Higher Nilpotent Analogues of A-infinity Structure

Higher nilpotent analogues of the $A-\mathrm{\infty }$-structure are explicitly defined on arbitrary simplicial complexes, generalizing explicit construction of /hep-th/0704.2609. These structures are associated with the higher nilpotent differential $d_n$, satisfying $d_n^n=0$, which is naturally defined on triangulated manifolds (tetrahedral lattices). The deformation $D_n = (I + \epsilon_n) d_n (I + \epsilon_n)^{-1}$isdefinedwiththehelpofthe$n$-versionsofdiscreteexteriorproduct$\wedge_n$andthe$K_n$-operator.