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

ISBN-10: 0817644717

ISBN-13: 9780817644710

ISBN-10: 0817645322

ISBN-13: 9780817645328

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

These ten unique articles by way of well-known mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, extensively 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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Probably the most inventive mathematicians of our instances, Vladimir Drinfeld bought the Fields Medal in 1990 for his groundbreaking contributions to the Langlands software and to the speculation of quantum teams. those ten unique articles through famous mathematicians, devoted to Drinfeld at the social gathering of his fiftieth birthday, widely mirror the diversity of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity thought.

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This quantity consists of invited expository articles via famous mathematicians in differential geometry and mathematical physics which were prepared in party of Hideki Omori's fresh retirement from Tokyo collage of technology and in honor of his basic contributions to those parts.

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

**Example text**

Any element D of W is thus a word µ1 · · · µl(D) in the letters from the alphabet ∪ − , where l(D) is its length. For α ∈ we shall denote by α¯ the opposite element from − . 4 The braid and Hecke semigroups Let B be the quotient of the semigroup W by αβα = βαβ αβαβ = βαβα αβαβαβ = βαβαβα ¯ α β¯ = βα, and α¯ β¯ α¯ = β¯ α¯ β¯ and α¯ β¯ α¯ β¯ = β¯ α¯ β¯ α¯ ¯ and α¯ β α¯ β¯ α¯ β¯ = β¯ α¯ β¯ α¯ β¯ α¯ (1) if Cαβ = Cβα = −1, if Cαβ = 2Cβα = −2, (2) if Cαβ = 3Cβα = −3. The semigroup B is called the braid semigroup.

One has H α (x) = exp(log(x)hα ). Let us introduce the group elements Eα = exp eα , Fα = exp fα . Replace the letters in D by the group elements using the rule α → Eα , α¯ → α F . For α ∈ and for any (αi ) ∈ J (D), insert H α (xiα ) somewhere between the corresponding walls. The choice in placing every H is nonessential since it commutes with all Es and F s unless they are marked by the same root. In other words the sequence of group elements is deﬁned by the following requirements: • • • • The sequence of Es and Fs reproduce the sequence of letters in the word D.

Moreover, for every γ ∈ , the subset of γ -decorated elements of J (D) has a natural linear order, induced by the ones on J (αi ), and the linear order of the word D. 2. , minimal and maximal) elements for the deﬁned above linear order on the γ -decorated part of J (D). The seed J(D) is obtained from the seed J(D) by reducing J0 (D) to the subset J0 (D). Observe that εαβ is integral unless both α and β are in J0 (D). Thus the integrality condition for εαβ holds. 3 An alternative deﬁnition of the seed J(D) The sets J0 (D) and J (D) Given a positive simple root α ∈ α and α¯ in the word D.

### Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

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