The Table of the Structure Constants for the Complex Simple Lie Algebra of Type F_4 and its Application to the Calculation of Commutators in the Chevalley Group of Type F_4 over Fields and Rings

Abstract

This work is the first in a series of papers devoted to constructing tables of structure constants for the complex simple Lie algebras and to finding an explicit form of Chevalley commutator formulas. The work consists of three parts. In the first part, expressions are found for the structure constants of the complex simple Lie algebra of type F_4 in the form of functions of structure constants corresponding to extraspecial pairs of roots. As a consequence, all Chevalley commutator formulas [x_r(u),x_s(y)] are calculated when the sum r+s is a root. Further, in the second part, tables of structure constants and Chevalley commutator formulas are given in the special case when all constants corresponding to extraspecial pairs are equal to one. Finally, in the third part, directed and weighted graphs associated with root systems are constructed. It is shown that the elements of the exponent of the adjacency matrices of directed graphs are the numbers P_{rs}, where P_{rs} is the number of representations of the root r in the form of a sum of the root s and fundamental roots such that any initial segment of the sum is a root. It is also shown that the elements of an exponent of the weight matrix of a weighted graph are the values of sums arising when calculating complex commutators in Chevalley groups.