TITLE: Geometric or conceptual way to understand supersymmetry algebra
QUESTION [10 upvotes]: Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ensuring the supersymmetry invariance and then writing down the supersymmetry algebra? For a geometric or algebraic way, I mean to (at least partially) derive the supersymmetry algebra from pure geometry (spinors, spin group, etc.).

REPLY [3 votes]: In this paper about a global theory of supermanifolds, Alice Rogers develops the theory of supermanifolds as they underly supersymmetric field theories from a rigorous but physicist-friendly differential-geometric point of view from topological scratch, and also puts some additional structures such as vector fields and tangent spaces on them. She also compares the $G^{\infty}$ or deWitt supermanifolds to the algebro-geometric approach of for example Kostant or Leites.
Alice Roger's 2007 textbook explains the supermathematics needed for doing superphysics (including Grassmann algebras, super Lie groups such as the super Poincare group, etc) and contains more applications to different physics topics such as $N=1$ supersymmetry, supergravity, some aspects of string theory, or Brownian motion from the same nice differential-geometric point of view.