Complex simple Lie algebras classification

[Reperio]

Does anybody can propose a good book on classification of the complex simple Lie algebras. I know that they fall into several simple (An, Bn, Cn, Dn) and exceptional groups (E, F, G), but I find only a pieces of information about these algebras.
Also, they could be represented by Coxeter–Dynkin diagram. In simples words, what are they and how to get an intuition to read them?

Good article or book about these diagrams and groups will be hellpfull...

Thank you

 

[Anonym]

Good article or book about these diagrams and groups will be hellpfull...

R.Gilmore, “Lie Groups,Lie Algebras,and Some of their Applications”, John Wiley&Sons, 1974

Regards, Dany.

 

[denian]

for your question. The classification of complex simple Lie algebras is a highly studied and important topic in mathematics and physics. These algebras are fundamental objects in the theory of Lie groups and have many applications in areas such as differential geometry, representation theory, and theoretical physics.

To answer your first question, there are several good books on the classification of complex simple Lie algebras. Some popular references include "Lie Algebras and their Representations" by J. P. Serre, "Introduction to Lie Algebras and Representation Theory" by James E. Humphreys, and "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" by Brian C. Hall. These books provide comprehensive and accessible treatments of the subject, with varying levels of mathematical rigor.

In simple terms, a complex simple Lie algebra is a type of algebraic structure that encodes the symmetries of a Lie group. These algebras are classified into different types based on their structure, which can be represented by Coxeter-Dynkin diagrams. These diagrams are graphical representations of the root system of a Lie algebra, which is a set of vectors that describe the symmetries of the algebra. The different types of simple Lie algebras (An, Bn, Cn, Dn, etc.) correspond to different patterns in the Coxeter-Dynkin diagrams.

To gain an intuition for reading these diagrams, it is helpful to understand their basic structure. The diagrams consist of nodes and lines connecting them, with the nodes representing the simple roots (basis elements) of the Lie algebra. The lines indicate the relationships between these roots, with certain patterns indicating specific types of symmetries. For example, a line with a single arrowhead represents a positive root, while a line with two arrowheads represents a negative root. By studying the patterns in the diagrams, one can gain a better understanding of the types of symmetries encoded by the Lie algebra.

I hope this explanation has provided some clarity on the classification of complex simple Lie algebras and the use of Coxeter-Dynkin diagrams. I highly recommend exploring the resources mentioned above for a more in-depth understanding of this fascinating topic.