Projective bases of division algebras and groups of central type

Natapov, Michael

doi:10.1007/bf02773539

Cited by 7 publications

(25 citation statements)

References 5 publications

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“…Based on earlier work of the authors [2], we in fact show that Q(U G,c ) is a division algebra if and only if the group G belongs to a certain explicit list Λ of groups. This is independent of the cocycle c, a fact we will return to below.…”

Section: It Is Clear That the Set Of Graded Identities Is An Ideal Ofmentioning

confidence: 75%

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: It Is Clear That the Set Of Graded Identities Is An Ideal Ofmentioning

confidence: 75%

Graded identities of matrix algebras and the universal graded algebra

Aljadeff

Haile

Natapov

2010

Trans. Amer. Math. Soc.

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“…By Proposition 9, B is isomorphic to the relative value group Γ D /Γ F with respect to the valuation v defined in (2)…”

Section: Rigiditymentioning

confidence: 99%

“…Indeed, by [2,Theorem 13], the algebra D 1 is isomorphic to a tensor product of two cyclic algebras, namely…”

Section: Rigiditymentioning

confidence: 99%

“…In [2,Corollary 3] there is a (short) list Λ of p-groups containing all p-groups which possibly are projective bases of division algebras. The list Λ consists of three families of groups G:…”

Section: Introductionmentioning

confidence: 99%

“…The main purpose of this paper is to complete the classification of projective bases of division algebras, begun in [2], by showing that every group on the list Λ is a projective basis for a suitable division algebra over an appropriate field. Thus, combining this with Theorems 1 and 2 we have the following result: Theorem 3.…”

Section: Introductionmentioning

confidence: 99%

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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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On groups of central type, non-degenerate and bijective cohomology classes

David

2009

Isr. J. Math.

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Projective bases of division algebras and groups of central type

Cited by 7 publications

References 5 publications

Graded identities of matrix algebras and the universal graded algebra

Graded identities of matrix algebras and the universal graded algebra

Projective bases of division algebras and groups of central type II

On groups of central type, non-degenerate and bijective cohomology classes

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