Supersymmetry as a quantum group
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The concept in theoretical physics of supersymmetry can be reinterpretated in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity.
[edit] (-1)F
Let's look at 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
- ε(( − 1)F) = 1
and the coproduct
and the antipode
- S( − 1)F = ( − 1)F
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
![\mathcal{R}=\frac{1}{2}\left[ 1 \otimes 1 + (-1)^F \otimes 1 + 1 \otimes (-1)^F - (-1)^F \otimes (-1)^F\right]](../../../math/7/c/c/7cc42e3061b1772be5307708138eded6.png)
In representation theory, +1 eigenstates of (-1)^F are called bosons and -1 eigenstates 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 gives us the essence of the boson/fermion distinction.
[edit] fermionic operators
We still haven't introduced any actual supersymmetry yet, but we had set the stage by introducing the concept of fermions. The Hopf algebra is graded and contains even and odd elements. Even elements commute with (-1)F; odd ones anticommute. The subalgebra not containing (-1)^F is supercommutative.
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 picks up a minus sign.
