Quiver Representations Respecting a Quiver Automorphism Andrew Hubery, Universiry of Leeds

A theorem of Kac [1] states that the dimension vectors of indecomposable representations of a quiver Q (finite and with no vertex loops) are precisely the positive roots of the corresponding symmetric Kac-Moody Lie algebra 𝔤(Q).

Here we extend this result to the case of a quiver with an admissible automorphism (Q, a) — that is, no arrow joins two vertices in the same orbit — and obtain the positive roots of an associated symmetrisable Kac-Moody Lie algeba.

Let (Q, a) be such a pair, and let I denote the vertex set of Q. We can construct a symmetrisable generalised Cartan matrix, indexed by the vertex orbits I, and hence a valued graph Γ (see [3]). Conversely, any valued graph can be obtained in this way.

There is a canonical bijection f : (ℤI)^{⟨a⟩} → ℤI from the fixed points of the root lattice of Q to the root lattice of Γ.

The automorphism a naturally acts on the category of representations of Q and we can study how it acts on the isomorphism classes of indecomposables. We form a representation by taking the direct sum of one representative from each isomorphism class in the orbit, and call such a representation an isomorphically invariant indecomposable.

Theorem 1 Let (Q, a) be a quiver with admissible automorphism and let k be an algebraically closed field of characteristic not dividing the order of a. Then

1. the images under f of the dimension vectors of the ii-indecomposables give the positive root system for the symmetrisable Kac-Moody algebra 𝔤(Γ);

2. if f(α) is real, there is a unique such ii-indecomposable of dimension vector α up to isomorphism. The number of direct summands of this iiindecomposable equals the length of the root f(α).

References

[1] V.G. Kac: Infinite Root Systems, Representations of Graphs and Invariant Theory, Invent. Math. 56 (1980), 57–92.

[2] V.G. Kac: Infinite Dimensional Lie Algebras (3rd ed.), Camb. Univ. Press (1990).

[3] G. Lusztig: Introduction to Quantum Groups, Birkhäuser (1993).