Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras

Abstract

In this paper, we introduce twisted and folded AR-quivers of type $A_{2n+1}$, $D_{n+1}$, $E_6$ and $D_4$ associated to (triply) twisted Coxeter elements. Using the quivers of type $A_{2n+1}$ and $D_{n+1}$, we describe the denominator formulas and Dorey's rule for quantum affine algebras $U_q^{\prime }(B_{n+1}^{(1)})$ and $U_q^{\prime }(C_n^{(1)})$, which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for $U_q^{\prime }(B_{n+1}^{(1)})$ (resp. $U_q^{\prime }(C_n^{(1)})$) using certain statistics on any folded AR-quiver of type $A_{2n+1}$ (resp. $D_{n+1}$) and Dorey's rule for $U_q^{\prime }(B_{n+1}^{(1)})$ (resp. $U_q^{\prime }(C_n^{(1)})$) applying the notion of minimal pairs in a twisted AR-quiver. By adopting the same arguments, we propose the conjectural denominator formulas and Dorey's rule for $U_q^{\prime }(F_4^{(1)})$ and $U_q^{\prime }(G_2^{(1)})$.

arXiv commentThis is a megred version of arXiv:1606.00076 and arXiv:1606.00102. This version is published in Journal of Algebra. We upload this for reder's convenience

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Cited by in corpus (9)

• Monoidal categorification and quantum affine algebras
• Q-data and representation theory of untwisted quantum affine algebras
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Continuous Quivers of Type A (II) The Auslander-Reiten Space
• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Categories over quantum affine algebras and monoidal categorification
• Categorical crystals for quantum affine algebras
• PBW theoretic approach to the module category of quantum affine algebras