Algebraic Geometry and Number Theory: In Honor of Vladimir by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

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

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

These ten unique articles through well-known mathematicians, devoted to Drinfeld at the party of his fiftieth birthday, largely replicate the diversity 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.

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Extra info for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday

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The bound of degree in d ensures that this transition preserves weight. This concludes the proof of Theorem 2. 1 Consider the operator 14 Alex Eskin and Andrei Okounkov W = exp n>0 α−2n−1 2n + 1 exp − n>0 α2n+1 . 2n + 1 (26) Because this operator is normally ordered, its matrix elements (Wv, w) are well defined for any vectors v and w of finite energy. The relevance of this operator for our purposes lies in the following. Theorem 4. The diagonal matrix elements of W are (Wvλ , vλ ) = w(λ), λ is balanced, 0 otherwise.

13] M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in The Moduli Space of Curves (Texel Island , 1994), Progress in Mathematics, Vol. 129, Birkhäuser Boston, Cambridge, MA, 1995, 165–172. Pillowcases and quasimodular forms 25 [14] S. Kerov and G. Olshanski, Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. , 319-2 (1994), 121–126. [15] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm.

8. Cluster X -varieties, amalgamation, and Poisson–Lie groups 45 It is useful to recall an explicit description of the amalgamation map XJ(A) × XJ(B) → XJ(AB) . Let {xiα }, {yiα } and {ziα } be the coordinates on XJ(A) , XJ(B) and XJ(AB) , respectively. Then the map is given by the formula: ⎧ α ⎪ ⎨xi α zi = xnαα (A) y0α ⎪ ⎩ α yi+nα (A) if i < nα (A), if i = nα (A), if i > nα (A). The crucial point is that, just by the construction, this map is compatible with the evaluation map ev to the group.

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