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A Basis for Z-Graded Identities of Matrices over Infinite Fields

Azevedo, Sergio (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z ), where α is the degree of the corresponding variable. This is a generalization of a result of Vasilovsky about the Z-graded identities...

A G -minimal model for principal G -bundles

Shrawan Kumar (1982)

Annales de l'institut Fourier

Sullivan associated a uniquely determined D G A | Q to any simply connected simplicial complex E . This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space E . In case E is the total space of a principal G -bundle, ( G is a compact connected Lie-group) we associate a G -equivariant model U G [ E ] , which is a collection of “ G -homotopic” D G A ’s | R with G -action. U G [ E ] will, in general, be different from the Sullivan’s minimal model of the space E . U G [ E ] contains the total rational...

Artin-Schelter regular algebras of dimension five

Gunnar Fløystad, Jon Eivind Vatne (2011)

Banach Center Publications

We show that there are exactly three types of Hilbert series of Artin-Schelter regular algebras of dimension five with two generators. One of these cases (the most extreme) may not be realized by an enveloping algebra of a graded Lie algebra. This is a new phenomenon compared to lower dimensions, where all resolution types may be realized by such enveloping algebras.

Bicrossproduct Hopf quasigroups

Jennifer Klim, Shahn Majid (2010)

Commentationes Mathematicae Universitatis Carolinae

We recall the notion of Hopf quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup k M ▹ ◂ k ( G ) from every group X with a finite subgroup G ⊂ X and IP quasigroup transversal M ⊂ X subject to certain conditions. We identify the octonions quasigroup G 𝕆 as transversal in an order 128 group X with subgroup ℤ 2 3 and hence obtain a Hopf quasigroup k G 𝕆 > ◂ k ( ℤ 2 3 ) as a particular case of our construction.

Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes

Frédéric Chapoton (2000)

Annales de l'institut Fourier

On définit une structure de bigèbre différentielle graduée sur la somme directe des complexes cellulaires des permutoèdres, qui contient une sous-bigèbre différentielle graduée dont le complexe sous-jacent est la somme directe des complexes cellulaires des polytopes de Stasheff. Ceci étend des constructions de Malvenuto et Reutenauer et de Loday et Ronco pour les sommets des mêmes polytopes.

Categorical methods in graded ring theory.

Angel del Río (1992)

Publicacions Matemàtiques

Let G be a group, R a G-graded ring and X a right G-set. We study functors between categories of modules graded by G-sets, continuing the work of [M]. As an application we obtain generalizations of Cohen-Montgomery Duality Theorems by categorical methods. Then we study when some functors introduced in [M] (which generalize some functors ocurring in [D1], [D2] and [NRV]) are separable. Finally we obtain an application to the study of the weak dimension of a group graded ring.

Certain partitions on a set and their applications to different classes of graded algebras

Antonio J. Calderón Martín, Boubacar Dieme (2021)

Communications in Mathematics

Let ( 𝔄 , ϵ u ) and ( 𝔅 , ϵ b ) be two pointed sets. Given a family of three maps ℱ = { f 1 : 𝔄 → 𝔄 ; f 2 : 𝔄 × 𝔄 → 𝔄 ; f 3 : 𝔄 × 𝔄 → 𝔅 } , this family provides an adequate decomposition of 𝔄 ∖ { ϵ u } as the orthogonal disjoint union of well-described ℱ -invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H * -algebras.

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