Notes for AKT-170317/0:10:47 - Revision history

Rolandvdv at 19:17, 21 March 2017

2017-03-21T19:17:56Z

← Older revision		Revision as of 15:17, 21 March 2017
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	Translating this to $G_0$ makes $G_0$ a Poisson manifold. That means there is a Poisson bracket on the space of functions		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		$F(G_0)$. 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 $df/dt = {f,H}$		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		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.		to be a condition for integrability of the classical mechanical system.

Rolandvdv at 18:58, 18 March 2017

2017-03-18T18:58:44Z

← Older revision		Revision as of 14:58, 18 March 2017
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	of upper triangular matrices with ones on the top left and bottom right.		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.		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).		(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		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		$F(G_0)$. 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 $df/dt = {f,H}$		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		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.		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).		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.		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.		Integrability survives the quantization and is now expressed in terms of the R-matrix satisfying Yang-Baxter.

Rolandvdv at 18:58, 18 March 2017

2017-03-18T18:58:07Z

← Older revision		Revision as of 14:58, 18 March 2017
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	Here's a way to think about quantum groups: they are integrable quantum mechanical systems that live on Lie groups.		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.		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.		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).		(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		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		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 $df/dt = {f,H}$		on G_0. They are $df/dt = {f,H}$

Rolandvdv at 18:55, 18 March 2017

2017-03-18T18:55:07Z

← Older revision		Revision as of 14:55, 18 March 2017
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	Here's a way to think about quantum groups: they are integrable quantum mechanical systems that live on Lie groups.		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.		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.		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		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		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}		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).
	It's easier to pass to the Lie-group G_0 that corresponds to
			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}}		{{Roland}}

Rolandvdv at 18:43, 18 March 2017

2017-03-18T18:43:29Z

← Older revision		Revision as of 14:43, 18 March 2017
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	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.		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		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		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}		on G_0. They are df/dt = {f,H}

Rolandvdv at 18:43, 18 March 2017

2017-03-18T18:43:19Z

← Older revision		Revision as of 14:43, 18 March 2017
Line 3:		Line 3:
	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.		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		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		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}		on G_0. They are df/dt = {f,H}

Rolandvdv at 18:41, 18 March 2017

2017-03-18T18:41:03Z

← Older revision		Revision as of 14:41, 18 March 2017
Line 3:		Line 3:
	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.		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		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		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}		on G_0. They are df/dt = {f,H}

Rolandvdv at 18:40, 18 March 2017

2017-03-18T18:40:35Z

← Older revision		Revision as of 14:40, 18 March 2017
Line 3:		Line 3:
	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.		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		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		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}		on G_0. They are df/dt = {f,H}

Rolandvdv at 18:40, 18 March 2017

2017-03-18T18:40:15Z

← Older revision		Revision as of 14:40, 18 March 2017
Line 3:		Line 3:
	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.		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		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		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}		on G_0. They are df/dt = {f,H}

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..."

2017-03-18T18:39:55Z

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..."

New page

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}}