Quantum Painlevé systems of type $A^{(1)}_l$

Details Quantum Painlevé systems of type $A^{(1)}_l$ Quantum Painlevé systems of type $A^{(1)}_l$

2004-02-17

arxiv.org/abs/math/0402281v2

Mathematics

Quantum Algebra

26 pages; v2, minor changes, corrected typos, add $l=1$ case to Section 2

Scientific paper

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.

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