Type A -fold supersymmetry and generalized Bender–Dunne polynomials

Abstract We derive the necessary and sufficient condition for type A N -fold supersymmetry by direct calculation of the intertwining relation and show the complete equivalence between this analytic construction and the sl (2) construction based on quasi-solvability. An intimate relation between the pair of algebraic Hamiltonians is found. The classification problem on type A N -fold supersymmetric models is investigated by considering the invariance of both the Hamiltonians and N -fold supercharge under the GL (2, K ) transformation. We generalize the Bender–Dunne polynomials to all the type A N -fold supersymmetric models without requiring the normalizability of the solvable sector. Although there is a case where weak orthogonality of them is not guaranteed, this fact does not cause any difficulty on the generalization. It is shown that the anti-commutator of the type A N -fold supercharges is expressed as the critical polynomial of them in the original Hamiltonian, from which we establish the complete type A N -fold superalgebra. A novel interpretation of the critical polynomials in view of polynomial invariants is given.