Degrees and exponents

introduction

• eigenvalues of Cartan matrices
  
• eigenvalues of incidence matrices of Dynkin diagram
  
• http://pythagoras0.springnote.com/pages/1938682/attachments/3170605

Cartan matrix

• h : Coxeter number
  
• eigenvalue
  \(4\sin^2(\frac{m_{i}\pi}{2h})\)
  
• \(m_{i}\) is called the exponents
  
• \(d_{i}=m_{i}+1\) is called a degree
  

adjacency matrix

• h : Coxeter number
  
• eigenvalue \(2\cos(\pi l_n/h)\)
  

1. Table[Simplify[2 Cos[Pi*l/5]], {l, 1, 4}]
   Table[Simplify[4 Sin[Pi*l/10]^2], {l, 1, 4}]

homological algebraic characterization

For a s.s. Lie algebra L

(a)H'(L) is a free super- commutative algebra with homogeneous generator in degrees 2m_1+1,\cdots,2m_l+1

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Coxeter_number
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

4909919

articles

• 논문정리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html[1]
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/