Unbounded symmetrysets
Root systems in \({\mathbb{R}}^ n\) are adequate for direct classification of complex semisimple Lie algebras. The classification problem for semisimple Lie algebras over a field of characteristic \(p>0\) is considerably more difficult. For a classical Lie algebra L of characteristic \(p>0\) with a Cartain subalgebra H we have a root system \(R=R(L,H)\subseteq ({\mathbb{Z}}/p)^ n\). It is shown by \textit{D. J. Winter} [Lect. Notes Math. 848, 41-62 (1981; Zbl 0469.17006)] that a certain structure of R enables to obtain a direct classification of classical Lie algebras. Let G be an additive abelian group with no 2-torsion, and R a finite subset in G containing zero. For \(a\in G\), \(b\in R\) denote by \(R_ b(a)\) the set of all elements b-ra, \(b-(r-1)a,...,b+qa\) in R, where r,q are non-negative integers. An orbit \(R_ b(a)\) is bounded if \(b-(r+1)a\), \(b+(q+1)a\not\in R\). In this case \(q+r\) is the length of \(R_ b(a)\). A set R is a symmetry set if for any \(a\in R\setminus 0\) there exists a bijection \(r_ a: R\to R\) with the following properties: (i) for any \(b\in R\) the map \(r_ a\) stabilizes \(R_ b(a);\) (ii) for \(x,y\in R\) a sum \(x+y\in R\) if and only if \(r_ a(x)+r_ a(y)\in R\) and \(r_ a(x)+r_ a(y)=r_ a(x+y).\) Theorem 1. Let R be a symmetry set such that 2a\(\not\in R\) for any \(a\in R\setminus 0\). Then each orbit \(R_ b(a)\) is bounded. - Put \(R^*=Hom(R,{\mathbb{Z}})\) and let \({\hat{\;}}: R\to R^{**}\) be a natural map. A symmetry set R is a Jacobi set if for a,b,c\(\in R\) such that \(a+b\), \(a+b+c\in R\), \(a\neq b\), one of the following conditions holds: (i) \(a+c\in R\), (ii) \(b+c\in R\) provided either \(\hat a+\hat b=0\), \(\hat c=0\) or \(\hat a+\hat b\neq 0\), \(\hat c\neq 0.\) Theorem 2. Let R be a Jacobi set, \(a\in R\) an unbounded element, that is \(R_ b(a)=b+{\mathbb{Z}}a\) for some \(b\in R\). Then \({\mathbb{Z}}a\subseteq R\). If \(c,c+ta\in R\), where \(t\in {\mathbb{Z}}\), then \(b+{\mathbb{Z}}(ta)\subseteq R.\) Theorem 3. Let \(R\subseteq ({\mathbb{Z}}/p)^ n\) be a Jacobi set such that every element in R is unbounded. Then R is an inner direct sum \(R=({\mathbb{Z}}/P)^{n_ 1}\cup... \cup ({\mathbb{Z}}/p)^{n_ k}.\)
scientific article; zbMATH DE number 3929228