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. (http://books.google.com.tw/books?id=U573NrppkA8C&printsec=frontcover), 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.

Reference:

  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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Bala-bala Lie algebras

August 16, 2008 at 1:40 am | Posted in Math Learning | 1 Comment
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The cover of "Introduction to Lie algebras and Representation Theory", translated by Jui-Ji Chang

Introduction to Lie algebras and Representation Theory, translated by Jui-Ji Chang

Gee was so happy to met Jyun-Ao , a student of  Shun-Jen, remind me to pick my memory about Lie algebras…  It had been several years since I was struggle for knowing what’s the key math touch my mind. Lie algebras was really a unexpected beauty for me. The history of me to touch Lie algebras maybe an accident(or not)…

  • Beginning, Wen Long Lin who taught Applied mathematics in department of physics shows me two books of Lie algebras in 2001. Which are:(A)Howard Gergi. Lie Algeras in Partical Physics(1999). ISBN:0-738-20233-9. (B)Robert . Chahn. Semi-Simle Lie Algebras & their Representations.BenJamin/cummings(1984). ISBN:0-805-31600-0. …He said “Department of Mathematics may be better than Department of Physics if you want to learn these.”… Did not agree with him that time because I never saw or learned Lie algebras in Department of Mathematics till met him.
  • Found what “root system”is of the paper on the webpage I saw in 2004 : Valentina Golubeva:Hamiltonians of the Calogeto-Sutherland Type Models Associated to the Root Systems and Corresponding Fock Spaces….For understanding this paper and others I did not mention here, I was been kicked out the door of Roger. It became a penal pain pricked my heart that hard to be cured in my memory…
  • Audit a course of Chia-Hsin Liu to know root systems, He used the Humphreys’ book… There is no feelings happened in my mind in his class, thus gave the course up immediately…Audit his class just one time…For knowing that course did open for those  interested in ideal and rings.
  • The story after baby lie groups(class of Chun Chung in 2004) and lie group(class of Ong, Ping-Zen) attract me, especially for the class of Chun Chung, Lie algebras just like the drum in heaven that drumed me…
  • Met I-Hsun in 2004. Wanted to go to his Differential Geometry course to listen more things about Lie algebras but fixed in the Lie algebras course of Shun-Jen for one year.
  • ??—to present.

m…m… let me talk about translation works of mathematics.  After doing “My oops translate“, I felt a little better. But offer such volunteer effort really not a work or plan that should be do. Anyway I get something I like to do from doing that… It’s a long time not touch math, this way can cure my life a moment…Good translation is not a easy work, especially for mathematics. Take my translating “18.238 Geometry and Quantum Field Theory, Fall 2002” for example, I never know what I type…Just a machine kala~ka-la.  Keep the questions in your stomach or you cannot finish the document.

Borrowed the book,「張瑞吉 譯。李代數與表現理論之導引。國立編譯館主編。台北:黎明文化(民70)。」which is translated from the famous book: “Jumes E. Humphreys. Introduction to Lie Algebras and Representation Theory, GTM9. Springer-Verlag New York,1972.” ,from Taitung County library.  Prof. Rui-Ji Chang translate the book very well. Didn’t know if the teachers in the university before like translate. Gee regard them are all tips on tops during those days. They wrote, edited and translated excellent books by themselves…seems a little different than present day. Anyway,Each period must exist its great events.Back to translate…

Because of the translation mistake always happened for general translation. Gee still like read the original version than the second. Even for this good book translated by Prof. Chang…

Example.(p.1)

J. E. Humphreys wrote…

Definition. A vector space L over a field F, with an operation L × L→L, dented (x,y) →[xy]and called the bracket or commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied:

(L1) The bracket operation is bilinear.
(L2) [xx]=0 for all x in L
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

Rui-Ji Chang translated

定義 設 L 為體 F上的向量空間,而 L × L→L為L中的一個運算,記為 (x,y) →[xy]稱為x與y的括弧或換位元素。再設這個括弧運算滿足下列公理:

(L1) 括弧運算為雙線性。
(L2)[xx]=0 對所有 x ∈ L。
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

則稱L為F上的李代數。

You catch the points here? That’s why I like to read the original book than the translated one, even though the Chinese words are so cute for me. Each translated book like this must have the tiny mistakes like these~ They will confuse me a lot to read the book (From these, you may understand that “to read a book well” is such a difficult hobby for me><).

For being an oops mathematic courses translator…Keep learning is an important things. Even though the problems also a lot, and lots problems like the upper statement…

“Just Go!” give myself the words!! I will be a good translator like Rui-Ji.

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