© | Dror Bar-Natan: Classes: 2013-14: AKT: | < > |

width: 400 720 |
download ogg/AKT-140224_400.ogg |

Videography by Iva Halacheva | troubleshooting |

# | Week of... | Notes and Links |
---|---|---|

1 | Jan 6 | About This Class (PDF). Monday: Course introduction, knots and Reidemeister moves, knot colourings. Tricolourability without Diagrams Wednesday: The Gauss linking number combinatorially and as an integral. Friday: The Schroedinger equation and path integrals. Friday Introduction (the quantum pendulum) |

2 | Jan 13 | Homework Assignment 1. Monday: The Kauffman bracket and the Jones polynomial. Wednesday: Self-linking using swaddling. Friday: Euler-Lagrange problems, Gaussian integration, volumes of spheres. |

3 | Jan 20 | Homework Assignment 2. Monday: The definition of finite-type and some examples. Wednesday: The self-linking number and framings. Friday: Integrating a polynomial times a Gaussian. Class Photo. |

4 | Jan 27 | Homework Assignment 3. Monday: Chord diagrams and weight systems. Wednesday: Swaddling maps and framings, general configuration space integrals. Friday: Some analysis of . |

5 | Feb 3 | Homework Assignment 4. Monday: 4T, the Fundamental Theorem and universal finite type invariants. The Fulton-MacPherson Compactification (PDF). Wednesday: The Fulton-MacPherson Compactification, Part I. Friday: More on pushforwards, , and . |

6 | Feb 10 | Homework Assignment 5. Monday: The bracket-rise theorem and the invariance principle. Wednesday: The Fulton-MacPherson Compactification, Part II. Friday: Gauge fixing, the beginning of Feynman diagrams. |

R | Feb 17 | Reading Week. |

7 | Feb 24 | Monday: A review of Lie algebras. Wednesday: Graph cohomology and . Friday: More on Feynman diagrams, beginning of gauge theory. From Gaussian Integration to Feynman Diagrams (PDF). |

8 | Mar 3 | Homework Assignment 6 (PDF) Monday: Lie algebraic weight systems. Wednesday: Graph cohomology and the construction of . Graph Cohomology and Configuration Space Integrals (PDF) Friday: Gauge invariance, Chern-Simons, holonomies. Mar 9 is the last day to drop this class. |

9 | Mar 10 | Homework Assignment 7 (PDF) Monday: The weight system. Wednesday: The universal property, hidden faces. Friday: Insolubility of the quintic, naive expectations for CS perturbation theory. |

10 | Mar 17 | Homework Assignment 8 (PDF) Monday: and PBW. Wednesday: The anomaly. Friday: Faddeev-Popov, part I. Gaussian Integration, Determinants, Feynman Diagrams (PDF). |

11 | Mar 24 | Homework Assignment 9 (PDF) Monday: is a bi-algebra. Wednesday: Understanding and fixing the anomaly. Friday: class cancelled. |

12 | Mar 31 | Monday, Wednesday: class cancelled. Friday: A Monday class: back to expansions. |

E | Apr 7 | Monday: A Friday class on what we mostly didn't have time to do. |

Add your name / see who's in! | ||

Dror's Notebook | ||

refresh

panel Managed by dbnvp: You can click on many of the blackboard shots / frame grabs to see their full-size versions.

panel Managed by dbnvp: You can click on many of the blackboard shots / frame grabs to see their full-size versions.

Using bilinearity,

$[y,z] = b_1c_1[\alpha,\alpha] + b_1c_2[\alpha, \beta] + b_2c_1[\beta,\alpha] + b_2c_2[\beta,\beta]$

Since $[\alpha, \alpha] = [\beta,\beta] = 0$, $[\alpha, \beta] = \alpha$, and $[\beta,\alpha] = -\alpha$, this evaluates to $$[y,z] = (b_1c_2 - b_2c_1)\alpha$$ Similar calculations yield $$[z,x] = (c_1a_2 - c_2a_1)\alpha$$ and $$[x,y] = (a_1b_2 - a_2b_1)\alpha$$ Thus, $$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = (a_2(b_1c_2 - b_2c_1) + b_2(c_1a_2-c_2a_1) + c_2(a_1b_2+a_2b_1))[\beta, \alpha]$$ $$= -\alpha(a_2b_1c_2 - a_2b_2c_1 + a_2b_2c_1 - a_1b_2c_2 + a_1b_2c_2 - a_2b_1c_2) = 0$$

As a result, the Jacobi identity holds.

1. **one-dimensional Lie algebras** are unique up to isomorphism. For if is a one dimensional Lie algebra, then since the bracket is antisymmetric, we have . Thus the bracket is zero and is unique up to isomorphism.

2. . is a two-dimensional Lie algebra. There are only two of such up to isomorphism, that is, the one with the bracket equal to zero and the other with bracket .