Enveloping algebra

November 11, 2013 at 12:48 pm | Posted in Uncategorized | Leave a comment
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Envelopes of Lines and Circles

Envelopes of Lines and Circles by J. W. Wilson

Picture  from: http://jwilson.coe.uga.edu/Texts.Folder/Envel/envelopes.html

  Being a quiet girl for such a long period of time, many people always encourage me to ask questions, to talk more and more if I can. But I still like writing. Just like the cute one who practiced drawing or asked the parallel in learning magazine before.

Now, my learning problems are still growing of some unknown rate function. After typing some notes today which about the universal enveloping algebra. Some feelings struck me that it really use lots of universal properties, but how about the properties of enveloping?

Look up the website right away. There is an envelope theorem in wikipedia state as: A curve in a two-dimensional space is best represented by the parametric equations like x(c) and y(c). The family of curves can be represented in the form g(x, y, c) = 0 where c is the parameter. Generally, the envelope theorem involves one parameter but there can be more than one para meter involved as well.

In what kind of situation that an universal (enveloping) algebra satisfy the envelope theorem? It seems needs to find the maximization after finding the object function. It’s quite different from the universal property.

Is there any kindly person want to help me to understand that?
Gee know that there must be here…

 

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2013Taipei Conference on Representation Theory IV

October 19, 2013 at 7:27 am | Posted in Uncategorized | Leave a comment
Taipei Conference on Representation Theory IV

Taipei Conference on Representation Theory IV

A Winter School in Representation Theory will be held in the Institute of Mathematics of the Academia Sinica, Taipei, Taiwan, from December 16-19, 2013. Prof. Anthony Licata (Australian National University) and Prof. Ivan Losev (Northeastern University) will each give an 8 hour lecture series. Following the Winter School the Institute of Mathematics will host the Taipei Conference on Representation Theory from December 20 to December 23, 2013. Below is a list of participants who have agreed to give an invited lecture.

Invited speakers:

Vyjayanthi Chari (University of California – Riverside)
Tsao-Hsien Chen (Northwestern University)
Jie Du (University of New South Wales)
Michael Finkelberg (National Research University Higher School of Economics)
Dennis Gaitsgory (Harvard University)
Xuhua He (Hong Kong University of Science and Technology / University of Maryland)
Seok-Jin Kang (Seoul National University)
Syu Kato (Kyoto University)
Alexander Kleshchev (University of Oregon)
Anthony Licata (Australian National University)
Ivan Losev (Northeastern University)
George Lusztig (Massachusetts Institute of Technology)
Arun Ram (University of Melbourne)
Peter Trapa (University of Utah)
Jinkui Wan (Beijing Institute of Technology)
Geordie Williamson (Max-Planck Institut für Mathematik)
Weiqiang Wang (University of Virginia)
Ben Webster (University of Virginia)

Organizers:
Shun-Jen Cheng(Institute of Mathematics Academia Sinica)
Weiqiang Wang(Department of Mathematics University of Virginia)

Link: http://www.math.sinica.edu.tw/chengsj/lie_2013.htm
Poster: lie_theory2013

Meet with Serre

July 24, 2013 at 3:18 pm | Posted in Uncategorized | Leave a comment
J. P. Serre

J. P. Serre and me

Modular, is a special topic for me to learn. Especially “Cours d’arithmétique”. I start read it when I was just an undergraduate student.

Gee is still working on (about) that, even the time I  met my advisor.

The circle is quite amazing. The coming of Serre, the coming of feelings in mathematics. It lead me to feel and see the author that influence me a lot while being a kid in math. And encourage me to keep reading and learning as usual…

Gee’s leaning magazine will continue again. For the beautiful gifts from you and lovely persons here.

Letter to P

August 31, 2009 at 1:40 am | Posted in Others | Leave a comment

P & Yeats

Dear P,

It’s a long time not talk to you. Do you know how happy I was when met you last week in NTNU, Taipei?  It always you give me some effort to keep my orientation(or central voice) going on and on. Trust the susceptibility between us will never decided.  My friend, my teacher forever.

Keep watching and looking at arXiv(http://arxiv.org/) to feel the field, thank you.

Select some words from W. B. Yeat, it maybe some mood:

We were the last romatics-chose for theme
Traditional sanctity and loveliness
Whatever’s written in what poet’s name
The book of the people; whatever most can bless
The mind of man or elevate a rhyme;
But all is changed, that high horse riderless,
Though mounted in that saddle Homer rode
Where the swan drifts upon a darkening flood.

Selectes from<Coole and Ballylee,1931>

Do You Know The Parallel in gragh Theory?

March 23, 2009 at 2:00 pm | Posted in Math Learning | 2 Comments
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A First Look at graph Theory

A First Look at graph Theory

“A First Look At Graph Theory” is  a good book for me if you want me to select books on the library.

Sorry for those like to vist these garden. There’s  some reasons made me leave off the blog for several days and not touched this one for some days. Thanks those can push me to the place I like. Appreciated here with lots of tears  for a hard and tiny life.

Graph Theory is the course that I never majored in before. But the interesting questions always attract some person.

But today, I am not to introduce the book. Gee just find some “interesting” and also “painful” things to share friends. Thus you can know the one, to be a mathematical student, is so great a person to distinguish these and study so much things that most people doesn’t underestand. You really deserve the applaus in several places.

Here are some examples from John Clark & Derek Allan Holon, 1999, p.7

Q1. Do you know “parallel” in junior high school?

Q2Do you know the “isolated” in complex variable?

Q3Do you know “simple” in Abstract algebra?

Think those questions and try to look at the following:

graph-definition

After look at these…

You really know what you learned before?

What is “parallel” “isolated“and “simple” in your mind?

Of course,…You can also choose to forget what the book wrote. That maybe the best way for most people, though the teachers who teach graph theory may stop you to do these stupid actions.><

“To be a student in the department of mathematics” is really a difficult and hard mission to overcome…Especially to  practice and describe one things with several ways in that fields like these.

Taipei Workshop in Lie Theory2008

September 16, 2008 at 4:57 am | Posted in Activities | Leave a comment
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Academia Sinica

Academia Sinica

Taipei Workshop in Lie Theory 2008

Page url: http://www.math.sinica.edu.tw/chengsj/lie_2008.htm

Sponsor: Academia Sinica
Date: December 28-30, 2008
Venue: Institute of Mathematics, Academia Sinica, Taipei, Taiwan

Invited speakers:
Tomoyuki Arakawa (Nara Women’s University, Japan)
Roman Bezrukavnikov* (MIT, USA)
Alexander Braverman (Brown University, USA)
Shun-Jen Cheng (Academia Sinica, Taiwan)
Meng-Kiat Chuah (National Tsing Hua University, Taiwan)
Jae-Hoon Kwon (University of Seoul, Korea)
Ching-Hung Lam (National Cheng-Kung University, Taiwan)
Zongzhu Lin (Kansas State University, USA)
Masahiko Miyamoto (University of Tsukuba, Japan)
Liangang Peng (Sichuan University, China)
Alexander Premet (University of Manchester, UK)
Vera Serganova (University of California, Berkeley, USA)
Eric Sommers (University of Massachusetts, Amherst, USA)
Toshiyuki Tanisaki (Osaka City University,Japan)
Weiqiang Wang (University of Virginia, USA)

Organizers:
Shun-Jen Cheng (Institute of Mathematics Academia Sinica,Taiwan)
Weiqiang Wang (Department of Mathematics University of Virginia, Charlottesville)

P.s. You may also click the Workshop on Algebraic Aspects of Lie Theory last year for more reference.

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.

ISEF ’08 Winner: Knot Theory Mathematics

August 20, 2008 at 1:57 pm | Posted in History and News | 2 Comments
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  Sana Raoof, 17 of Jericho High School in Jericho, New York, is one of the winners of Intel Foundation Scientist Award from Intel International Science & Engineering Fair (ISEF) on May 16, 2008. (from the Winner Announcement and News Release). And will attend Harvard University this fall.

   The vedio shows us how the intelligent girl introduce her research, “Computation of the Alexander-Conway polynomial on the Chord Diagrams of Singular Knots” to ISEF.  The lucky girl motivated by something from MIT, maybe in some graduate summer program, which provided her a great introduction of knot theory and lots of paper for this subject. She focus on computation of the δ-function that can distinguish if those on Chord Diagrams. 

 Congratulation to the shiny star…She won the prize to the royal road!!

 

P.s.

  • Want to know more about “Chord Digrams“, you may also see the article of Greg in The Everything Seminar to discuss with them.
  • You can also click “knot theory and biochemistry” by Jesse Johnson to study the subject more.
  • Gee didn’t think the result application of protein or DNA in biochemistry can be helpful… We all know that the identified methods of biochemistry are quite differenent from the knot computation. Knot model allowed knots to be contracted, distorted and twisted infinitely. That’s not the same for DNA nor protein. The force energies between slight masses of protein or DNA need more detailed discussion or computation. It’s more more important than to compute the free knot polynomial of the model. … Just my little experience in the program of Institue of Bio-Chemistry to learn something about protein in Academia sinica, 2008.

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.

Equations Editor in WordPress

August 11, 2008 at 12:27 pm | Posted in Math Learning | Leave a comment
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  To Tell the truth, there are not only amazing persons, but also good tools here that support a good  environment to learn mathematics in WordPress.

  Yes! WordPress can include \LaTeX code in the post. See the announcement “Can I put Math or Equations in my Posts? ” of FAQ. You can type the mathematical formula on the blog. Just as the article show us :
So, taking these two definitions for example,

Definition3.6 The Jones polynomial V(L) of an oriented link L is the Laurent polynomial in t1/2, with integer coefficients, defined by

V(L)=((-A))^{-3(D)})\langle{D}\rangle)_{t^{1/2}=A^{-2}}\in{Z[t^{-1/2},t^{1/2}]}

where D is any oriented diagram for L .

Definition6.6  The rth Alexander ideal of an oriented link L is the rth elementary ideal of the Z[t-1,t] module H1[X;Z]. The rth Alexander polynomial of L is a generator of the smallest principal ideal of Z[t-1,t] that contains the rth Alexander polynomial. The first Alexander polynomial is the called the Alexander polynomial and is written ΔL(t)

(Definition Resourse: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM175. New York:Springer-Verlag (1997). p. 26,55.)

Using this useful Latex code… expressing and discussing mathematic are never difficulity problems here. You can also try it. But if you haven’t known what Latex would be, you may also click”LaTeX–A document preparation system” to keep close and to attach it.

 

p.s. As for the two definitions in the post, Gee still not catch their true meanings in my mind after typing them. If anybody can explain them more clearly…Please and please tell me.

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