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)^F2 = 1

with the counit

ϵ((−1)^F) = 1

and the coproduct

Δ(−1)^F = (−1)^F ⊗ (−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 ℤ₂. Supersymmetry comes in when introducing the nontrivial quasitriangular structure

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

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 ℤ₂ 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,

Δx = x ⊗ 1 + 1 ⊗ x

Δy = y ⊗ 1 + (−1)^F ⊗ y

This is compatible with ℛ.

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