Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

• 来源期刊: 代数集刊（英文版）
• 关键词: multiplicity-free grading symplectic abelian group skew root system simple Lie algebra simple Jordan algebra
• 年，卷（期）: 2019,（1）
• 页码范围: 123-138
• 页数: 16页
• 语种: 英文
• DOI: 10.1142/S1005386719000129

摘要

A G-grading on an algebra,where G is an abelian group,is called multiplicity-free if each homogeneous component of the grading is 1-dimensional.We introduce skew root systems of Lie type and skew root systems of Jordan type,and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively.Under certain conditions the corresponding Lie (resp.,Jordan) algebras are simple.Two families of skew root systems of Lie type (resp.,of Jordan type) are constructed and the corresponding Lie (resp.,Jordan) algebras are identified.This is a new approach to study abelian group gradings on Lie and Jordan algebras.