Group-graded Algebras Research Articles

Group-Graded Systems and Algebras

In the paper, we discuss some problems concerning the structural properties of crossed products. While expansions of C*-algebras under group actions have been studied rather extensively, known difficulties in the transition to irreversible dynamical systems require the development of new methods. The first step in this direction is to study actions of inverse semigroups, whose properties are closest to those of groups. The main tool to achieve the goal is the concept of grading. The detection of the grading structure in the corresponding constructions seems to be very promising.

Modular G-graded algebras and G-algebras of endomorphisms

We study Clifford Theory and field extensions for strongly group-graded algebras. In [Turull, A., Clifford theory and endoisomorphisms, J. Algebra 371 (2012), 510–520] and [Turull, A., Endoisomorphisms yield mo-dule and character correspondences, J. Algebra 394 (2013), 7–50] the author introduced the notion of endoisomorphism showing that there is a natural connection between it and Clifford Theory of finite group algebras. An endoisomorphism is an isomorphism between G-algebras of endomorphisms, where G is a finite group. We consider here endoisomorphisms between modules over strongly G-graded algebras. An endoisomorphism induces equivalences of categories with some good compatibility properties (see Theorem ?? and Theorem ?? below).

Nilpotent Subsets of Graded Algebras with Color Involution

In this article, we examine group-graded algebras R with color involution. If S is the symmetric elements and K the skew symmetric elements, it is shown that if S n = 0 then R 3n = 0, whereas if K n = 0 then (R[R, R]ε R) n = 0, where R[R, R]ε R is the ε-commutator ideal corresponding to a bicharacter ε. It is then shown that the bounds found for the degrees of nilpotence of R and R[R, R]ε R are best possible. We also examine the situation where R is graded by a finite group and only the identity component of either S or K is assumed to be nilpotent. We conclude by looking at some questions related to the Nagata-Higman theorem.

Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications

Using results on algebras that are graded by p-groups, we study representations of infinitesimal groups \( \mathcal{G} \) that possess a normal subgroup \( \mathcal{N}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } \mathcal{G} \) with a diagonalizable factor group \( \mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{N}}} \right. \kern-\nulldelimiterspace} \mathcal{N} \). When combined with rank varieties, Auslander-Reiten theory and Premet's work on SL(2)1-modules, these techniques lead to the determination of the indecomposable modules of the infinitesimal groups of domestic representation type.

On Galois extension of Hopf algebras

Let H be a cosemisimple Hopf algebra over a field k, and π: A → H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B = LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.

On a Group Graded Version of BGG

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr Λ ≅ 𝒟b(Coh ℙn), where gr Λ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟b(Coh ℙn) is the derived category of bounded complexes of coherent sheaves on projective space ℙn. Generalizations of this result were obtained in Martínez-Villa and Saorín (2004) and from a different point of view, the theorem has been extended by Yanagawa (2004) to ℤn-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra. The aim of the article is to extend the results of Martínez-Villa and Saorín (2004) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987).

Induced modules of strongly group-graded algebras

Induced modules of strongly group-graded algebras

Killing of supports on graded algebras

Killing of supports along subsets U of a group G and regradings along certain maps of groups φ : G ′ → G are studied, in the context of group-graded algebras. We show that, under precise conditions on U and φ, the module theories over the initial and the final algebras are functorially well-connected. Special attention is paid to G = Z , in which case the results can be applied to n-Koszul algebras.

A Note on Symmetrizing Forms for Fully Group-Graded Algebras

A Note on Symmetrizing Forms for Fully Group-Graded Algebras

A module-theoretic approach to Clifford theory for blocks

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

The representation theory of fully group-graded algebras

Let G be a finite group, k an algebraically closed field of characteristic p, and R a fully G-graded k-algebra of finite dimension. The most trivial example of such an algebra is kG. This paper attempts to generalize the p-modular representation theory of the group algebra kG to the groupgraded algebra R. It is shown that most of the theory is the same for R as for kG. Two areas in which the representation theory of R differs from that of kG are the dimensions of induced modules and the properties of the defect groups of the blocks of R. If H is a subgroup of G and V is an R,-module, then the dimension of VR is not necessarily divisible by the dimension of V. If B is a block of R and D is a defect group of B, then D is a p-group as would be expected. However, it is not necessarily true that D is a Sylow intersection, nor that D = O,(N,(D)). In the first section we define what it means for an algebra to have a G-graded structure. We show how to give such a structure to the algebra M,(k) of r by r matrices over the field k whenever IGl <r. A number of basic results about fully G-graded algebras are then given. The second section deals with R-modules. Proofs are given of the Mackey decomposition formula, the existence of vertices and sources, and the Green correspondence. These three results were already known to hold for R-for example, they are mentioned in [6]-but proofs have not previously been given in the literature. They are proved here to make this paper more or less self-contained. Many of the proofs of supporting results for these theorems will not appear, as they are easy generalizations of the known proofs for R = kG. The reader will often be referred to [ 1 ] for these proofs. The third and last section contains new results. The Burry-Carlson-Puig theorem is generalized, and a new definition of the defect groups of a block

The Jacobson radical of monoid-graded algebras

It is always a pleasant surprise to find that certain well-known results seemingly of a different nature can be obtained as a consequence of a general approach which absorbs and unifies all the existing methods. This right viewpoint is often the main difficultyin any subject. It certainly applies to the topic under consideration here, which is the study of the Jacobson radical of monoid-graded algebras. These algebras include such classicalobjects as group-graded algebras, crossed products, twisted monoid rings, skew monoid rings, polynomial rings, skew polynomial rings, etc. The correct approach which we shall adopt is to consider the graded radical of a module and its important special case, namely, the graded Jacobson radical of a graded algebra. A detailedaccount of all relevant background for group-graded algebras can be found in NSsta'sescu and Van Oystaeyen (1982a). As examples of successful applications of graded radicals we mention the works of NastSsescu (1984), NSst&sescu and Van Oystaeyen (1982b) and Jespers and Puczylowski (1990). The purdose of this paper is to prove a number of general results concerning the Jacobson radical of monoid-graded and group-graded algebras. One of the main theorems provides a large class of groups G for which any G-graded algebra has the property that its Jacobson radical is a graded ideal. We also demonstrate that most of what is known concerning the Jacobson radical of polynomial rings and skew polynomial rings is an easy consequence of our results.

On Extending Projectives of Finite Group-Graded Algebras

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

Graphs with Relations, Coverings and Group-Graded Algebras

The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced.

Graphs with relations, coverings and group-graded algebras

The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced.