deformation parameter. This $\star$-product is different from the Moyal-Weyl $\star$-product [@mw] where the expansion in powers of the deformation parameter leads to an infinite power series. One should also note that the $\star$-product (\[star\]) is hermitian, $$({\rm F}\star {\rm G})^* = {\rm G}^* \star {\rm F}^* , \label{complconj}$$ where $*$ denotes the usual complex conjugation.

The $\star$-product (\[star\]) implies $$\begin{aligned}
\{ \theta^\alpha \ds \theta^\beta \} &=& C^{\alpha\beta}, \quad \{
\bar{\theta}_{\dot\alpha}\ds \bar{\theta}_{\dot\beta}\} =
\bar{C}_{\dot{\alpha}\dot{\beta}}, \quad \{ \theta^\alpha \ds
\bar{\theta}_{\dot\alpha} \} = 0 ,\nonumber \\
\lbrack x^m \ds x^n \rbrack &=& 0 , \quad [x^m \ds \theta^\alpha ]
= 0, \quad [x^m \ds \bar{\theta}_{\dot\alpha} ] = 0 .
\label{thetastar}\end{aligned}$$ Relations (\[thetastar\]) enable us to define the deformed superspace or “nonanticommutative superspace”. It is generated by the usual bosonic and fermionic coordinates (\[undefsupsp\]) while the deformation is contained in the new product (\[star\]).

The next step is to apply the twist (\[twist\]) to the Hopf algebra of SUSY transformations. We will not give details here, they can be found in [@miSUSY]. We just state the most important results.

The deformed infinitesimal SUSY transformation is defined in the following way $$\delta^\star_\xi {\rm F} = \big(\xi Q + \bar{\xi}\bar{Q} \big) {\rm F}.
\label{defsusytr}$$ The twist 