Editing Root system (section)

{{Short description|Geometric arrangements of points, foundational to Lie theory}}
{{About|root systems in mathematics|[[plant]] root systems|Root}}
{{Lie groups |Semi-simple}}

In [[mathematics]], a '''root system''' is a configuration of [[vector space|vector]]s in a [[Euclidean space]] satisfying certain geometrical properties. The concept is fundamental in the theory of [[Lie group]]s and [[Lie algebra]]s, especially the classification and representation theory of [[semisimple Lie algebra]]s. Since Lie groups (and some analogues such as [[algebraic group]]s) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.  Further, the classification scheme for root systems, by [[Dynkin diagram]]s, occurs in parts of mathematics with no overt connection to Lie theory (such as [[singularity theory]]).  Finally, root systems are important for their own sake, as in [[spectral graph theory]].<ref>{{cite journal | doi=10.1016/S0024-3795(02)00377-4 | volume=356 | issue=1–3 | title=Graphs with least eigenvalue −2; a historical survey and recent developments in maximal exceptional graphs | journal=Linear Algebra and Its Applications | pages=189–210| year=2002 | last1=Cvetković | first1=Dragoš | doi-access=free }}</ref>