Affine type <math> <mi>C</mi> </math> Temperley-Lieb

Affine type C Temperley-Lieb

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 June 2010

Affine type C Temperley-Lieb

Let H~ be the quotient of ℂ B~ by the relations gi2 = q- q -1 gi +1,  for  1≤i≤n-1, g02 = s- s -1 g0 +1,  and   gn2 = t- t -1 gn +1. Then let ei =q- gi ,  for  1≤i≤n-1, e0 =s- g0 ,  and   en =t- gn .

  1. The relation gi2 = q- q -1 gi +1  is equivalent to   ei2 = qq -1 ei .
  2. Assuming the relations gi2 = q- q -1 gi +1, the relation gi g i+1 gi = g i+1 gi g i+1   is equivalent to   e0 e1 e0 - e1 e0 e0 = s q -1 +q s -1 e0 e1 - e1 e0 .

Define an algebra Tn generated by e0 ,…, en with relations e12 = s+ s -1 e1 , ei2 = q+ q -1 2 , e02 = t+ t -1 e0 , e2 e1 e2 = s q -1 +q s -1 e2 , ei e i-1 ei = ei , ei e i+1 ei = ei , en e0 en = t q -1 +q t -1 en , where 2≤i≤n. This algebra is a surjective ring of H^ wirth kernel generated by ? Putting I= ∏ i  even ei   and  J= ∏ i  odd ei and imposing the relations IJI=bI  and  JIJ=bJ makes this into a finite dimensional algebra (see the work of Rittenberg, Nichols, de Gier and Pyatov.)

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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