Supersymmetry as a quantum group

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.

(-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

:$\{(-1)^F\}^2=1$

with the counit

:$epsilon((-1)^F)=1$

and the coproduct

:$Delta\; (-1)^F=(-1)^F\; otimes\; (-1)^F$

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 $mathbb\{Z\}\_2$. Supersymmetry comes in when introducing the nontrivial quasitriangular structure

:$mathcal\{R\}=frac\{1\}\{2\}left\; [\; 1\; otimes\; 1\; +\; (-1)^F\; otimes\; 1\; +\; 1\; otimes\; (-1)^F\; -\; (-1)^F\; otimes\; (-1)^F\; ight]$

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.

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 $mathbb\{Z\}\_2$ 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,

:$Delta\; x\; =\; x\; otimes\; 1\; +\; 1\; otimes\; x$:$Delta\; y\; =\; y\; otimes\; 1\; +\; (-1)^F\; otimes\; y$

This is compatible with $mathcal\{R\}$.

Supersymmetry is the symmetry over systems where interchanging two fermions picks up a minus sign.