# Transformation algebra of supersymmetry transformation

Given supersmmetry transformation:

$$δX^\mu = i\barξψ^\mu.δψ^\mu = ξγ^a(∂_aX^\mu-\frac i2\barχ_a ψ^\mu). δχ_a = 2 D_aξ.$$

How to calculate the transformation algebra of $$X^\mu$$ and $$\psi^\mu$$?

Does that mean calculating $$\{δ_1,δ_2\}X^\mu$$ and $$\{δ_1,δ_2\}\psi^\mu$$?

Reference: Superstring theory (by GSW), Section $$4.1.1$$ Global World-Sheet Symmetry.

It just means calculating $$[δ_1,δ_2]X^\mu$$ and $$[δ_1,δ_2]\psi^\mu$$... Perhaps you should first practice with susy invariance on a simple scalar multiplet, no doubt covered in your relevant course, and only then apply your expertise to this one.
Recall that $$\delta \phi \equiv i[\bar \xi Q,\phi]$$ for an arbitrary field $$\phi$$, boson or fermion. So δ may be thought of as a boson operator, and it is its commutators with other fields or operators that are meaningful, not its anticommutators. The fermionic nature of the supercharge Q is neutralized by the fermionic parameter ξ to yield a bosonic δ. Confirm this is so by comparing the l.h.s. of your transformations to the r.h.sides, and their respective statistics.