On Lie gradings. I
Although gradings have been a part of the theory of Lie algebras virtually from the beginning of the theory, a general systematic study of gradings has not been undertaken. This paper, and another promised, present such a study. A Lie grading is defined as a decomposition \(\gamma: L=\oplus_{g\in G}L(g)\) of a Lie ring into the direct sum of non-zero submodules \(L(g)\) of \(L\). \(G\) is an index set and one demands that either \(L(g)L(g')=0\) or \(L(g)L(g')\subseteq L(g'')\) for some \(g''\in G\). There corresponds to \(\gamma\) an abelian semigroup \(\gamma\) generated by the components. Refinements of gradings are studied with an eye toward gradings which cannot be properly refined. These are called fine gradings, an example of which is the Cartan decomposition of \(L\) over \(\mathbb C\). Automorphisms and derivations and their behavior with respect to gradings is studied. If \(L\) is finite dimensional over \(\mathbb C\), then \(T\) is fine precisely if \(\text{Diag}(\gamma)\) is maximal diagonal in \(\Aut(L)\) where \(\text{Diag}(\gamma)\) is determined by the non-zero characters of \(L\). Compatibility of \(\gamma\) and representation space gradings is considered. The authors show that if \(L\) is a fully reducible linear Lie algebra of finite dimension and over a field \(F\) of characteristic 0, then every subgroup of the normalizer group of \(L\) in \(\text{GL}(d,f)\) with diagonal transformation action on \(L\) normalizes a Cartan subalgebra of \(L\). Finally the fine gradings of \(\mathfrak{sl}(3,\mathbb C)\) are described.
scientific article; zbMATH DE number 4105167
scientific article