Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

Abstract

Let $J$ be a set of pairs consisting of good modules over an affine quantum algebra and invertible elements. The distribution of poles of the normalized R-matrices yields Khovanov-Lauda-Rouquier algebras $R^J$. We define a functor $F$ from the category $S_J$ of finite-dimensional graded $R^J$-modules to the category of finite-dimensional integrable $U_q(g)$-modules. The functor $F$ sends convolution products of $R^J$-modules to tensor products of $U_q(g)$-modules. It is exact if $R^J$ is of finite type A,D,E. When $J$ is the vector representation of $A_{n-1}^{(1)}$, we recover the affine Schur-Weyl duality. Focusing on this case, we obtain an abelian rigid graded tensor category $T_J$ by localizing the category $S_J$. The functor $F$ factors through $T_J$. Moreover, the Grothendieck ring of the category $C_J$, the image of $F$, is isomorphic to the Grothendieck ring of $T_J$ at $q=1$.

80 pages. arXiv:1209.3536 is merged to this paper. Version 2: We proved that the Grothendieck group $K(T_J)$ is isomorphic to the t-deformation of $K(C_J)$. Version 3: We made corrections mainly according to Correction in Invent. Math. 216 (2019), no. 2, 597--599

References in corpus (3)

• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Quantum Grothendieck rings and derived Hall algebras
• PBW bases and KLR algebras

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