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

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

with the counit

ε(( - 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

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

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.

[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 $\mathbb{Z}_2$ 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,

$\Delta x = x \otimes 1 + 1 \otimes x$