Drorbn - User contributions [en]

Notes for AKT-170317/0:10:47

2017-03-21T19:17:56Z

Rolandvdv:

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. [http://www.ihes.fr/~maxim/TEXTS/DefQuant_final.pdf]

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

Notes for AKT-170317/0:10:47

2017-03-18T18:58:44Z

Rolandvdv:

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. [http://www.ihes.fr/~maxim/TEXTS/DefQuant_final.pdf]

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

Notes for AKT-170317/0:10:47

2017-03-18T18:58:07Z

Rolandvdv:

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. [http://www.ihes.fr/~maxim/TEXTS/DefQuant_final.pdf]

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

Notes for AKT-170317/0:10:47

2017-03-18T18:55:07Z

Rolandvdv:

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.
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. [http://www.ihes.fr/~maxim/TEXTS/DefQuant_final.pdf]

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

Notes for AKT-170317/0:10:47

2017-03-18T18:43:29Z

Rolandvdv:

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 to a Lie group, call it G_0.
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.
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 $H:G_0 \to R$ is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:10:47

2017-03-18T18:43:19Z

Rolandvdv:

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 to a Lie group, call it G_0.
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.
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 $H:G_0 -> R$ is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:10:47

2017-03-18T18:41:03Z

Rolandvdv:

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 to a Lie group, call it G_0.
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.
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 <math>H:G_0 -> R</math> is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:10:47

2017-03-18T18:40:35Z

Rolandvdv:

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 to a Lie group, call it G_0.
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.
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 <latex>H:G_0 -> R</latex> is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:10:47

2017-03-18T18:40:15Z

Rolandvdv:

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 to a Lie group, call it G_0.
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.
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 $$H:G_0 -> R$$ is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:10:47

2017-03-18T18:39:55Z

Rolandvdv: Created page with "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 algebr..."

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 to a Lie group, call it G_0.
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.
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 $H:G_0 -> R$ is enough to write the equations of motion
on G_0. They are df/dt = {f,H}

It's easier to pass to the Lie-group G_0 that corresponds to

{{Roland}}

Notes for AKT-170317/0:13:57

2017-03-18T18:00:43Z

Rolandvdv:

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.
[https://en.wikipedia.org/wiki/Logos see the wiki entry]
{{Roland}}

Notes for AKT-170317/0:13:57

2017-03-18T18:00:04Z

Rolandvdv:

The word Logos in Greek has various related meanings, everything from 'word' to 'principle' or 'order' it's one of the key buzz-words of Greek and early Christian philosophy.
[https://en.wikipedia.org/wiki/Logos see the wiki entry]
{{Roland}}

Notes for AKT-170317/0:13:57

2017-03-18T17:59:52Z

Rolandvdv:

Notes for AKT-170317/0:13:57

2017-03-18T17:59:36Z

Rolandvdv: Created page with "The word Logos in Greek has various related meanings, everything from 'word' to 'principle' or 'order' it's one of the key buzz-words of Greek and early Christian philosophy. ..."

Notes for AKT-170217/0:03:07

2017-02-17T22:17:29Z

Rolandvdv:

<math>g_0</math> may be the 0th in the class of Lie algebras we're playing with but its enveloping algebra can be made simpler still.
Consider the algebra H obtained from <math>U(g_0)</math> by inverting h and quotienting by the relation <math>l=-ef/h</math>
(check that <math>l+ef/h</math> is a central element in <math>U(g_0)</math>). And while we're at it, why not scale out the h entirely?
Does H reproduce the theory of <math>\Gamma</math> calculus?
Also, the screen is impossible to see on video but the accompanying mathematica file makes up for it twice.
{{Roland}}

Notes for AKT-170217/0:03:07

2017-02-17T22:10:23Z

Rolandvdv: Created page with "<math>g_0</math> may be the 0th in the class of Lie algebras we're playing with but its enveloping algebra can be made simpler still. Consider the algebra H obtained from <ma..."

<math>g_0</math> may be the 0th in the class of Lie algebras we're playing with but its enveloping algebra can be made simpler still.
Consider the algebra H obtained from <math>U(g_0)</math> by inverting h and quotienting by the relation <math>l=-ef/h</math>
(check that <math>l+ef/h</math> is a central element in <math>U(g_0)</math>). And while we're at it, why not scale out the h entirely?
Also, the screen is impossible to see on video but the accompanying mathematica file makes up for it twice.
{{Roland}}

Notes for AKT-170117-2/0:13:00

2017-01-17T22:48:36Z

Rolandvdv:

{{Roland}} At 13:18 there is the statement that the width of a knot diagram is <math>\mathcal{O}(\sqrt{n})</math> where n is the number of crossings. I think this is a consequence of the more general planar-graph result called the Planar Separator Theorem, or rather the edge version:
https://en.wikipedia.org/wiki/Planar_separator_theorem
Here's a link to the relevant article:
http://www.sciencedirect.com/science/article/pii/S0196677483710138?via%3Dihub

Notes for AKT-170117-2/0:13:00

2017-01-17T22:15:10Z

Rolandvdv: Created page with "{{Roland}} At 13:18 there is the statement that the width of a knot diagram is <math>\mathcal{O}(\sqrt{n})</math> where n is the number of crossings. I think this is a consequ..."

Notes for AKT-170113/0:50:48

2017-01-14T13:33:52Z

Rolandvdv:

{{Roland}} At 38:12 Dror mentions a solution to CYBE already gives a knot invariant by setting <math>R_{ij} = 1 + hr_{ij} + \frac{1}{2!}h^2r_{ij}^2</math> and working modulo <math >h^3 </math>.
Notice I added the <math>h^2</math> term to make the inverse <math>R^{-1}</math> be identical but with negative <math>h</math>, the factorial is just for fun.
I wanted to test this idea in <math>U(sl_2)</math> where you can check that <math>r_{ij} = E_iF_j + \frac{1}{4} H_iH_j</math> is a solution to CYBE.
If I got it right the positive Reidemeister 1 curl yields the value <math>1+ h(EF+\frac{1}{4}H^2)+\frac{1}{2}h^2(2E^2F^2 + EH^2F+EFH+\frac{H^4}{8})</math> bad news, we need the element <math>S</math> to get an invariant in this case.
Taking <math>\tilde{r}_{ij} = r_{ij}+r_{ji}</math> we may do a little better in that now the curl yields <math>1+ h(EF+FE+\frac{1}{2}H^2)+\mathcal{O}(h^2)</math>
indicating our ambiguity may now be a central element (the Casimir at order h).

Notes for AKT-170113/0:50:48

2017-01-14T13:18:25Z

Rolandvdv:

{{Roland}} At 38:12 Dror mentions a solution to CYBE already gives a knot invariant by setting <math>R = 1 + hr + \frac{1}{2!}h^2r^2</math> and working modulo <math >h^3 </math>.
Notice I added the <math>h^2</math> term to make the inverse <math>R^{-1}</math> be identical but with negative <math>h</math>, the factorial is just for fun.
I wanted to test this idea in <math>U(sl_2)</math> where you can check that <math>r_{ij} = E_iF_j + \frac{1}{4} H_iH_j</math> is a solution to CYBE.
If I got it right the positive Reidemeister 1 curl yields the value <math>1+ h(EF+\frac{1}{4}H^2)+\frac{1}{2}h^2(2E^2F^2 + EH^2F+EFH+\frac{H^4}{8})</math> bad news, we need the element <math>S</math> to get an invariant in this case.
Taking <math>\tilde{r}_{ij} = r_{ij}+r_{ji}</math> we may do a little better in that now the curl yields <math>1+ h(EF+FE+\frac{1}{2}H^2)+\mathcal{O}(h^2)</math>
indicating our ambiguity may now be a central element (the Casimir at order h).

Notes for AKT-170113/0:50:48

2017-01-14T13:09:07Z

Rolandvdv:

{{Roland}} At 38:12 Dror mentions a solution to CYBE already gives a knot invariant by setting <math>R = 1 + hr + \frac{1}{2!}h^2r^2</math> and working modulo <math >h^3 </math>.
I put the <math>h^2</math> term to make the inverse <math>R^{-1}</math> be identical but with negative <math>h</math>, the factorial is just a hint of more to come.
I thought it was fun to have an example of this in <math>U(sl_2)</math> where you can check that <math>r_{ij} = E_iF_j + \frac{1}{4} H_iH_j</math> is a solution to CYBE.
If I got it right the positive Reidemeister 1 curl yields the value <math>1+ h(EF+\frac{1}{4}H^2)+\frac{1}{2}h^2(2E^2F^2 + EH^2F+EFH+\frac{H^4}{8})</math> bad news, we need the element <math>S</math> to get an invariant in this case.

Notes for AKT-170113/0:50:48

2017-01-14T13:07:37Z

Rolandvdv:

{{Roland}} At 38:12 Dror mentions a solution to CYBE already gives a knot invariant by setting <math>R = 1 + hr + \frac{1}{2!}h^2r^2</math> and working modulo <math >h^3 </math>.
I put the <math>h^2</math> term to make the inverse <math>R^{-1}</math> be identical but with negative <math>h</math>, the factorial is just a hint of more to come.
I thought it was fun to have an example of this in <math>U(sl_2)</math> where you can check that <math>r_{ij} = E_iF_j + \frac{1}{4} H_iH_j</math> is a solution to CYBE.
If I got it right the positive Reidemeister 1 curl yields the value <math>1+ h(EF+\frac{1}{4}H^2)+\frac{1}{2}h^2(2E^2F^2 + EH^2F+EFH+\frac{H^4}{8})</math> bad news, we need the element <math>S</math> to fix it.

Notes for AKT-170113/0:50:48

2017-01-14T12:22:01Z

Rolandvdv:

Notes for AKT-170113/0:50:48

2017-01-14T12:21:28Z

Rolandvdv: Created page with "{{Roland}} At 38:12 Dror mentions a solution to CYBE already gives a knot invariant by setting <math>R = 1 + hr + \frac{1}{2!}h^2r^2</math> and working modulo <math >h^3 </mat..."

Notes for AKT-170110-1/0:43:57

2017-01-11T12:36:08Z

Rolandvdv:

Kauffman often defines his bracket using the variable <math>A</math>, it is not invariant under Reidemeister 1, a positive curl spits out <math>-A^3</math>.
Multiplying through the relation for the <math>\pm</math> crossing by <math>-A^{\mp 3}</math> and absorbing that factor into the crossing,
we get Dror's Kauffman bracket with <math>q = -A^{-2}</math>. {{Roland}}

Notes for AKT-170110-1/0:43:57

2017-01-11T12:35:12Z

Rolandvdv:

Notes for AKT-170110-1/0:43:57

2017-01-11T12:34:07Z

Rolandvdv:

Kauffman often defines his bracket using the variable <math>A</math>, it is not invariant under Reidemeister 1, a positive curl spits out <math>-A^3</math>.
Multiplying through the relation for the <math>\pm</math> crossing by <math>-A^{\mp 3}</math> and setting
<math>q = -A^{-2}</math> one gets Dror's Kauffman bracket.

Notes for AKT-170110-1/0:43:57

2017-01-11T12:33:38Z

Rolandvdv: Created page with "Kauffman often defines his bracket using the variable A, it is not invariant under Reidemeister 1, a positive curl spits out <math>-A^3</math>. Multiplying through the relati..."

Kauffman often defines his bracket using the variable A, it is not invariant under Reidemeister 1, a positive curl spits out <math>-A^3</math>.
Multiplying through the relation for the <math>\pm</math> crossing by <math>-A^{\mp 3}</math> and setting
<math>q = -A^{-2}</math> one gets Dror's Kauffman bracket.

Template:Roland

2017-01-11T12:25:19Z

Rolandvdv:

[[User:Rolandvdv|Roland]]

Template:Roland

2017-01-11T12:25:03Z

Rolandvdv:

[[User:Drorbn|Dror]]

Template:Roland

2017-01-11T12:24:19Z

Rolandvdv: Created page with "User:Rolandvdv"

User:Rolandvdv

2017-01-11T12:22:45Z

Rolandvdv: Roland van der Veen (Leiden University)

Roland van der Veen (Leiden University)
http://www.rolandvdv.nl

User:Rolandvdv

2017-01-11T12:22:24Z

Rolandvdv: Roland van der Veen (Leiden University)

Roland van der Veen (Leiden University)
[http://www.rolandvdv.nl]

Notes for AKT-170110-1/0:48:55

2017-01-11T12:20:53Z

Rolandvdv:

Roland van der Veen wrote {{Dror}}:

Dear Dror,

I enjoyed watching your first hour of AKT and had a minor gripe about your sentence at around 48:55
Your statement
"Relative to difficult to compute things, the Jones polynomial is in fact surprisingly easy to compute"
is in some sense very wrong as it ignores the fact that computing Jones is #P-hard, shown by Jaeger. Toda's theorem implies that using an oracle computing Jones one can obtain a polynomial time algorithm to solve all NP problems and even much more difficult problems
https://en.wikipedia.org/wiki/Toda's_theorem

I suppose what you mean is you can write a short program to do the job.

Best,
Roland

{{Dror}}: Thanks for the comment, Roland! Do you have a link to the Jaeger result?
{{Roland}}: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100068936

No, I mean that I can write a short program that ''does the job well'', in practice even if not in theory. So even though the Jones polynomial is in-theory complicated, in practice it can be computed for knots with about 200 crossings.

Aside. Jones once claimed to me (in person) that the Jones polynomial can be computed with knots with up to 200 crossings, but I've never verified it with my directly. Yet I've seen Khovanov homology computed (using my algorithm!) on knots with about 70 crossings, and in the light of that, 200 for Jones seems reasonable.

--[[User:Drorbn|Drorbn]] ([[User talk:Drorbn|talk]]) 06:55, 11 January 2017 (EST)