On the Hopf-Schur group of a field

Eli Aljadeff; Juan Cuadra; Shlomo Gelaki; Ehud Meir

doi:10.1016/j.jalgebra.2008.01.029

On the Hopf-Schur group of a field

Eli Aljadeff, Juan Cuadra, Shlomo Gelaki^*, Ehud Meir

^*Corresponding author for this work

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

1 Scopus citations

Abstract

Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.

Original language	English
Pages (from-to)	5165-5177
Number of pages	13
Journal	Journal of Algebra
Volume	319
Issue number	12
DOIs	https://doi.org/10.1016/j.jalgebra.2008.01.029
State	Published - 15 Jun 2008
Externally published	Yes

Keywords

Brauer group
Hopf algebras
Hopf-Schur group
Projective group

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10.1016/j.jalgebra.2008.01.029

Cite this

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title = "On the Hopf-Schur group of a field",

abstract = "Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.",

keywords = "Brauer group, Hopf algebras, Hopf-Schur group, Projective group",

author = "Eli Aljadeff and Juan Cuadra and Shlomo Gelaki and Ehud Meir",

note = "Funding Information: The second named author{\textquoteright}s research was supported by projects MTM2005-03227 from MEC and FEDER and P06-FQM-01889 from Junta de Andaluc{\'i}a. The third named author{\textquoteright}s research was supported by the Israel Science Foundation (grant No. 125/05).",

year = "2008",

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doi = "10.1016/j.jalgebra.2008.01.029",

language = "英语",

volume = "319",

pages = "5165--5177",

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TY - JOUR

T1 - On the Hopf-Schur group of a field

AU - Aljadeff, Eli

AU - Cuadra, Juan

AU - Gelaki, Shlomo

AU - Meir, Ehud

N1 - Funding Information: The second named author’s research was supported by projects MTM2005-03227 from MEC and FEDER and P06-FQM-01889 from Junta de Andalucía. The third named author’s research was supported by the Israel Science Foundation (grant No. 125/05).

PY - 2008/6/15

Y1 - 2008/6/15

N2 - Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.

AB - Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.

KW - Brauer group

KW - Hopf algebras

KW - Hopf-Schur group

KW - Projective group

UR - http://www.scopus.com/inward/record.url?scp=43449107946&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2008.01.029

DO - 10.1016/j.jalgebra.2008.01.029

M3 - 文章

AN - SCOPUS:43449107946

SN - 0021-8693

VL - 319

SP - 5165

EP - 5177

JO - Journal of Algebra

JF - Journal of Algebra

IS - 12

ER -

On the Hopf-Schur group of a field

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