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

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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

$$\begin{array}{c} t0 x(t0) t x θ \\ \text{Figure 1. The line (the graph of movement) of a particle.}\end{array}$$ $$\begin{array}{c} t x θ1 θ2 θ3 θ4 x1 x2 x3 x4 \\ \text{Figure 2. The collision of four particles.}\end{array}$$ $$\begin{array}{ccc} i3,θ1 i2,θ2 i1,θ3 θ3,j3 θ2,j2 θ1,j1 u v u+v & =& i3,θ3 i2,θ2 i1,θ1 θ3,j3 θ2,j2 θ1,j1 u v u+v \\ \text{Figure 3a.}& & \text{Figure 3b.}\\ \multicolumn{3}{c}{\text{Figure 3. The independence of the three-particle}\hspace{0.17em}S\text{-matrix of the initial points.}}\end{array}$$ $$\begin{array}{c} θ1 θ2 θ3 θ4 θ5 θ1 θ2 θ3 θ4 θ5 w=s2 s3 s2 s4 s3 s1 s2= =(4,5,1,3,2)=(1 2 3 4 54 5 1 3 2) \\ \text{Figure 4. Collisions and reduced decompositions.}\end{array}$$ $$\begin{array}{ccc} u3 u2 u1 u0=λ s1 s0 s2 s1 s0 & =& u3 u2 u1 λ=u0 s2 s1 s0 s2 s1 \\ \text{Figure 5a.}& & \text{Figure 5b.}\\ \multicolumn{3}{c}{\text{Figure 5. Some version of Fig. 3 with "parallel" lines.}}\end{array}$$ $$\begin{array}{c} i1,θ1 i2,-θ2 i3,-θ3 θ1,j1 θ2,j2 θ3,j3 w∼=s1 s2 s1 π s1 s2 π s1= =(1,-2,-3)=(1231 -2 -3) \\ \text{Figure 6. Decomposing of collisions with reflection.}\end{array}$$ $$\begin{array}{ccc} i j t -θ1 θ1 | |ij(θ1) & i1 i2 j1 j2 θ = Si1i2j1 j2(θ) i1 i2 j1 j2 θ & i1 i2 j1 j2 θ Sˆi1i2j1 j2(θ) \\ \text{Figure 7a.}& \text{Figure 7b.}& \text{Figure 7c.}\\ \multicolumn{3}{c}{\text{Figure 7. The elementary processes on the half-line.}}\end{array}$$ $$\begin{array}{ccc} u v u+v 2u+v θ1 θ2 -θ1 -θ2 & =& u v u+v 2u+v θ1 θ2 -θ1 -θ2 \\ \text{Figure 8a.}& & \text{Figure 8b.}\\ \multicolumn{3}{c}{\text{Figure 8. The fundamental identity for reflections and intersections.}}\end{array}$$ $$\begin{array}{ccc} i j θ Xij(θ) & & i j 0 Xˆij(-θ) \\ \text{Figure 9a.}& & \text{Figure 9b.}\\ \multicolumn{3}{c}{\text{Figure 9. The transmission through the glass.}}\end{array}$$

page history