Descent, fields of invariants, and generic forms via symmetric monoidal categories

Research output: Contribution to journalArticle

1 Citation (Scopus)
3 Downloads (Pure)

Abstract

Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.
Original languageEnglish
Pages (from-to)2077-2111
Number of pages35
JournalJournal of Pure and Applied Algebra
Volume220
Issue number6
Early online date10 Nov 2015
DOIs
Publication statusPublished - Jun 2016

Fingerprint

Monoidal Category
Descent
Invariant
Comodule
Subfield
Algebra
Central Simple Algebra
Associative Algebra
Commutative Algebra
Finite Dimensional Algebra
Cocycle
Specialization
Algebraic Structure
Hopf Algebra
Classify
Form
Generalise
Zero

Cite this

Descent, fields of invariants, and generic forms via symmetric monoidal categories. / Meir, Ehud.

In: Journal of Pure and Applied Algebra, Vol. 220, No. 6, 06.2016, p. 2077-2111.

Research output: Contribution to journalArticle

@article{e8a8f65530b54cc1a223eb4612092f25,
title = "Descent, fields of invariants, and generic forms via symmetric monoidal categories",
abstract = "Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.",
author = "Ehud Meir",
note = "The author was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. The author is grateful to Eli Aljadeff, Apostolos Beligiannis, Julien Bichon, Pavel Etingof, Gaston Garcia, Christian Kassel, Bernhard Keller, Henning Krause, Akira Masuoka and Jan Stovicek for guidance and for fruitful discussions.",
year = "2016",
month = "6",
doi = "10.1016/j.jpaa.2015.10.016",
language = "English",
volume = "220",
pages = "2077--2111",
journal = "Journal of Pure and Applied Algebra",
issn = "0022-4049",
publisher = "Elsevier",
number = "6",

}

TY - JOUR

T1 - Descent, fields of invariants, and generic forms via symmetric monoidal categories

AU - Meir, Ehud

N1 - The author was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. The author is grateful to Eli Aljadeff, Apostolos Beligiannis, Julien Bichon, Pavel Etingof, Gaston Garcia, Christian Kassel, Bernhard Keller, Henning Krause, Akira Masuoka and Jan Stovicek for guidance and for fruitful discussions.

PY - 2016/6

Y1 - 2016/6

N2 - Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.

AB - Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.

U2 - 10.1016/j.jpaa.2015.10.016

DO - 10.1016/j.jpaa.2015.10.016

M3 - Article

VL - 220

SP - 2077

EP - 2111

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 6

ER -