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

t0 x(t0) t x θ Figure 1. The line (the graph of movement) of a particle. t x θ1 θ2 θ3 θ4 x1 x2 x3 x4 Figure 2. The collision of four particles. 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 Figure 3a.Figure 3b. Figure 3. The independence of the three-particleS-matrix of the initial points. θ1 θ2 θ3 θ4 θ5 θ1 θ2 θ3 θ4 θ5 w=s2 s3 s2 s4 s3 s1 s2= =(4,5,1,3,2)=(1234545132) Figure 4. Collisions and reduced decompositions. u3 u2 u1 u0=λ s1 s0 s2 s1 s0 = u3 u2 u1 λ=u0 s2 s1 s0 s2 s1 Figure 5a.Figure 5b. Figure 5. Some version of Fig. 3 with "parallel" lines. 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) Figure 6. Decomposing of collisions with reflection. i j t -θ1 θ1 ||ij(θ1) i1 i2 j1 j2 θ = Si1i2j1j2(θ) i1 i2 j1 j2 θ i1 i2 j1 j2 θ Sˆi1i2j1j2(θ) Figure 7a.Figure 7b.Figure 7c. Figure 7. The elementary processes on the half-line. u v u+v 2u+v θ1 θ2 -θ1 -θ2 = u v u+v 2u+v θ1 θ2 -θ1 -θ2 Figure 8a.Figure 8b. Figure 8. The fundamental identity for reflections and intersections. i j θ Xij(θ) i j 0 Xˆij(-θ) Figure 9a.Figure 9b. Figure 9. The transmission through the glass.

page history