In 1 + 1 dimensions the N = 1 supersymmetry algebra (also known as 
because we have one left-moving SUSY generator and one right moving one) has the following generators:
- supersymmetric charges:
- supersymmetric central charge:
- time translation generator:
- space translation generator:
- boost generator:
- fermionic parity:
- unit element:
The following relations are satisfied by the generators:
![\begin{align}
& \{ \Gamma,\Gamma \} =2I && \{ \Gamma, Q \} =0 && \{ \Gamma, \bar{Q} \} =0\\
&\{ Q,\bar{Q} \}=2Z && \{ Q, Q \}=2(H+P) && \{ \bar{Q}, \bar{Q} \} =2(H-P) \\
& [N,Q]=\frac{1}{2} Q && [N,\bar{Q} ]=-\frac{1}{2} \bar{Q} && [N,\Gamma]=0 \\
& [N,H+P]=H+P && [N,H-P]=-(H-P) &&
\end{align}](http://wpcontent.answers.com/math/6/8/a/68ab9795baa1a8f92341c293734292e1.png)
is a central element.
The supersymmetry algebra admits a 
-grading. The generators 
are even (degree 0), the generators 
are odd (degree 1).
2(H − P) gives the left-moving momentum and 2(H + P) the right-moving momentum.
Basic representations of this algebra are the vacuum, kink and boson-fermion representations, which are relevant e.g. to the supersymmetric (quantum) sine-Gordon model.
References
- K. Schoutens, Supersymmetry and factorized scattering, Nucl.Phys. B344, 665–695, 1990
- T.J. Hollowood, E. Mavrikis, The N = 1 supersymmetric bootstrap and Lie algebras, Nucl. Phys. B484, 631–652, 1997, arXiv:hep-th/9606116
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