AKT-09/HW2 - Revision history

Drorbn at 23:17, 28 October 2009

2009-10-28T23:17:00Z

← Older revision		Revision as of 19:17, 28 October 2009
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	{{AKT-09/Navigation}}		{{AKT-09/Navigation}}

	'''Solve the following problems''' and submit them in class by ~~October~~ 29, 2009:		'''Solve the following problems''' and submit them in class by November 3, 2009:

	'''Problem 1.''' Let <math>{\mathfrak g}_1</math> and <math>{\mathfrak g}_2</math> be finite dimensional metrized Lie algebras, let <math>{\mathfrak g}_1\oplus{\mathfrak g}_2</math> denote their direct sum with the obvious "orthogonal" bracket and metric, and let <math>m</math> be the canonical isomorphism <math>m:{\mathcal U}({\mathfrak g}_1)\otimes{\mathcal U}({\mathfrak g}_2)\to{\mathcal U}({\mathfrak g}_1\oplus{\mathfrak g}_2)</math>. Prove that		'''Problem 1.''' Let <math>{\mathfrak g}_1</math> and <math>{\mathfrak g}_2</math> be finite dimensional metrized Lie algebras, let <math>{\mathfrak g}_1\oplus{\mathfrak g}_2</math> denote their direct sum with the obvious "orthogonal" bracket and metric, and let <math>m</math> be the canonical isomorphism <math>m:{\mathcal U}({\mathfrak g}_1)\otimes{\mathcal U}({\mathfrak g}_2)\to{\mathcal U}({\mathfrak g}_1\oplus{\mathfrak g}_2)</math>. Prove that

Drorbn at 23:22, 19 October 2009

2009-10-19T23:22:25Z

← Older revision		Revision as of 19:22, 19 October 2009
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	{{AKT-09/Navigation}}		{{AKT-09/Navigation}}
	{{In Preparation}}

	'''Solve the following problems''' and submit them in class by October 29, 2009:		'''Solve the following problems''' and submit them in class by October 29, 2009:

Drorbn at 17:14, 19 October 2009

2009-10-19T17:14:57Z

← Older revision		Revision as of 13:14, 19 October 2009
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	'''Solve the following problems''' and submit them in class by October 29, 2009:		'''Solve the following problems''' and submit them in class by October 29, 2009:

			'''Problem 1.''' Let <math>{\mathfrak g}_1</math> and <math>{\mathfrak g}_2</math> be finite dimensional metrized Lie algebras, let <math>{\mathfrak g}_1\oplus{\mathfrak g}_2</math> denote their direct sum with the obvious "orthogonal" bracket and metric, and let <math>m</math> be the canonical isomorphism <math>m:{\mathcal U}({\mathfrak g}_1)\otimes{\mathcal U}({\mathfrak g}_2)\to{\mathcal U}({\mathfrak g}_1\oplus{\mathfrak g}_2)</math>. Prove that
⚫	'''Problem 1.'''
			{{Equation*\|<math>{\mathcal T}_{{\mathfrak g}_1\oplus{\mathfrak g}_2} = m\circ({\mathcal T}_{{\mathfrak g}_1}\otimes{\mathcal T}_{{\mathfrak g}_2})\circ\Box</math>,}}
			where <math>\Box:{\mathcal A}(\uparrow)\to{\mathcal A}(\uparrow)\otimes{\mathcal A}(\uparrow)</math> is the co-product and <math>{\mathcal T}_{{\mathfrak g}}</math> denotes the <math>{\mathcal U}({\mathfrak g})</math>-valued "tensor map" on <math>{\mathcal A}</math>. Can you relate this with the first problem of [[AKT-09/HW1\|HW1]]?

		⚫	'''Problem 2.'''
	# Find a concise algorithm to compute the weight system <math>W_{so}</math> associated with the Lie algebra <math>so(N)</math> in its defining representation.		# Find a concise algorithm to compute the weight system <math>W_{so}</math> associated with the Lie algebra <math>so(N)</math> in its defining representation.
	# Verify that your algorithm indeed satisfies the <math>4T</math> relation.		# Verify that your algorithm indeed satisfies the <math>4T</math> relation.

	'''Problem 2.''' The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref\|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations		'''Problem 3.''' The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref\|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations

	{{Equation*\|<math>L(s_\pm)=a^{\pm 1}L(s))</math>}}		{{Equation*\|<math>L(s_\pm)=a^{\pm 1}L(s))</math>}}
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	and by the initial condition <math>L(\bigcirc)=1</math>. State and prove the relationship between <math>F</math> and <math>W_{so}</math>.		and by the initial condition <math>L(\bigcirc)=1</math>. State and prove the relationship between <math>F</math> and <math>W_{so}</math>.

	'''Problem 3.'''

	'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo\|photo page]].		'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo\|photo page]].

Drorbn at 16:50, 19 October 2009

2009-10-19T16:50:59Z

New page

{{AKT-09/Navigation}}
{{In Preparation}}

'''Solve the following problems''' and submit them in class by October 29, 2009:

'''Problem 1.'''
# Find a concise algorithm to compute the weight system <math>W_{so}</math> associated with the Lie algebra <math>so(N)</math> in its defining representation.
# Verify that your algorithm indeed satisfies the <math>4T</math> relation.

'''Problem 2.''' The ''Kauffman polynomial'' <math>F(K)(a,z)</math> (see {{ref|Kauffman}}) of a knot or link <math>K</math> is <math>a^{-w(K)}L(K)</math> where <math>w(L)</math> is the writhe of <math>K</math> and where <math>L(K)</math> is the regular isotopy invariant defined by the skein relations

{{Equation*|<math>L(s_\pm)=a^{\pm 1}L(s))</math>}}

(here <math>s</math> is a strand and <math>s_\pm</math> is the same strand with a <math>\pm</math> kink added) and

{{Equation*|<math>L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)</math>}}

and by the initial condition <math>L(\bigcirc)=1</math>. State and prove the relationship between <math>F</math> and <math>W_{so}</math>.

'''Problem 3.'''

'''Mandatory but unenforced.''' Find yourself in the class photo and identify yourself as explained in the [[AKT-09/Class Photo|photo page]].

[[Image:AKT-09-ClassPhoto.jpg|center|400px]]

{{note|Kauffman}} L. H. Kauffman, ''An invariant of regular isotopy'', Trans. Amer. Math. Soc. '''312''' (1990) 417-471.