Although the idea of having a partnership between bosons and
fermions (called supersymmetry, SUSY, closely related to the grading
of algebras) seems to have failed completely in particle physics, it
found a new life as a guide to a partnership of certain exactly
solvable and almost isospectral potentials $V^{(\pm)}(x, a_0)$ in
nonrelativistic quantum mechanics. Their shape invariance means that
$V^{(+)}(x, a_0)$ happens to be equal to $ V^{(-)}(x, a_1)$ up to an
additive $x-$independent shift $R(a_0)$. In the paper the possible
algebraic structure is sought for this shape invariance mapping
after $k$ iterations. In a search for sufficiently simple special
cases the author succeeds in finding a user-friendly structure
(called potential algebra and equivalent to the generalized deformed
oscillator algebra having an interesting grading structure) after
having imposed certain extra relations in the problem. A dark side
of the beauty of his/her algebraic construction lies, as often, in
the parallel loss of isospectrality discovered by Jevicki and
Rodrigues [25]. This requires, even at $k=2$, either the use of an
extremely ugly brute-force regularization procedure based on an {\it
ad hoc} change of boundary conditions (cf. the author's comment on
the three last lines of page~11), or a not too much less drastic
Dyson-mapping change of the representation of the Hilbert space of
states which I explained in more detail, e.g., in J. Phys. A: Math.
Gen. 37 (2004) 10209 - 10222.


MR2540389 Su, Wang-Chang Algebraic shape invariant potentials as the
generalized deformed oscillator. J. Phys. A 42 (2009), no. 38,
385202, 17 pp. 81Q05