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AKT-170317 Video

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Notes on AKT-170317:    [edit, refresh]

Deriving and testing the Logos following 170317-g1Invariant.nb (pre-class: pdf, nb; post-class: pdf, nb) and 170317-TestingTheLogos.nb (pre-class: pdf, nb; post-class: pdf, nb).


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# Week of... Notes and Links
1 Jan 9 dbnvp Tuesday Hour 1: Course introduction as in About This Class (pdf, html).
dbnvp Tuesday Hour 2: The Jones polynomial via the Kauffman bracket, following 170110-JonesPoly.nb (pre-class: pdf, nb; post-class: pdf, nb).
dbnvp Friday: Knots, algebras, Yang-Baxter, CYBE, Lie algebras, universal enveloping algebras, formulas.
2 Jan 16 dbnvp Tuesday Hour 1: More on the Jones polynomial via the Kauffman bracket, following 170117-MoreJones.nb (pre-class: pdf, nb; post-class: pdf, nb).
dbnvp Tuesday Hour 2: Continuation, 170117-FastSlowRace.nb (pre-class: pdf, nb; post-class: pdf, nb), knot genus.
dbnvp Friday: The Lie algebra {\mathfrak g}_0 following 170120-g0.nb (pre-class: pdf, nb; post-class: pdf, nb).
3 Jan 23 dbnvp Tuesday Hour 1: Tangles and meta-monoids.
dbnvp Tuesday Hour 2: The fundamental group, meta-Hopf algebras, algebraic knot theory.
dbnvp Friday: The Lie algebra {\mathfrak g}_0 following 170127-g0.nb (pre-class: pdf, nb; post-class: pdf, nb).
Class Photo.
4 Jan 30 dbnvp Tuesday Hour 1: Tangles, links, and 3-manifolds.
dbnvp Tuesday Hour 2: Genus, ribbon, slice.
dbnvp Friday: Normal orderings following 170203-g0dsO.nb (pre-class: pdf, nb; post-class: pdf, nb).
5 Feb 6 dbnvp Tuesday Hour 1: \Gamma-calculus following 170207-GammaCalculus.nb (pre-class: pdf, nb; post-class: pdf, nb), part I.
dbnvp Tuesday Hour 2: \Gamma-calculus following 170207-GammaCalculus.nb (pre-class: pdf, nb; post-class: pdf, nb), part II.
dbnvp Friday: The main {\mathfrak g}_0 theorem following 170210-g0MainTheorem.nb (pre-class: pdf, nb; post-class: pdf, nb).
6 Feb 13 dbnvp Tuesday Hour 1: Expansions in general; also using 170214-ProgressiveScanning.nb (pre-class: pdf, nb; post-class: pdf, nb).
dbnvp Tuesday Hour 2: Expansions for tangles, finite type invariants.
dbnvp Friday: Computing the {\mathfrak g}_0 invariant following 170217-g0Invariant.nb (pre-class: pdf, nb; post-class: pdf, nb).
R Feb 20 Reading Week.
7 Feb 27 dbnvp Tuesday Hour 1: Expansions and the fundamental theorem for {\mathcal K}^u.
dbnvp Tuesday Hour 2: Algebraic structures on {\mathcal A}^u.
dbnvp Friday: Conclusion of the {\mathfrak g}_0 discussion following 170303-g0LemmaAndInvariant.nb (pre-class: pdf, nb; post-class: pdf, nb).
8 Mar 6 dbnvp Tuesday Hour 1: The polished {\mathfrak g}_0 invariant following 170307-g0Polished.nb (pre-class: pdf, nb; post-class: pdf, nb) and {\mathfrak g}_1 lemmas following 170307-geps.nb (pre-class: pdf, nb; post-class: pdf, nb).
dbnvp Tuesday Hour 2: The logos for {\mathfrak g}_1 following 170307-geps.nb (pre-class: pdf, nb; post-class: pdf, nb).
dbnvp Friday: Algebraic structures on {\mathcal A}^u, trivalent diagrams.
9 Mar 13 Monday is the last day to drop this class.
dbnvp Friday: Deriving and testing the Logos following 170317-g1Invariant.nb (pre-class: pdf, nb; post-class: pdf, nb) and 170317-TestingTheLogos.nb (pre-class: pdf, nb; post-class: pdf, nb).
10 Mar 20 dbnvp Tuesday Hour 1: The {\mathfrak g}_1 invariant following 170321-g1Invariant.nb (pre-class: pdf, nb; post-class: pdf, nb) and 170321-Polishedg1Invariant.nb (pre-class: pdf, nb; post-class: pdf, nb).
Tuesday Hour 2: offline, though see BBS/AKT17-170321-145350.jpg and BBS/AKT17-170321-150540.jpg.
dbnvp Friday: Associators, also following 170324-Associator.nb (pre-class: pdf, nb; post-class: pdf, nb) and 170324-MutipleZetaValues.pdf.
11 Mar 27 Classes canceled due to an MSRI workshop.
12 Apr 3 dbnvp Tuesday Hour 1: Diagrams to Universal Enveloping Algebras, the u-case.
dbnvp Tuesday Hour 2: Virtual knots and diagrams to Universal Enveloping Algebras, the v-case.
dbnvp Friday: Sjabbo Schaveling: The Quantum Double towards sl_3, following 170407-Classical_Algebra_E0.nb (pdf, nb), 170407--Quantum_Algebra_E0.nb (pdf, nb), and 170407--E_machten_Algebra_E0_v2.nb (pdf, nb).
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0:00:10 [edit] Testing.
0:10:47 [edit] Here's a way to think about quantum groups: they are integrable quantum mechanical systems that live on Lie groups. To make sense of this let's first upgrade our 4d Lie algebra $g_0$ to a Lie group, call it $G_0$. In $GL_3$ it is the group of upper triangular matrices with ones on the top left and bottom right. I'm not sure it was mentioned in the course but $g_0$ is not just a Lie algebra, it is also a Lie bialgebra. (it has a compatible bracket on the dual the trivial one in the case of $g_0$). Translating this to $G_0$ makes $G_0$ a Poisson manifold. That means there is a Poisson bracket on the space of functions $F(G_0)$. Given such a Poisson bracket {.,.} and a Hamiltonian function $H\in F(G_0)$ is enough to write the equations of motion on $G_0$. They are $df/dt = \{f,H\}$ With all this in place we can first remark the fact that our r-matrix is closely related to the bialgebra structure and CYBE to be a condition for integrability of the classical mechanical system. Next we may quantize with respect to the Poisson structure to obtain a deformation of the algebra of functions $F_h(G_0)$. We would then like to think of this non-commutative algebra as the algebra of functions on the (non-existent) quantum group. Integrability survives the quantization and is now expressed in terms of the R-matrix satisfying Yang-Baxter. Dually one may also consider universal enveloping algebra and its deformation as a Hopf algebra.

Quantizing often seems a little ad-hoc but Kontsevich gave a general procedure for (deformation) quantizing with respect to any Poisson structure. [1]

A reference for such things would be A guide to quantum groups by Chari and Pressley. Roland

0:13:57 [edit] The word Logos in Greek has various related meanings, everything from 'word' to 'principle' or 'order' it's one of the key terms in Greek and early Christian philosophy. see the wiki entry Roland