We consider aane group schemes G over a eld k of characteristic p > 0. Equivalently , we consider nitely generated commutative k-algebras kG] (the coordinate algebra of G) endowed with the structure of a Hopf algebra. The group scheme G is said to be nite if kG] is nite dimensional (over k) and a nite group scheme is said to be innnitesimal if the ((nite dimensional) algebra kG] is local. A rat...

The genus Hypericum is one of the most important medicinal plants that contain 17 species in Iran, three of them are endemics. This paper reports the essential oil composition of eight Hypericum species from Iran. The essential oil analysis of a number of the studied plants has already been reported but their report from Iran may be valuable for scientists.  Samples collected from different pla...

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of inte-grals: for an arbitrary Hopf algebra, the integral space has dimension ≤ 1, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode fol...

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of preshe...

Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category DY (C)AA of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group (A,A,R) the correspo...

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A,Γ). That is, we construct the derived category Stable(Γ) of (A,Γ) as the homotopy category of a Quillen model structure on Ch(Γ), the category...

We build upon our recently introduced concept of an update structure to show that it is a generalisation very-well-behaved lenses, is, there bijection between strict subset structures and vwb lenses in cartesian categories. are also sufficiently general capture quantum observables, pinpointing the additional assumptions required make two coincide. In doing so, we shift focus from special commut...

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to a q-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of i...

Let C be a (small) category and let F : C −→ Malgf be a functor, where Malgf is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra Aaut(F ) such that F factorizes through a functor F : C −→ Mcoalgf(Aaut(F )), where Mcoalgf(Aaut(F )) is the category of finite-dimensional measured Aaut(F )-comodule algebras. This gene...

Cyclic cohomology of Hopf algebras admitting a modular pair was first defined in [2] and further developed in [3] and [4] in the context of transverse geometry. Their results are followed by several papers computing Hopf cyclic (co)homology of certain Hopf algebras such as [15], [5] and [13]. In a series of papers [1], [10], [11], and [6] the authors developed a theory of cyclic (co)homology wh...