Math 4010 & 5530 Section 101 Homepage

4010 Syllabus and 5530 Syllabus
Schedule
asULearn (Online gradebook, chatroom, etc.)
News & Announcements

Term Project Ideas: ...or pick one of your own!
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 Physics! Quantum Harmonic Oscillator                                [Ethan]
 Lie Groups and their Connection with Geometry                       [Russell]
 Type C: The Symplectic Algebras                                     [Stephen]
 Introduction to Leibniz algebras                                    [Carter]
 Cyclic Leibniz Algebras                                             [John]
 Tensor Products and Clebsh-Gordon                                   [Leah]
 Coxeter Groups (A Generalization of Weyl Groups)                    [Bri and Cecily]
 Constructing the exceptional algebra G2.                            [Sam]
 The universal enveloping algebra and PBW theorem.
 The infinite dimensinoal Heisenberg algebra.
 Formal calculus (background for vertex algebras)
 Kac-Moody Lie algebras (generalized Cartan matrices)
 Theory of solvable Lie algebras (like low dim. classification).
 Theory or nilpotent Lie algebras (like low dim. classification).
 Simple Lie algebras of type B_n = so(2n+1) (or others) in detail.
 The connection between a Lie group and a Lie algebra.               
 Representation theory of sl3=A2 (especially weight multiplicities). 
 Ado's Theorem


12/11 Sam's Construction of G2 Talk (.nb) (Mathematica file) -- Thanks Sam!

12/01 Root System Demo (preliminary version)

      Root Systems in Maple (.mw)
      This worksheet will plot Dynkin diagrams, weights for irreducible representations
      for simple Lie algebras of ranks 1, 2, and 3. It will also find Cartan matrices,
      simple roots, fundamental weights, highest long roots, and do Weyl group computations.



11/08 Homework #7 (.pdf) [Source: (.tex)] is due Wednesday, November 15^th.

10/31 Homework #6 (.pdf) [Source: (.tex)] is due Wednesday, November 8^th.

10/23 Homework #5 (.pdf) [Source: (.tex)] is due Wednesday, November 1^st.



09/27 Homework #4 (.pdf) [Source: (.tex)] is due Wednesday, October 4^th.

      We decided that Test #1 will be Monday, October 9^th with a take home part made
      available by at least the Friday before. 

09/16 Homework #3 (.pdf) [Source: (.tex)] is due Monday, September 25^th.

09/06 Homework #2 (.pdf) [Source: (.tex)] is due Wednesday, September 13^th.

09/01 Possibly relevant handouts from Graduate Linear Algebra...
      RREF and Linear Correspondence (.pdf) [Source: (.tex)]
      Computing Bases (.pdf) [Source: (.tex)]
      Eigenhandout (.pdf) [Source: (.tex)]
      Coordinate Matrix Example (.pdf) [Source: (.tex)]
      Kernel, Range, & Composition Example (.pdf) [Source: (.tex)]
      Coordinates vs. Coordinate Matrices (.pdf) [Source: (.tex)]

08/28 Homework #1 (.pdf) [Source: (.tex)] is due Wednesday, September 6^th.

08/18 Syllabus [Syllabus for MAT 5530] and tentative schedule have been posted.

07/03 Course Data
      MAT 4010 Section 101 & MAT 5530 Section 101
      INTRO TO LIE ALGEBRAS
      Meeting Times MWF 11:00am-11:50am
      Room WA 309

      [Also, MAT 4011 Section 101 = Capstone]

      Course Title & Description:
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      MAT 4010-101 & 5530-101 "Introduction to Lie Algebras" 3 credits
 
      Prerequisite: 3110

      Title: "Introduction to Lie Algebras"

      Around 1860 Sophus Lie developed the theory of Lie groups and Lie algebras
      to study solvability of differential equations. Later these theories were 
      used in the development of quantum theories in physics. Many areas of modern 
      mathematics and physics require some knowledge of Lie theory. In this course 
      we will study the theory of Lie algebras (but only briefly hint at some 
      applications).
 
      In particular, after some “intermediate” linear algebra background, we will 
      look at the basic theory of Lie algebras including the definition, examples, 
      subalgebras, homomorphisms, and quotients. Then we will transition to some 
      representation theory discussing the definition of a representation/module, 
      module maps, Schur's lemma, sl2 representation theory, Weyl's theorem, the 
      root space decomposition of simple Lie algebras, root systems, Weyl groups, 
      Cartan matrices, Dynkin diagrams, and the classification of simple Lie 
      algebras.
     
      Time permitting, we will also look at irreducible representations of simple 
      Lie algebras, the weight space decomposition, and the classification of 
      these irreducible representations.
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      A note about background: We will be building up the theory of Lie algebras from 
      basically nothing. I will assume that everyone already knows some basic concepts 
      from linear algebra (like definitions and basics about vector spaces, linear 
      independence, spanning, bases, and dimension). More "serious" linear algebra 
      background will be either be presented in class or reviewed as necessary. Graduate 
      linear algebra will NOT be assumed. Also, I will not be using group and ring 
      theory learned in MAT 3110 (Modern Algebra) in a direct way. The modern algebra 
      prerequisite serves to guarantee some maturity in the area of writing abstract 
      algebra style proofs as well as making sure that everyone has already been 
      exposed to the ideas of sub-thing tests, homomorphisms, isomorphisms, and 
      quotients.
 
      Any questions about this class?
      Send me an email at cookwj@appstate.edu
 
      -Bill Cook

Term Project Ideas: More suggestions later...or come up with one of your own!
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 The connection between a Lie group and a Lie algebra.
 The universal enveloping algebra and PBW theorem.
 Constructing the exceptional algebra G2.
 Representation theory of sl3=A2 (especially weight multiplicities).
 Formal calculus (background for vertex algebras)
 Kac-Moody Lie algebras (generalized Cartan matrices)
 Solvable Lie algebras and Lie's Theorem
 Nilpotent Lie algebras & Engel's Theorem
 Formal Calculus & Vertex Algebras
 Universal enveloping algebras & the PBW Theorem
 Coxeter Groups (A Generalization of Weyl Groups)
 Connections between Lie groups and algebras
 The Killing form and Cartan's Criterion (I and II)
 Heisenberg Lie Algebra(s)
 Simple Lie algebras of type B_n = so(2n+1)


Term Project Topics Reserved:

 None so far


Fall 2013 stuff...

      Root Systems in Maple (.mw) 
      This worksheet will plot Dynkin diagrams, weights for irreducible representations
      for simple Lie algebras of ranks 1, 2, and 3. It will also find Cartan matrices,
      simple roots, fundamental weights, highest long roots, and do Weyl group computations.

      Final Project Rubric

      LaTeX Examples

      Big Quiz #2 (.pdf) and source Big Quiz #2 (.tex)

      Homework #2X is due Monday, November 18^th.
      [For those who LaTeX, Homework #2X (.tex).]

      Homework #1X is due Friday, November 8^th.
      [For those who LaTeX, Homework #1X (.tex).]

      Schedule for classes 1-15 (including suggested homeworks & readings)

      Homework #1 due Wed. Sept. 4^th

      Schedule for classes 1-5 (including suggested homeworks & readings)

      Some old 2240 handouts: 
      Gauss-Jordan Elimination & The Linear Correspondance
      Finding & Extending Bases Example
      Coordinate Matrix Example
      Kernel, Range, Composition of Linear Transformations Example
      Eigenhandout