Weak coalgebra-Galois extensions are studied. A notion of an invertible weak entwining structure is introduced. It is proven that, within an invertible weak entwining structure, the surjectivity of the canonical map implies bijectivity provided the structure coalgebra C is either coseparable or projective as a C-comodule.

We define comodule algebras and Galois extensions for actions of bialgebroids. Using just module conditions we characterize the Frobenius extensions that are Galois as depth two and right balanced extensions. As a corollary, we obtain characterizations of certain weak and ordinary Hopf-Galois extensions without reference to action in the hypothesis. 2000 AMS Subject Classification: 13B05, 16W30

We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. consider fully faithful as well adjoint pairs functors. give several applications to Grothendieck categories and, in particular, (graded) module and comodule

An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual knots. Constructions of such system are presented.

For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.

1 Markov chains and random walks in classical probability . . 3 2 Quantum Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Random walks on comodule algebras . . . . . . . . . . . . . . . . . . . . . . 7 4 Random walks on finite quantum groups . . . . . . . . . . . . . . . . . . 11 5 Spatial Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . ....

We introduce an algebra $\mathcal{K}_n$ which has a structure of left comodule over the quantum toroidal type $A_{n-1}$. Algebra is higher rank generalization $\mathcal{K}_1$, provides uniform description deformed $W$ algebras associated with Lie (super)algebras types BCD. show that possesses family commutative subalgebras.