Categorification of tensor powers of the vector by Antonio Sartori

By Antonio Sartori

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The elements {vηa | η ∈ {0, 1} } are called the standard basis vectors of V(a). 8) vη♦a = vηa11 ♦ · · · ♦vηa . The bilinear form Fix a sequence of positive numbers a = (a1 , . . , a ). 9) (vηa , vγa )a = q β1η + · · · + β η γ1 δη1 · · · δηγ β1η , . . 10) and h1 + · · · + h h1 , . . 11) η = [h1 + · · · + h ]! [h1 ]! · · · [h ]! 9) is a polynomial in q with constant term 1. 13) [h]0 ! 15) a+b a h1 + · · · + h h1 , . . , h h(h−1) 2 a+b a = q ab 0 =q [h]! i=j hi hj 0 h1 + · · · + h . h1 , . .

0, 1} we Chapter 3. Graphical calculus for Uq (gl(1|1))–representations 27 let vηa = vηa11 ⊗ · · · ⊗ vηa . The elements {vηa | η ∈ {0, 1} } are called the standard basis vectors of V(a). 8) vη♦a = vηa11 ♦ · · · ♦vηa . The bilinear form Fix a sequence of positive numbers a = (a1 , . . , a ). 9) (vηa , vγa )a = q β1η + · · · + β η γ1 δη1 · · · δηγ β1η , . . 10) and h1 + · · · + h h1 , . . 11) η = [h1 + · · · + h ]! [h1 ]! · · · [h ]! 9) is a polynomial in q with constant term 1. 13) [h]0 !

3. For n ≥ 1, the Super Temperley-Lieb Algebra STLn is the unital associative C(q)–algebra generated by {Ci | i = 1, . . 9e) ((q + q −1 ) − Ci−1 )((q + q −1 ) − Ci+1 )Ci Ci−1 Ci+1 = 0. Since the first three relations are just the relations that the generators Ci = Hi + q satisfy in the Hecke algebra, it follows that STLn is a quotient of Hn . Moreover, by the discussion above, we have STLn ∼ = EndUq (V ⊗n ). 12) (V ⊗n )k = {v ∈ V ⊗n | qh v = q h,kε1 +(n−k)ε2 v}. Clearly, every weight space is a module for the Hecke algebra.

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