ⊂ E₃ ⊂ E₂ ⊂ E₁ ⊂ E

with each quotient object Eⁱ⁻¹/Eⁱ simple. Once again, this filtration cannot continue indefinitely, so after a finite number of steps we get E_n = 0. Renumbering by setting E_i := Eⁿ⁻ⁱ for 1 ≤ i ≤ n gives a Jordan-Hölder filtration for E. The basic reason for finiteness is the assumption that all representations of Q are finite-dimensional. This means that there can be no infinite descending chains of subrepresentations or quotient representations, since a proper subrepresentation or quotient representation has strictly smaller dimension.

In many geometric and algebraic contexts, what is of interest in representations of a quiver Q are morphisms associated to the arrows that satisfy certain relations. Formally, a quiver with relations (Q, R) is a quiver Q together with a set R = {r_i} of elements of its path algebra, where each r_i is contained in the subspace A(Q)_aibi of A(Q) spanned by all paths p starting at vertex aiand finishing at vertex b_i. Elements of R are called relations. A representation of (Q, R) is a representation of Q, where additionally each relation r_i is satisfied in the sense that the corresponding linear combination of homomorphisms from V_ai to V_bi is zero. Representations of (Q, R) form an abelian category Rep(Q, R).

A special class of relations on quivers comes from the following construction, inspired by the physics of supersymmetric gauge theories. Given a quiver Q, the path algebra A(Q) is non-commutative in all but the simplest examples, and hence the sub-vector space [A(Q), A(Q)] generated by all commutators is non-trivial. The vector space quotientA(Q)/[A(Q), A(Q)] is seen to have a basis consisting of the cyclic paths a_na_n−1 · · · a₁ of Q, formed by composable arrows a_i of Q with h(a_n) = t(a₁), up to cyclic permutation of such paths. By definition, a superpotential for the quiver Q is an element W ∈ A(Q)/[A(Q), A(Q)] of this vector space, a linear combination of cyclic paths up to cyclic permutation.

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