## Notes on Affine Hecke Algebras I.(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

Last update: 23 April 2014

## Notes and References

This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.

## Figures

${t}_{0} x\left({t}_{0}\right) t x \theta Figure 1. The line (the graph of movement) of a particle.$ $t x {\theta }_{1} {\theta }_{2} {\theta }_{3} {\theta }_{4} {x}_{1} {x}_{2} {x}_{3} {x}_{4} Figure 2. The collision of four particles.$ ${i}_{3},{\theta }_{1} {i}_{2},{\theta }_{2} {i}_{1},{\theta }_{3} {\theta }_{3},{j}_{3} {\theta }_{2},{j}_{2} {\theta }_{1},{j}_{1} u v u+v = {i}_{3},{\theta }_{3} {i}_{2},{\theta }_{2} {i}_{1},{\theta }_{1} {\theta }_{3},{j}_{3} {\theta }_{2},{j}_{2} {\theta }_{1},{j}_{1} u v u+v Figure 3a.Figure 3b. Figure 3. The independence of the three-particle S-matrix of the initial points.$ ${\theta }_{1} {\theta }_{2} {\theta }_{3} {\theta }_{4} {\theta }_{5} {\theta }_{1} {\theta }_{2} {\theta }_{3} {\theta }_{4} {\theta }_{5} w={s}_{2} {s}_{3} {s}_{2} {s}_{4} {s}_{3} {s}_{1} {s}_{2}= =\left(4,5,1,3,2\right)=\left(\genfrac{}{}{0}{}{1 2 3 4 5}{4 5 1 3 2}\right) Figure 4. Collisions and reduced decompositions.$ ${u}_{3} {u}_{2} {u}_{1} {u}_{0}=\lambda {s}_{1} {s}_{0} {s}_{2} {s}_{1} {s}_{0} = {u}_{3} {u}_{2} {u}_{1} \lambda ={u}_{0} {s}_{2} {s}_{1} {s}_{0} {s}_{2} {s}_{1} Figure 5a.Figure 5b. Figure 5. Some version of Fig. 3 with "parallel" lines.$ ${i}_{1},{\theta }_{1} {i}_{2},-{\theta }_{2} {i}_{3},-{\theta }_{3} {\theta }_{1},{j}_{1} {\theta }_{2},{j}_{2} {\theta }_{3},{j}_{3} \stackrel{\sim }{w}={s}_{1} {s}_{2} {s}_{1} \pi {s}_{1} {s}_{2} \pi {s}_{1}= =\left(1,-2,-3\right)=\left(\genfrac{}{}{0}{}{1\phantom{\rule{1em}{0ex}}2\phantom{\rule{1em}{0ex}}3}{1 -2 -3}\right) Figure 6. Decomposing of collisions with reflection.$ $i j t -{\theta }_{1} {\theta }_{1} {| |}_{i}^{j}\left({\theta }_{1}\right) {i}_{1} {i}_{2} {j}_{1} {j}_{2} \theta = {S}_{{i}_{1}{i}_{2}}^{{j}_{1} {j}_{2}}\left(\theta \right) {i}_{1} {i}_{2} {j}_{1} {j}_{2} \theta {i}_{1} {i}_{2} {j}_{1} {j}_{2} \theta {\stackrel{ˆ}{S}}_{{i}_{1}{i}_{2}}^{{j}_{1} {j}_{2}}\left(\theta \right) Figure 7a.Figure 7b.Figure 7c. Figure 7. The elementary processes on the half-line.$ $u v u+v 2u+v {\theta }_{1} {\theta }_{2} -{\theta }_{1} -{\theta }_{2} = u v u+v 2u+v {\theta }_{1} {\theta }_{2} -{\theta }_{1} -{\theta }_{2} Figure 8a.Figure 8b. Figure 8. The fundamental identity for reflections and intersections.$ $i j \theta {X}_{i}^{j}\left(\theta \right) i j 0 {\stackrel{ˆ}{X}}_{i}^{j}\left(-\theta \right) Figure 9a.Figure 9b. Figure 9. The transmission through the glass.$