We propose quantum Painlev\'e systems of type $A_l^{ (1)}$. These systems, for $l=1$ and $l\ge 2$, should be regarded as quantizations of the second Painlev\'e equation and the differential systems with the affine Weyl group symmetries of type $A_l^{ (1)}$ studied by M. Noumi and Y. Yamada \cite{ NYhigherorder}, respectively. These quantizations enjoy the affine Weyl group symmetries of type $A_l^{ (1)}$ as well as the Lax representations. The quantized systems of type $A_1^{ (1)}$ and type $A_l^{ (1)}$ ($l= 2n$) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type $A_2^{ (1)}$ and $A_{ l+1}^{ (1)}$, respectively.
Categories: math.QA