By Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

One of the main inventive mathematicians of our occasions, Vladimir Drinfeld got the Fields Medal in 1990 for his groundbreaking contributions to the Langlands software and to the speculation of quantum groups.

These ten unique articles through well known mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, commonly replicate the variety of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity theory.

Contributors: A. Eskin, V.V. Fock, E. Frenkel, D. Gaitsgory, V. Ginzburg, A.B. Goncharov, E. Hrushovski, Y. Ihara, D. Kazhdan, M. Kisin, I. Krichever, G. Laumon, Yu.I. Manin, A. Okounkov, V. Schechtman, and M.A. Tsfasman.

**Read Online or Download Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday PDF**

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The images of evα , evα¯ , ev∅ and the union of the images of evαα ¯ and evα α¯ are pairwise disjoint. The complement to their union in the whole group is of codimension two. It consists of the antidiagonal matrices. 7. The images of evα , evα¯ , ev∅ are Poisson subvarieties with respect to the standard Drinfeld–Jimbo Poisson–Lie structure on PGL(2, C). The Poisson bracket is given by {z0 , z1 } = z0 z1 ; {w0 , w1 } = w0 w1 . In the above constructions one can replace C× by Gm , PGL(2, C) by the group scheme PGL(2), and upgrade all maps to the maps of the corresponding schemes.

Xn ) = 1 Z(∅, ∅; q) (29) × [y10 · · · yn0 ] tr q H ψ(x1 y1 )ψ ∗ (y1 ) · · · ψ(xn yn )ψ ∗ (yn )W. 2 By the main result of [9], we have w(λ) ≤ 1 (30) for any partition λ. In other words, all diagonal matrix elements of W are bounded by 1. For the off-diagonal elements, we prove the following cruder bound. Proposition 3. Let M = max{|λ|, |µ|}. Then ⎛ ⎜1 (Wvλ , vµ ) ≤ exp ⎝ 2 M−1 2 n=0 ⎞ 1 ⎟ 1/4 ⎠ ∼ const · M . 2n + 1 (31) To see this note that (Wvλ , vµ ) = (W[M] vλ , vµ ), where W[M] is the truncated operator ⎛ ⎞ ⎛ α−2n−1 ⎠ exp ⎝ exp ⎝− 2n + 1 2n+1≤M ⎞ 2n+1≤M α2n+1 ⎠ .

If Cαβ = −3, Cβα = −1, then αα → α. αβ → βα. αβα → βαβ. Eα H α (x)Eβ H β (y)Eα H α (z)Eβ H β (w)Eα Eβ = H β (a )H α (b )Eβ H β (y )Eα H α (x )Eβ H β (w )Eα H α (z )Eβ H β (q )Eα H α (p ), (13) where a , b , x , y , z , w , p , q are rational functions of x, y, z, w given by (24). Further, one has αα ¯ → α α. ¯ ⎛ Fα H α (x)Eα = ⎝ β=α ⎞ H β (1 + x)−Cαβ⎠H α (1+x −1 )−1 Eα H α (x −1 )Fα H α (1+x −1 )−1 . 46 V. V. Fock and A. B. Goncharov αβ ¯ → β α. ¯ Fα Eβ = Eβ Fα if α = β. Applying the antiautomorphism of g which acts as the identity on the Cartan subalgebra, and interchanges Fα and Eα , we obtain similar formulas for α¯ α¯ → α, ¯ α¯ β¯ → β¯ α, ¯ α¯ β¯ α¯ → β¯ α¯ β¯ and α¯ β¯ α¯ β¯ → β¯ α¯ β¯ α.