Visualizing Lie Subalgebras using Root and Weight Diagrams

Aaron Wangberg

Department of Mathematics Winona State University University Winona, MN 55987

awangberg@winona.edu

Tevian Dray

Department of Mathematics Oregon State University Corvallis, OR 97331

tevian@math.oregonstate.edu

Abstract

While Dynkin diagrams are useful for classifying Lie algebras, it is the root
and weight diagrams that are most often used in applications, such as when
describing the properties of fundamental particles. This paper illustrates
how to construct root and weight diagrams from Dynkin diagrams, and how the
root and weight diagrams can be used to identify subalgebras. In particular,
we show how this can be done for some algebras whose root and weight diagrams
have dimension greater than $3$, including the exceptional Lie algebras $F_4$
and $E_6$.