We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra H 0 in the category of Yetter-Drinfeld modules that can be obtained by modifying the multiplication in a proper way is quantum commutative. We give a Structure Theorem for Hopf modules in the...

We consider a pair (a, A) consisting of a quasi-hereditary algebra A and a (positively) graded subalgebra a. We present conditions which guarantee that the algebra grA obtained by grading A by its radical filtration is quasi-hereditary and Koszul. In such cases, we also show that the standard and costandard modules for A have a structure as graded modules for a. These results are applied to obt...

Self-organization is a phenomenon found in biomolecular self-assembly by which proteins are spontaneously driven to assemble and attain various functionalities. This study reports on self-organized behavior in which distributed centimeter-sized modules stochastically aggregate and exhibit a translational wheeling motion. The system consists of two types of centimeter-sized water-floating module...

Let k be an algebraically closed field, let X be a topological space, let A be a sheaf of associative k-algebras on X, and let F = {F1, . . . ,Fp} be a finite family of sheaves of left A-modules on X. In this paper, we develop a noncommutative deformation theory for the family F . This theory generalizes the noncommutative deformation theory for finite families of left modules over an associati...

It is known that the Fundamental Theorem of Hopf modules can be used to characterize Hopf algebras: a bialgebra H over a field is a Hopf algebra (i.e. it is endowed with a so-called antipode) if and only if every Hopf module M over H can be decomposed in the form M coH ⊗ H , where M coH denotes the space of coinvariant elements in M . A partial extension of this equivalence to the case of quasi...