Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Frobenius semisimplicity for convolution morphisms

Mathematische Zeitschrift - This article concerns properties of mixed $$ell $$ -adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a...     

The Frobenius rank equality for morphisms

Let $\text{U}$(

,ε) be an abelian category with Serre class

and Euler-Poincaré mapping ε. For η a morphism in $\text{U}$(

,ε) with Imη a member of

, let rank _εη =ε(Imη). A proof is given of the Frobenius rank equality: if αβγ is a composition of three morphisms in $\text{U}$(

,ε) and Imβ is a member of

, then rank_εαβ+rank_εβγ+rank_ε(kerβγ)β(cokαβ)=rank_εβ+rank_εαβγ.     

Graphical Frobenius representations

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A graphical Frobenius representation (GFR) of a Frobenius group G is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to G. The problem of classifying which Frobenius groups admit a GFR is a natural extension of the classification of groups that have a graphical regular representation (GRR), which occupied many authors from 1958 through 1982. In this paper, we review for graph theorists some standard and deep results about finite Frobenius groups, determine classes of finite Frobenius groups and individual groups that do and do not admit GFRs, and classify those Frobenius groups of order at most 300 having a GFR. Because a Frobenius group, as opposed to a regular permutation group, has a highly restricted structure, the GFR problem emerges as algebraically more complex than the GRR problem. This paper concludes with some further questions and a strong conjecture.     

Double Frobenius algebras

Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of (strict or non-strict) double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a (strict) double Frobenius algebra is given.     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

Frobenius induction for algebras

Let B→A be a homomorphism of Hopf algebras and let C be an algebra. We consider the induction from B to A of C in two cases: when C is a B-interior algebra and when C is a B-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.     

Frobenius structural matrix algebras

We discuss when the incidence coalgebra of a locally finite preordered set is right co-Frobenius. As a consequence, we obtain that a structural matrix algebra over a field k is Frobenius if and only if it consists, up to a permutation of rows and columns, of diagonal blocks which are full matrix algebras over k.     

Deformations of Loday-type algebras and their morphisms

We study formal deformations of multiplication in an operad. This closely resembles Gerstenhaber's deformation theory for associative algebras. However, this applies to various algebras of Loday-type and their twisted analogs. We explicitly describe the cohomology of these algebras with coefficients in a representation. Finally, deformation of morphisms between algebras of the same Loday-type is also considered.     

Hochschild cohomology of Frobenius algebras

Let be a field, a finite-dimensional Frobenius -algebra and , the Nakayama automorphism of with respect to a Frobenius homomorphism . Assume that has finite order and that has a primitive -th root of unity . Consider the decomposition of , obtained by defining , and the decomposition of the Hochschild cohomology of , obtained from the decomposition of . In this paper we prove that and that if the decomposition of is strongly -graded, then acts on and .

     

Extending involutions on Frobenius algebras

 Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C ₂ -module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution σ, we characterize the central simple algebras A of degree n that contain B and carry involutions extending σ. Received: 29 October 2001 / Revised version: 2 February 2002     

L-Infinity algebra morphisms and symmetric brace algebras

Morphisms of L-infinity algebras and symmetric brace algebras are studied. This is intended to be an expository note. Much of what is discussed here arose from joint work with M. Markl and from discussions with T. Kadeishvili. We wish to work primarily with symmetric brace algebras but will begin with brace algebras.     

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl₂(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5 pp. 734–740, November, 1999.     

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions of subalgebras of a Hopf algebra are -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra , we show that when are Hopf algebras in the Yetter-Drinfeld category for , the extension is -Frobenius provided is finite over and the extension of biproducts is cleft.

More generally we give conditions for an extension to be -Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

     

Projective cell modules of Frobenius cellular algebras

In this note we give a criterion of projectiveness of the simple cell modules over finite dimensional Frobenius cellular algebras.