High Quality Content by WIKIPEDIA articles! The concept in theoretical physics of supersymmetry can be reinterpreted 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 ?(( ? 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)^Fright] +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.

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