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 . Supersymmetry comes in when introducing the nontrivial quasitriangular structure
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.
Read more about this topic: Supersymmetry As A Quantum Group