On representations of Clifford algebras of ternary cubic forms
type: Item
instance of: scholarly article
zbMATH Open document ID: 1271.14057
author: Emre Coskun
author: Rajesh S. Kulkarni
author: Yusuf Mustopa
published in: New Trends in Noncommutative Algebra
publication date: 2012-07-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.1506
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 6052233
zbMATH Keywords: Generalized Clifford algebra
zbMATH Keywords: Ulrich bundles
MaRDI profile type: Publication
OpenAlex ID: W2101820895
arXiv ID: 1107.1506
DOI: 10.1090/CONM/562/11132
title: On representations of Clifford algebras of ternary cubic forms
review text: Given a homogeneous nondegenerate form \(f\) of degree \(d\geq2\) in \(n\) variables, let \(C_f\) be the generalized Clifford algebra associated to \(f\), which is the natural generalization of the classical Clifford algebra associated a quadratic form. Let \(X_f\) be the smooth hypersurface in \(\mathbb{P}^n\) defined by the equation \(w^d-f=0\).NEWLINENEWLINE\textit{M. Van den Bergh} in [J. Algebra 109, 172--183 (1987; Zbl 0631.13007)] established a one-to-one correspondence between equivalence classes of \(dr\)-dimensional representations of the Clifford algebra \(C_f\) and isomorphism classes of rank \(r\) Ulrich bundles on \(X_f\), which are those vector bundles on \(X_f\) whose direct images under the projection map to \(\mathbb{P}^{n-1}\) are the trivial vector bundles of rank \(dr\).NEWLINENEWLINEIn the paper under review, the authors prove that irreducible representations correspond to stable Ulrich bundles on \(X_f\).NEWLINENEWLINEThey focus then on the case \(n=3\), where \(X_f\) is the cubic surface in \(\mathbb{P}^3\). In this case, using [\textit{M. Casanellas}, \textit{R. Hartshorne}, \textit{F. Geiss} and \textit{F.-O. Schreyer}, Int. J. Math. 23, No. 8, Article ID 1250083, 50 p. (2012; Zbl 1255.14034)], they prove the existence of irreducible representations of \(C_f\) of every possible dimension.NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].
reviewed by: Maria Chiara Brambilla
description: scientific article; zbMATH DE number 6052233