2015
DOI: 10.4134/jkms.2015.52.2.239
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Generalized McKay Quivers, Root System and Kac-Moody Algebras

Abstract: Abstract. Let Q be a finite quiver and G ⊆ Aut( Q) a finite abelian group. Assume that Q and Γ are the generalized Mckay quiver and the valued graph corresponding to (Q, G) respectively. In this paper we discuss the relationship between indecomposable Q-representations and the root system of Kac-Moody algebra g(Γ). Moreover, we may lift G to G ⊆ Aut(g( Q)) such that g(Γ) embeds into the fixed point algebra g( Q) G and g( Q) G as a g(Γ)-module is integrable.

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Cited by 2 publications
(5 citation statements)
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“…In 2015, lots of references have gone appearing regarding this topic. Among them, B. Hou and S. Yang dealt with generalized McKay quivers, root system and Kac-Moody algebras [124], R. Lv and Y. Tan studied the link between some gim algebras and the associated indefinite Kac-Moody algebras [176] and D. Allcock [4] found root systems for Lorentzian Kac-Moody algebras in rank 3.…”
Section: Kac-moody Algebras: Other Related Topicsmentioning
confidence: 99%
“…In 2015, lots of references have gone appearing regarding this topic. Among them, B. Hou and S. Yang dealt with generalized McKay quivers, root system and Kac-Moody algebras [124], R. Lv and Y. Tan studied the link between some gim algebras and the associated indefinite Kac-Moody algebras [176] and D. Allcock [4] found root systems for Lorentzian Kac-Moody algebras in rank 3.…”
Section: Kac-moody Algebras: Other Related Topicsmentioning
confidence: 99%
“…By [9], Propsoition 3.6, there is an action of G on Γ Q such that Γ Q = Γ Q = Γ Q . The proof is completed.…”
Section: Proof Of Main Theoremmentioning
confidence: 99%
“…Following from [15], we can define a linear action of G on kQ * G by g(λh) = χ g (h)λh, g ∈ G, λh ∈ kQ * G. Then G ⊆ Aut(kQ * G) and under this action, we can prove that (kQ * G) * G is Morita equivalent to kQ (see [9], Proposition 3. For our quivers Q and Q, we denote by Γ and Γ the corresponding valued graphs (Q, G) and ( Q, G), respectively.…”
Section: An Examplementioning
confidence: 99%
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