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 ${\mathcal {A}}(\Gamma )\equiv {\mathcal {A}}(\uparrow _{b_{1}(\Gamma )})$.

• ${\mathcal {A}}(\uparrow _{n})$ is a VS-algebra (see more at VS, TS and TG Algebras).

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

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.