arXiv subjects

G. Militaru

Publications and source records attributed to G. Militaru.

43 records · Page 3Linked to original sources

arXiv2001

Maschke functors, semisimple functors and separable functors of the second kind. Applications

We introduce separable functors of the second kind (or $H$-separable functors) and $H$-Maschke functors. $H$-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An $H$-separable functor is always $H$-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to $H$-separability. Our concepts are applied to modules, comodules, entwined modules, quantum Yetter-Drinfeld modules, relative Hopf modules.

math.RA↗

arXiv2001

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.

math.QA↗

arXiv2000

Heisenberg duoble, pentagon equation, structure and classification of finite dimensional Hopf algebras

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and the set of the orbits of the resticted Jordan action $GL_n(k) \times M_n(k)\otimes M_n(k) \to M_n(k) \otimes M_nk$ $(u, R) \to (u\otimes u)R (u\otimes u)^{-1}$, the representatives of wich are invertible solutions of length n of the pentagon equation.

math.QA↗

arXiv2000

Frobenius and Mashke type Theorems for Doi-Hopf modules and entwined modules revisited: a unified approach

We study when induction functors (and their adjoints) between categories of Doi-Hopf modules and, more generally, entwined modules are separable, resp. Frobenius. We present a unified approach, leading to new proofs of old results by the authors, as well as to some new ones. Also our methods provide a categorical explanation for the relationship between separability and Frobenius properties.

math.RA↗

arXiv1999

The structure of Frobenius algebras and separable algebras

We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}R^{23}=R^{23}R^{13}=R^{13}R^{12}$, called the FS-equation. Given a solution of the FS-equation satisfying certain normalizing conditions, we can construct a Frobenius algebra or a separable algebra A(R). The main result of this paper in the structure of this two fundamental types of algebras: a finitely generated projective Frobenius or separable $k$-algebra $A$ is isomorphic to such A(R). If $A$ is free as a $k$-module, then A(R) can be described using generators and relations. A new characterisation of Frobenius extensions is given: $B\subset A$ is Frobenius if and only if $A$ has a $B$-coring structure such that the comultiplication $Δ$ is an $A$-bimodule map.

math.QA↗

arXiv1999

Factorisation structures of algebras and coalgebras

We consider the factorisation problem for bialgebras: when a bialgebra $K$ factorises as $K=HL$, where $H$ and $L$ are algebras and coalgebras (but not necessarly bialgebras). Given two maps $R: H\ot L\to L\ot H$ and $W:L\ot H\to H\ot L$, we introduce a product $L_W\bowtie_R H$ and give necessary and sufficient conditions for $L_W\bowtie_R H$ to be a bialgebra. It turns out that $K$ factorises as $K=HL$ if and only if $K\cong L_W\bowtie_R H$ for some maps $R$ and $W$. As examples of this product we recover constructions introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed bu Beattie, D\u asc\u alescu and Grünenfelder appear as special cases.

math.QA↗

arXiv1998

Separable functors for Doi-Hopf modules. Applications

There are theorems in the literature which are referred as Maschke type theorems, because of the attempt to generalize Maschke's classical theorem. Our remark is that all these theorems are in fact providing sufficient conditions for the separability of a certain forgetful functor. But, unlike classical Maschke's theorem, where the sufficient condition is also necessary, none of these theorems gives any necessary condition for the separability of the forgeful functor in question. Our job is to find a necessary and sufficient condition for the separability of the forgetful functor from the category of Doi-Hopf modules forgetting the comodule structure. As in the classical case, this functor is separable iff exists a certain "integral map" with a normalizing property. As particular cases, improved versions of "Maschke type" theorems are given. Other various applications are given. For example, keeping in mind that the category of Yetter-Drinfel'd modules is also a category of Doi-Hopf modules, we obtain the notion of quantum integral. For H a finite dimensional Hopf algebra, the existence of such a quantum integral is a necessary and sufficient condition for the Drinfel'd double to be a separable extension of H.

math.QA↗