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

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## Abstract

Let *J* be a set of pairs consisting of good \(U'_q(\mathfrak g)\)-modules and invertible elements in the base field \(\mathbb C(q)\). The distribution of poles of normalized R-matrices yields Khovanov–Lauda–Rouquier algebras \(R^J(\beta )\) for each \(\beta \in \mathsf {Q}^+\). We define a functor \(\mathcal F_\beta \) from the category of graded \(R^J(\beta )\)-modules to the category of \(U'_q(\mathfrak g)\)-modules. The functor \(\mathcal F= \bigoplus _{\beta \in \mathsf {Q}^+} \mathcal {F}_\beta \) sends convolution products of finite-dimensional graded \(R^J(\beta )\)-modules to tensor products of finite-dimensional \(U'_q(\mathfrak g)\)-modules. It is exact if \(R^J\) is of finite type *A*, *D*, *E*. If \(V(\varpi _1)\) is the fundamental representation of \(U_q'({\widehat{\mathfrak {sl}}_N})\) of weight \(\varpi _1\) and \(J=\left\{ \bigl (V(\varpi _1), q^{2i} \bigr ) \mid i \in \mathbb Z \right\} \), then \(R^J\) is the Khovanov–Lauda–Rouquier algebra of type \(A_{\infty }\). The corresponding functor \(\mathcal {F}\) sends a finite-dimensional graded \(R^J\)-module to a module in \(\mathcal {C}_J\), where \(\mathcal {C}_J\) is the category of finite-dimensional integrable \(U_q'({\widehat{\mathfrak {sl}}_N})\)-modules *M* such that every composition factor of *M* appears as a composition factor of a tensor product of modules of the form \(V(\varpi _1)_{q^{2s}}\) \((s \in {\mathbb {Z}})\). Focusing on this case, we obtain an abelian rigid graded tensor category \({\mathcal T}_J\) by localizing the category of finite-dimensional graded \(R^J\)-modules. The functor \(\mathcal {F}\) factors through \({\mathcal T}_J\). Moreover, the Grothendieck ring of the category \(\mathcal {C}_J\) is isomorphic to the Grothendieck ring of \({\mathcal T}_J\) at \(q=1\).

## Mathematics Subject Classification

81R50 16G 16T25 17B37## Notes

### Acknowledgements

We would like to express our gratitude to Bernard Leclerc for his kind explanations of his works and many fruitful discussions. The first and the third author gratefully acknowledge the hospitality of RIMS (Kyoto) during their visit in 2011 and 2012.

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