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
SN - 0022-4049
VL - 220
SP - 2077
EP - 2111
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 6
ER -
