On fine gradings on central simple algebras

Aljadeff, Eli; Haile, Darrell; Natapov, Michael

doi:10.1201/9781420010961.ch1

Cited by 5 publications

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“…Leaving the other generators unchanged we obtain an equivalent set of generators Φ (3) such that the elements Now, let c ∈ Z 2 (G, C × ) be a nondegenerate two-cocycle and write C c G = g∈G Cu g . As in the previous steps, put σ, τ, γ 1 , . .…”

Section: (Iii) Next We Consider the Groupmentioning

confidence: 99%

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Graded identities of matrix algebras and the universal graded algebra

Aljadeff

Haile

Natapov

2010

Trans. Amer. Math. Soc.

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Abstract. We consider fine group gradings on the algebra M n (C) of n by n matrices over the complex numbers and the corresponding graded polynomial identities. Given a group G and a fine G-grading on M n (C), we show that the T -ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra U (depending on the group G and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of U is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M n (C). Finally, we consider the ring of central quotients of U which is a central simple algebra over the field of quotients of the center of U . Using earlier results of the authors we show that this is a division algebra if and only if the group G is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example.

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Section: (Iii) Next We Consider the Groupmentioning

confidence: 99%

“…It follows that the corresponding subalgebra F (t σ u σ , t y u y ) of the universal central simple algebra Q(U G,c ) = F s(c) G is a symbol algebra (a, b) 3 , and by Aljadeff, et al [3,Lemma 6], a and b are roots of unity. It follows that F (t σ u σ , t y u y ) is split, that is, isomorphic to M 3 (F ).…”

Section: (Iii) Next We Consider the Groupmentioning

confidence: 99%

Graded identities of matrix algebras and the universal graded algebra

Aljadeff

Haile

Natapov

2010

Trans. Amer. Math. Soc.

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“…Hence, by an index argument we have that (a, b) p s ⊗  k=1 (a 2k−1 , a 2k ) p f k is a division algebra, and, furthermore, it is isomorphic to the algebra obtained in (3). Then, applying Proposition 7, we get…”

Section: Rigiditymentioning

confidence: 86%

Projective bases of division algebras and groups of central type II

Natapov

2008

Isr. J. Math.

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Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k α G for some α ∈ H 2 (G, k × ), where the action of G on k × is trivial. In a preceding paper by Aljadeff, Haile and the author it was shown that if a group G is a projective basis in a k-central division algebra then G is nilpotent and every Sylow p-subgroup of G is on the short list of p-groups, denoted by Λ. In this paper we complete the classification of projective bases of division algebras by showing that every group on that list is a projective basis for a suitable division algebra.We also consider the question of uniqueness of a projective basis of a k-central division algebra. We show that basically all groups on the list Λ but one satisfy certain rigidity property.

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“…Simply-graded algebras are vastly investigated, e.g. see [1][2][3][4][5][6][7][8]. Let (1) be a Gsimple grading.…”

Section: Gradingsmentioning

confidence: 99%