Séminaires : Séminaire d'Algèbre

Let $g_0$ be a simple Lie algebra of type $ADE$ and let $U′_q(g)$ be the corresponding untwisted quantum affine algebra. We found an action of the braid group $B(g_0)$ on the quantum Grothendieck ring $K_t(g)$ of Hernandez-Leclerc's category $C^0_g$. In the case of $g_0=A_{N−1}$, we construct a monoidal autofunctor $S_i$ for each integer $i$ on a category $T_N$ arising from the  quiver Hecke algebra of type $A_\infty$. 
Since there is an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t(A^(1)_{N−1})$, the functors $S_i$, $(i=1, ..., N-1)$, recover the action of the braid group $B(A_{N−1})$. 
This is a joint work with Masaki Kashiwara, Euiyong Park and Se-jin Oh.