On groupoid gradings
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.
scientific article; zbMATH DE number 6820383
scientific article