Division algebras with a projective basis

Eli Aljadeff; Darrell Haile

doi:10.1007/BF02802503

Division algebras with a projective basis

Eli Aljadeff^*, Darrell Haile

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let k be any field and G a finite group. Given a cohomology class α ∈ H²(G, k*), where G acts trivially on k*, one constructs the twisted group algebra k^αG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Z_n × Z_n). This paper has two main results: First we prove that if D = k^αG is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G′ , the commutator subgroup of G, is cyclic. Next we show that unless char(k) = 0 and √-1 ∉ k, the division algebra D = k^αG is a product of cyclic algebras. Furthermore, if D_p is a p-primary factor of D, then D_p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and √-1 ∉ k, the same result holds for D_p, p odd. If p = 2 we show that D₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z₂ × Z_2n.

Original language	English
Pages (from-to)	173-198
Number of pages	26
Journal	Israel Journal of Mathematics
Volume	121
DOIs	https://doi.org/10.1007/BF02802503
State	Published - 2001
Externally published	Yes

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title = "Division algebras with a projective basis",

abstract = "Let k be any field and G a finite group. Given a cohomology class α ∈ H2(G, k*), where G acts trivially on k*, one constructs the twisted group algebra kαG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Zn × Zn). This paper has two main results: First we prove that if D = kαG is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G′ , the commutator subgroup of G, is cyclic. Next we show that unless char(k) = 0 and √-1 ∉ k, the division algebra D = kαG is a product of cyclic algebras. Furthermore, if Dp is a p-primary factor of D, then Dp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and √-1 ∉ k, the same result holds for Dp, p odd. If p = 2 we show that D2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z2 × Z2n.",

author = "Eli Aljadeff and Darrell Haile",

year = "2001",

doi = "10.1007/BF02802503",

language = "英语",

volume = "121",

pages = "173--198",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

}

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T1 - Division algebras with a projective basis

AU - Aljadeff, Eli

AU - Haile, Darrell

PY - 2001

Y1 - 2001

N2 - Let k be any field and G a finite group. Given a cohomology class α ∈ H2(G, k*), where G acts trivially on k*, one constructs the twisted group algebra kαG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Zn × Zn). This paper has two main results: First we prove that if D = kαG is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G′ , the commutator subgroup of G, is cyclic. Next we show that unless char(k) = 0 and √-1 ∉ k, the division algebra D = kαG is a product of cyclic algebras. Furthermore, if Dp is a p-primary factor of D, then Dp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and √-1 ∉ k, the same result holds for Dp, p odd. If p = 2 we show that D2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z2 × Z2n.

AB - Let k be any field and G a finite group. Given a cohomology class α ∈ H2(G, k*), where G acts trivially on k*, one constructs the twisted group algebra kαG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Zn × Zn). This paper has two main results: First we prove that if D = kαG is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G′ , the commutator subgroup of G, is cyclic. Next we show that unless char(k) = 0 and √-1 ∉ k, the division algebra D = kαG is a product of cyclic algebras. Furthermore, if Dp is a p-primary factor of D, then Dp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and √-1 ∉ k, the same result holds for Dp, p odd. If p = 2 we show that D2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z2 × Z2n.

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U2 - 10.1007/BF02802503

DO - 10.1007/BF02802503

M3 - 文章

AN - SCOPUS:0035655638

SN - 0021-2172

VL - 121

SP - 173

EP - 198

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -

Division algebras with a projective basis

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