Weak supersymmetry

We explore ``weak'' supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1--particle states in the spectrum. Coleman--Mandula and Haag--Lopuszanski--Sohnius theorems forbid the presence of such extra bosonic charges in {\it interacting} theory for $d \geq 3$. However, these theorems do not apply in one or two dimensions. For $d=1$, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to ``n--fold'' supersymmetric systems and to quasi-exactly solvable systems studied earlier.