The concept in theoretical physics of supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity.
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Unitary (-1)F operator
Following is the essence of supersymmetry, which is encapsulated within the following minimal quantum group. We have the two dimensional Hopf algebra generated by (-1)F subject to
with the counit
and the coproduct
and the antipode
Thus far, there is nothing supersymmetric about this Hopf algebra at all; it is isomorphic to the Hopf algebra of the two element group
where +1 eigenstates of (-1)F are called bosons and -1 eigenstates are called fermions.
This describes a fermionic braiding; don't pick up a phase factor when interchanging two bosons or a boson and a fermion, but multiply by -1 when interchanging two fermions. This provides the essence of the boson/fermion distinction.
Fermionic operators
The previous analysis only introduced the concept of fermions, and is not actual supersymmetry. The Hopf algebra is
Let's say we are dealing with a super Lie algebra with even generators x and odd generators y.
Then,
This is compatible with
Supersymmetry is the symmetry over systems where interchanging two fermions attain a minus sign.