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Super Lax Pairs and Infinite Symmetries in The $1/r^2$ System

B. Sriram Shastry·Bill Sutherland
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Abstract

We present an algebraic structure that provides an interesting and novel link between supersymmetry and quantum integrability. This structure underlies two classes of models that are exactly solvable in 1-dimension and belong to the $1/r^2 $ family of interactions. The algebra consists of the commutation between a ``Super- Hamiltonian'', and two other operators, in a Hilbert space that is an enlargement of the original one by introducing fermions. The commutation relations reduce to quantal Ordered Lax equations when projected to the original subspace, and to a statement about the ``Harmonic Lattice Potential'' structure of the Lax operator. These in turn lead to a highly automatic proof of the integrability of these models. In the case of the discrete $SU(n)-1/r^2$ model, the `` Super-Hamiltonian'' is again an $SU(m)-1/r^2$ model with a related $m$, providing an interesting hierarchy of models.

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B. Sriram Shastry, Bill Sutherland. 1992-12-19. Super Lax Pairs and Infinite Symmetries in The $1/r^2$ System. https://doi.org/10.1103/physrevlett.70.4029

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