Abstract Lie Algebras by David J Winter

Author:David J Winter
Language: eng
Format: epub
Publisher: Dover Publications
Published: 1972-06-05T16:00:00+00:00

We now strengthen 3.7.3.7.6.

3.7.3.7.10 Proposition

Let S be a standard subset of Σ, and let b ∈ S. Then the total number of lines joining b to adjacent points of S is at most three.

PROOF. Let a1, . . ., an be the distinct roots in S adjacent to b. Since there is no cycle in S, (ai, aj) = 0 for i ≠ j. It follows that the rai commute and that the root

where is the number of lines joining ai and b in DS. But |b*(c)| ≤ 1 since b and c have the same length, by 3.7.3.3.1. Thus, .

In the next two theorems, we follow a convention for describing a vector , with S a standard subset of 2, whereby the coefficients ca are written next to the points of DS. For example

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