On groupoid gradings
type: Item
instance of: scholarly article
title: On groupoid gradings
zbMATH Open document ID: 1406.16040
author: Candido Martín González
author: Daouda Ndoye
author: Antonio J. Calderón Martín
published in: Journal of Geometry and Physics
publication date: 2017-12-20T00:00:00Z
review text: The theory of group-graded algebras has a long tradition (see, for instance, the book by \textit{C. Nastasescu} and \textit{F. van Oystaeyen} [Graded and filtered rings and modules. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0418.16001)]), but the paper under review proposes to study gradings by groupoids and more generally, gradings by sets, less known than gradings by groups. Recall that a \textit{groupoid} is a small category in which every morphism is an isomorphism. Roughly speaking, it is a set with a partially defined multiplication satisfying group-like properties. The study is supported by a collection of examples of groupoid gradings which are not group gradings, on matrix algebras as well as on some Lie algebras. For these examples, groupoids are given by directed graphs (the elements in the groupoid are the paths in the graph). Other examples of groupoid gradings are proved to be the orthogonal fine gradings on complex matrix algebras. In particular, one of the points the authors are interested in, is the conjecture in Lie theory about whether every set grading on a finite-dimensional simple Lie algebra is in fact a group grading. They do not give a definitive answer to such question, but their partial result in a more general setting is that any groupoid grading on a simple commutative or anti-commutative algebra is necessarily a group grading. Some work is also done about set gradings, for instance proving that they admit coarsenings which are groupoid gradings, whose homogeneous components are ideals of the algebra.
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 6820383
zbMATH Keywords: groupoid gradings
zbMATH Keywords: set gradings
zbMATH Keywords: orthogonal gradings
MaRDI profile type: Publication
full work available at URL: j.geomphys.2017.08.015
OpenAlex ID: W2751718363
cites work: Gradings on simple Jordan and Lie algebras
cites work: Group Gradings onG2
cites work: ON THE STRUCTURE OF GRADED LIE SUPERALGEBRAS
cites work: Models of the Lie algebra \(F_4\)
cites work: Gradings on \(\mathfrak {g}_2\)
cites work: Gradings on the Albert algebra and on \(\mathfrak{f}_4\)
cites work: Fine gradings on simple classical Lie algebras
cites work: On Lie gradings. II
cites work: On Lie gradings. III: Gradings of the real forms of classical Lie algebras
cites work: Gradings on finite-dimensional simple Lie algebras
cites work: On Lie gradings. I
cites work: PHANTOM ENERGY FROM GRADED ALGEBRAS
cites work: NONCOMMUTATIVE GEOMETRY AND GRADED ALGEBRAS IN ELECTROWEAK INTERACTIONS
cites work: Graded differential geometry of graded matrix algebras
cites work: On maximally graded algebras and Walsh functions
cites work: Graded algebras and geometry based on Yang–Baxter operators
cites work: Lagrangian formalism over graded algebras
cites work: Classification of Lie algebras with naturally graded quasi-filiform nilradicals
cites work: From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets
cites work: Infinitesimal deformations of naturally graded filiform Leibniz algebras
cites work: Gradings on simple Lie algebras
cites work: From Groups to Groupoids: a Brief Survey
cites work: On graded associative algebras.
DOI: 10.1016/J.GEOMPHYS.2017.08.015
description: scientific article; zbMATH DE number 6820383