Projective bases of division algebras and groups of central type II

Natapov, Michael

doi:10.1007/s11856-008-0020-7

Cited by 2 publications

(7 citation statements)

References 16 publications

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“…The equivalence of (b) and (c) follows from the previous proposition. Finally the equivalence of (c) and (d) follows from Aljadeff et al [2,Corollary 3] and Natapov [17,Theorem 3].…”

Section: Proposition 17 Let G Be a Group Of Central Type Of Order N mentioning

confidence: 76%

“…As in the previous case, a construction of a K-central division algebra of the form K β G 1 , analogous to that in Natapov [17], yields a 4 × 4 matrix algebra isomorphic to…”

Section: ) 2kmentioning

confidence: 87%

“…, which is analogous to that in Natapov [17]. Let K = Q(s, t) be the subfield of C generated by algebraically independent elements s and t. Let L = K(v π )/K be a cyclotomic extension, where v It is easy to see that D is isomorphic to a twisted group algebra of the form K β G 1 .…”

Section: (Iii) Next We Consider the Groupmentioning

confidence: 88%

“…The ring R s (r 1 ) G is an Azumaya algebra over the discrete valuation ring R (r 1 ) and hence is isomorphic to M t (A), where A is an Azumaya algebra such that A ⊗ F = D (see Reiner [18,Theorem 21.6 The question for which groups G of central type there is a cocycle c such that Q(U G,c ) is a division algebra was answered in Aljadeff, et al [2] and Natapov [17]. We consider the following list of p-groups, called Λ p :…”

Section: Proposition 16 the Index Of Q(u G ) Is Equal To The Maximummentioning

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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“…The equivalence of (b) and (c) follows from the previous proposition. Finally the equivalence of (c) and (d) follows from Aljadeff et al [2,Corollary 3] and Natapov [17,Theorem 3].…”

Section: Proposition 17 Let G Be a Group Of Central Type Of Order N mentioning

confidence: 76%

“…As in the previous case, a construction of a K-central division algebra of the form K β G 1 , analogous to that in Natapov [17], yields a 4 × 4 matrix algebra isomorphic to…”

Section: ) 2kmentioning

confidence: 87%

Section: (Iii) Next We Consider the Groupmentioning

confidence: 88%

Section: Proposition 16 the Index Of Q(u G ) Is Equal To The Maximummentioning

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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“…The question for which groups G of central type there is a cocycle c such that Q(U G,c ) is a division algebra was answered in Aljadeff, et al [2] and Natapov [15]: We consider the following list of p-groups, called Λ p :…”

Section: Proposition 7 (A)mentioning

confidence: 99%

Graded identities of matrix algebras and the universal graded algebra

Aljadeff,

Haile,

Natapov

2007

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In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of M n (C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on M n (C) and their corresponding graded identities.Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼ = ⊕ g∈G R g is a grading and dim k (R g ) ≤ 1. Thus any component is either 0 or isomorphic to k as a k-vector space. It is easy to show that Supp(R), the subset of elements of G for which R g is not 0, is a subgroup of G. Moreover

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Projective bases of division algebras and groups of central type II

Cited by 2 publications

References 16 publications

Graded identities of matrix algebras and the universal graded algebra

Graded identities of matrix algebras and the universal graded algebra

Graded identities of matrix algebras and the universal graded algebra

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