06-1350/Class Notes for Thursday November 16

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Today's Agenda

Formulas are a Chore (Bore?)

• Sweeping clean a tree and [math]\displaystyle{ {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}) }[/math].

• [math]\displaystyle{ {\mathcal A}(\uparrow_n) }[/math] is a VS-algebra (see more at VS, TS and TG Algebras).

• In the coordinates above, write the [math]\displaystyle{ TR\Phi B }[/math] relations in various algebraic notations.
  • R4: [math]\displaystyle{ (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B }[/math] or [math]\displaystyle{ (B_{1a}B_{2a}\Phi_{1a}; B_{1b}B_{2b}; B_{1c}B_{2a}\Phi_{1b}; B_{2c}\Phi_{1c}) = (\Phi_{2a}B_{3a}; \Phi_{2a}B_{3b}; \Phi_{2b}B_{3c}; \Phi_{2c}B_{3c}) }[/math].
  • R3: [math]\displaystyle{ (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B }[/math] or [math]\displaystyle{ (B_{1a}B_{2a}B_{3a}; B_{1b}B_{2b}; B_{1c}B_{2a}B_{3b}; B_{2c}B_{3c}) = (B_{4a}B_{5a}B_{6a}; B_{4a}B_{5b}B_{6b}; B_{4b}B_{6c}; B_{4c}B_{5c}B_{6a}) }[/math].
  • R2: [math]\displaystyle{ (123)^\star B^\pm\cdot(132)^\star B^\mp=1_3 }[/math] or [math]\displaystyle{ (B^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1) }[/math].
  • R1: [math]\displaystyle{ (B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2}) }[/math].
• But for now, skip the writing of the following relations:
  • Symmetry of [math]\displaystyle{ \Phi }[/math] and of [math]\displaystyle{ B^{\pm} }[/math].
  • [math]\displaystyle{ u }[/math], [math]\displaystyle{ d }[/math] and [math]\displaystyle{ \# }[/math]
  • Idempotence for [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math].
  • [math]\displaystyle{ B^{\pm} }[/math] in terms of [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ R }[/math] and [math]\displaystyle{ R }[/math] in terms of [math]\displaystyle{ T }[/math].

Exponentiation is a Miracle

• Description of the problem.
• Beads and strands.
• The perturbative approach, linearization.
• The syzygies: relations between the errors.
• The Hochschild complex and homology.