We introduce the notions of silting comodules and finitely silting comodules in quasi-finite category, and study some properties of them. We investigate the torsion pair and dualities which are related to finitely silting comodules, and give the equivalences among silting comodules, finitely silting comodules, tilting comodules and finitely tilting comodules.

In this communication, I recall the main results [BDK1] in the classification of finite Lie pseudoalgebras, which generalize several previously known algebraic structures, and announce some new results [BDK2] concerning their representation theory.

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical $S{U}_q\left(2\right)$ studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $𝕊$-modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.Let $𝒪$ denote the forgetful functor from $𝕊$-modules to graded vector spaces. Left modules over an operad $𝒫$ are treated as $𝒫$-algebras in the category of $𝕊$-modules. We generalize the results obtained...