Lie groups to Lie algebras

September 4, 2008 at 2:15 pm | Posted in History and News, Math Learning | Leave a comment
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Differentail Geometry, Lie Groups, and Symmetric Spaces by Sigurder Helgason

Differentail Geometry, Lie Groups, and Symmetric Spaces by Sigurdur Helgason

Translating 18.755 Introduction to Lie groups, Fall 2004 remind me the book. It’s I-hsun told me the book and my kindly friend ,Yu-Xuan who studied Computer Science in NCYU now, sent me the sweet gift. He plundered lots of books from the general cleaning of the library of the departmet of mathematics in NTU that year, and I the lucky guy.  We cannot find the book in the bookstore now because of the copyright arguments. The book from 凡異-Press is out of print recently.

Definition.  A Lie group is a group G which is also an analytic manifold such that the mapping (σ,τ) →στ-1 of the product manifold G × G into G is analytic.

Definition. A Lie algebra is an algebra g which operation[,] satisfied [x,x]=0 and Jacobi identity…(please click Bala-bala Lie Algebras).

It always confused me what the relation between Lie groups and Lie algebras. What is the difference between them? What’s the property of them? They must have some relations…what are them? Is Lie groups can be an Lie algebra?

In wikipedia, Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie algebra is said to be an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. But we know that Lie algebra not only can be used in geometic objects, but can be discussed in an independent algebraic, graphing or combination view today.  They contribute to modern physics a lot. Imagine the Lie group just like the rotation system, and Lie algebras are angular momentums. With some differetialble function, we must can know the relation between them.(courtesy of my friend, Meng-Syun)

Let me told a brief history of Lie group of Sophus Lie. Something occurred to him in 1874 that he tried to create a theory for transformation groups which might do for differential equations what Galios theory did for algebraic equations. It implies that Lie group must play an important role to analytic method on algebraic structure. And these works of he also brings us the theory and definition of Lie groups. Back to the history, after he got an idea to take effort on the theory in 1867, he went to Berlin and Paris, Klein and Jordan influence he a lot. They affect his thoughts in a geometric view to build more about Lie theory.  As for Lie algebras, We should say thanks to Jacobi, He generated further solutions of differential equations from a translation group to an “infinitesimal group” which we called it “Lie algebra” today. This is the beginning of Lie algebra. Here be mentioned that Killing took efforts on the association between Lie groups and Lie algebras, and Cartan completed the classification of semi-simple Lie algebras, Lie theory was finally developed on its own. And now, You may used these them all to the final structure of rotation momentums.

If you don’t know how Lie algebras be used…Try back to study Lie group is an good way…Because I just look at the text book but not notice the history of them; just roughly book collection but not suck the essence of them, it confused me that why should learn so many symbols with no meanings. Will I not get lost if I learn these with no ground on?  When back to the history, you will find the natural motivation problem to solved that build the world. Just show my experience.

You may also try to read the book, Anthony W. Knapp. Lie Groups Beyond an Introduction: Beyond an Introduction, 2ndedt. Birkhäuser, 2002, ISBN:0817642595, 9780817642594. (, first if you did not touch the introduction of Lie groups and Lie algebras, You will know what I mean… To know the knowledge structure and the history thoughts are quite important for mathematical learning, or you may get lost in the ocean like me.


  1. Singurdur Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978. recopy by 凡異-Press.
  2. Sophus Lie‘s history from MacTotor History of Mathematics archive.
  3. Lie group and Lie group in Wikipedia.

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