# Download Communications in Mathematical Physics - Volume 240 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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**Extra resources for Communications in Mathematical Physics - Volume 240**

**Example text**

Drinfel’d realisation. 2. 21) 2 0 ± ± f0 , f (u) f (u) 5 = − f02 , f ± (u) − 2u f02 , f ± (u) − f0 f ± (u)f0 − {f0 , f1 } , f ± (u) . 41), adding superscripts ± to e(u), f (u) and h(u). 1. 6), adding a superscript to the generating functions with parameter u, and a superscript to the generating functions with parameter v, where , = ±. 8) with are also valid in DA, but not sufficient, because they do not couple enough positive modes with the negative ones. 18) with, in this case, k, l ∈ Z and with the following additional relations, for k ∈ Z : 2[h−1 , ek ] − 2ek−1 = [h−1 , ek−2 ] − {h−1 , ek−1 }, 2[h−1 , fk ] − 2fk−1 = [h−1 , fk−2 ] + {h−1 , fk−1 }.

Z≥0 }. In addition, the pairing restricted of the subalgebras H− and H+ is not degenerated. 4. A corollary of the previous results is that the pairing between A− and A+ is not degenerated. Then, thanks to the isomorphisms φ ± and , neither is the pairing between Y − (R) and Y + (R). 4. 2). Thus, it is isomorphic, as a Hopf algebra, to the quantum double of Y (osp(1|2)), denoted DY (osp(1|2)). Similarly, A+ (R)⊗A− (R) is the quantum double of A+ (R). 48 D. Arnaudon, N. Cramp´e, L. Frappat, E. Ragoucy ˙ ± (u)⊗L ˙ ± (u), the cross-multiplication in Proof.

2) It is seen that ψ is non-decreasing and satisfies ψ( 21 ) = 1, ψ(1− ) = 0, and especially 1 ψ(x) dx = ζ (2) − 1 π2 1 = − . 3) 1/2 We can now state the main result. 1. For each t > 0, there exists H (t) = lim Hε (t). Moreover, one has ε→0+ 1 Hε (t) = H (t) + Oδ (ε 8 −δ ) for any (small) δ > 0, where 2t if 0 < t < 21 ; 1 − ζ (2) 1 2 H (t) = ψ(x) dx if 21 < t < 1; ζ (2) t 0 if t > 1. ˜ = h(t)dt and In particular µ˜ ε → µ˜ weakly as ε → 0+ , where d µ(t) 1 if 0 < t ≤ 21 ; 2 h(t) = −H (t) = · ψ(t) if 21 ≤ t ≤ 1; ζ (2) 0 if t ≥ 1.