Understanding E8 Root Systems: A Comprehensive Guide to Prime Roots and Patterns

Thread starter JeremyEbert
Start date Feb 1, 2011
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Root System

In summary, the conversation revolves around understanding root systems and the suggestion of books for further knowledge. The discussion also includes the presentation of an E8 root system and a walkthrough of a smaller example (G2) to explain the concept. However, it is noted that working out E8 may take a considerable amount of time.

Feb 1, 2011

#1

JeremyEbert

I'm trying to understand root systems better. They seem to be tied to what I've been working on. Any suggestions on books?

here is an e8 root system

http://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/E8_graph.svg/500px-E8_graph.svg.png

here is some of my root system. the prime roots in mine form essentially the same patterns that are showing up in root systems. Zoom in if needed.

http://2.bp.blogspot.com/_u6-6d4_gs...AAAE8/_hov_b0sno4/s1600/prime-+square+12a.png
also a pdf
https://www.physicsforums.com/attachment.php?attachmentid=31636&d=1296068205

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Jul 13, 2019

#2

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E8 is quite big to start with 8 basis roots and 120 positive roots.
A walkthrough with a small example (G2) is in this series:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/It explains how it is done, but to work out E8 will take you quite some time.

1. What is the E8 root system?

The E8 root system is a mathematical construct used in the study of Lie algebras, which are abstract structures that represent symmetries in mathematics. The E8 root system is particularly interesting because it has the highest possible dimension, consisting of 240 roots in 8-dimensional space.

2. How is the E8 root system used in science?

The E8 root system has various applications in physics, particularly in the study of particle physics and string theory. It is also used in the classification of crystals and in the study of finite groups. Additionally, the E8 root system has connections to other areas of mathematics, such as combinatorics and geometry.

3. What is the significance of the E8 root system?

The E8 root system is significant because it is a fundamental part of the larger classification of root systems, which has implications in various areas of mathematics and science. It also has connections to other important mathematical structures, such as the exceptional Lie group E8 and the Cartan matrix.

4. How is the E8 root system related to other root systems?

The E8 root system is part of the larger classification of root systems, which includes the A, B, C, D, and F root systems. These root systems have different properties and are used in different contexts, but they all share similar algebraic structures and symmetries.

5. Can the E8 root system be visualized?

While the E8 root system cannot be visualized in its entirety due to its high dimensionality, there are various visual representations of it that can aid in understanding its structure. This includes the Dynkin diagram, which is a graphical representation of the E8 root system, and virtual models that can be manipulated and rotated to explore its properties.

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