Lattices of covariant quadratic forms
type: Item
instance of: scholarly article
title: Lattices of covariant quadratic forms
zbMATH Open document ID: 1081.11031
author: Wilhelm Plesken
published in: L'Enseignement Mathématique. 2e Série
publication date: 2005-04-04T00:00:00Z
review text: The author studies the \({\mathbb Z}\)-lattice \(\text{ Bil}_\Lambda(L)\) of all bilinear forms on a lattice \(L\) that are covariant with respect to a positive involution on the order \(\Lambda \) of the lattice. He studies the discriminant of \(\text{ Bil}_\Lambda(L)\), the autoequivalence group \(\text{ Aut}^e(\text{ Bil}_\Lambda(L))\), its subgroup fixing the center of \(\text{ End}_\Lambda(L)\), the group of inner automorphisms, and the outer group of equivalences. The notions of e-depth and e-\(*\)-depth for \(\text{ Bil}_\Lambda(L)\) are defined. They measure how far \(\text{ End}_\Lambda(L)\) and \(\text{ End}_\Lambda(L\oplus L^*)\) are away from being hereditary. The \(*\)-depth zero situations are often classifiable. In the depth zero situation structural results on the outer group of autoequivalences are given. Some examples are studied in detail, e. g. if \(\text{ End}_\Lambda(L)\) is a \(\mathbb Z\)-order in the algebra \({\mathbb Q}^{2\times 2}\).
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 2151826
zbMATH Keywords: lattices
zbMATH Keywords: covariant quadratic forms
zbMATH Keywords: orders
zbMATH Keywords: endomorphism rings
zbMATH Keywords: autoequivalence group
reviewed by: Kanat Abdukhalikov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2151826