Small Bialgebras with a Projection: Applications

In this paper we continue the investigation started in [A.M.St.-Small], dealing with bialgebras $A$ with an $H$-bilinear coalgebra projection over an arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as deformed bosonizations R#_{\xi} H of a pre-bialgebra $R$ by $H$ with a cocycle $\xi$. Here we describe the behavior of $\xi$ in the case when $R$ is f.d. and thin i.e. it is connected with one dimensional space of primitive elements. This is used to analyze the arithmetic properties of $A$. Meaningful results are obtained when $H$ is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations (e.g. the multiplication of $R$ is not $H$-colinear or $\xi$ is non-trivial)

Integrals, quantum Galois extensions and the affineness criterion for quantum Yetter-Drinfel'd modules

We introduce and study a general concept of integral of a threetuple (H, A, C), where H is a Hopf algebra acting on a coalgebra C and coacting on an algebra A. In particular, quantum integrals associated to Yetter-Drinfel'd modules are defined. Let A be an H-bicomodule algebra, $^H {\cal YD}_A$ be the category of (generalized) Yetter-Drinfel'd modules and $B$ the subalgebra of coinvariants of the Verma structure of $A$. We introduce the concept of quantum Galois extensions and we prove the affineness criterion in a quantum version.Comment: latex 32 pg. J. Algebra, to appea

Categories of comodules and chain complexes of modules

Let \lL(A) denote the coendomorphism left $R$-bialgebroid associated to a left finitely generated and projective extension of rings $R \to A$ with identities. We show that the category of left comodules over an epimorphic image of \lL(A) is equivalent to the category of chain complexes of left $R$-modules. This equivalence is monoidal whenever $R$ is commutative and $A$ is an $R$-algebra. This is a generalization, using entirely new tools, of results by B. Pareigis and D. Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the non commutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.Comment: The title has been changed, the first part is removed and the construction of the coendomorphim bialgebroid is now freely used in the statement of the main Theorem

When is R-gr equivalent to the category of modules?

AbstractIn the first part of this paper, we characterize graded rings R=⊕σ∈GRσ for which the category R-gr is equivalent with a category of modules over a certain ring.In the second part, sufficient conditions are given for the following implication to hold: if R-gr is equivalent with R1-mod (1 is the unit element of G), then R is a strongly graded ring

Braided Bialgebras of Type One

Braided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly $\mathbb{N}$-graded both as an algebra and as a coalgebra

Braided Bialgebras of Hecke-type

The paper is devoted to prove a version of Milnor-Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from $2$, we prove that, for a given connected braided bialgebra $(A,\mathfrak{c}_A)$ which is infinitesimally $\lambda$-cocommutative for some element $\lambda \neq 0$ that is not a root of one in the base field, then the infinitesimal braiding of $A$ is of Hecke-type of mark $\lambda$ and $A$ is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements

Small bialgebras with a projection

AbstractLet A be a bialgebra with an H-bilinear coalgebra projection over an arbitrary subbialgebra H with antipode. In characteristic zero, we completely describe the bialgebra structure of A whenever H is either f.d. or cosemisimple and the H-coinvariant part R of A is connected with one-dimensional space of primitive elements

Modulation by internal protons of native cyclic nucleotide-gated channels from retinal rods

Ion channels directly activated by cyclic nucleotides are present in the plasma membrane of retinal rod outer segments. These channels can be modulated by several factors including internal pH (pH(i)). Native cyclic nucleotide-gated channels were studied in excised membrane patches from the outer segment of retinal rods of the salamander. Channels were activated by cGMP or cAMP and currents as a function of voltage and cyclic nucleotide concentrations were measured as pH(i) was varied between 7.6 and 5.0. Increasing internal proton concentrations reduced the current activated by cGMP without modifying the concentration (K(1/2)) of cGMP necessary for half-activation of the maximal current. This effect could be well described as a reduction of single-channel current by protonation of a single acidic residue with a pK(1) of 5.1. When channels were activated by cAMP a more complex phenomenon was observed. K(1/2) for cAMP decreased by increasing internal proton concentration whereas maximal currents activated by cAMP increased by lowering pH(i) from 7.6 to 5.7-5.5 and then decreased from pH(i) 5.5 to 5.0. This behavior was attributed both to a reduction in single-channel current as measured with cGMP and to an increase in channel open probability induced by the binding of three protons to sites with a pK(2) of 6

Design of controllers for hybrid linear systems with impulsive inputs and periodic jumps

In this study, the problem of designing a controller for a hybrid system with impulsive input and periodic jumps is addressed. In particular, it is shown that any hybrid system with impulsive inputs and periodic jumps can be recast into a discrete-time, linear, time-invariant system, which, in turn, can be used to design a controller by using classical methods. Furthermore, it is shown that, once such a controller has been designed, it can be readily used to control the hybrid system by mean of an interfacing system that is based just on the continuous-time dynamics of the plant to be controlled. Several examples, spanning from aerospace to biomedical applications, are reported in order to corroborate the theoretical results