Communications in Mathematical Physics - Volume 301 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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36(6), 1108–1141 (1995) 32. : Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998) Communicated by P. Constantin Commun. Math. Phys. A. A. edu Received: 13 March 2009 / Accepted: 18 July 2010 Published online: 7 October 2010 – © The Author(s) 2010. com Abstract: Let M be a smooth, simply-connected, closed oriented manifold, and L M the free loop space of M. Using a Poincaré duality model for M, we show that the reduced equivariant homology of L M has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.

Proof of Theorem 1. (i) If (L , d) is a DG Lie algebra, then its universal enveloping U (L) with the induced differential is a DG Hopf algebra; denote the induced differential by b. 1]). Therefore, by Theorems 9 and 15, H∗ (A, b) quantizes the Lie bialgebra H C∗ (C)[m − 1]. (ii) This is immediate from Theorem 5 and Theorem 1 (i) . Acknowledgements. The first author would like to thank Professor Yongbin Ruan for his encouragement during the preparation of this paper. The third author was partially supported by NSF grant DMS-0726154.

3. Proof of DG coalgebra. For an element X in the form of (35), let P := PX := {(i, j) | 1 ≤ i ≤ k, 1 ≤ j ≤ pi }. If (i, j) ∈ P, we let (i, j) + (0, 1) = (i, j + 1) if j < pi , (i, 1) if j = pi . Let n be an integer greater than or equal to 2. Now let I be any subset of P such that #I is even, and let φ : I → I be an involutive, fixed point-free map, where by being involutive we mean φ 2 = id. We call (I, φ, f ) an n-labeling of X if f : P → {1, 2, . . , n} is a map such that: f (i, j) = f ((i, j) + (0, 1)), if (i, j) ∈ / I; f (φ(i, j) + (0, 1)), if (i, j) ∈ I, (56) and f (i, j) > f (φ(i, j)) if and only if h i, j > h φ(i, j) , for (i, j) ∈ I.

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