By Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)
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 via famous mathematicians, devoted to Drinfeld at the social gathering of his fiftieth birthday, extensively mirror 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.
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Additional info for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday
Let µk be the kth letter of D and let sgn(k) be +1 if µk ∈ and −1 otherwise. Let ﬁnally |µk | = sgn(µk )µk ∈ . 3. Let D be a word of W; then • • the multipliers are given by the rule d(αi ) (D) = d α ; the integers ε(α )(β ) are deﬁned by the formula i j ε(α )(β ) β j i (αi )(j ) = 1 2 ∂ ∂ ∧ ∂ log xiα ∂ log x β j l(D) sgn(µk )Cµ(k)α k=1 α ⎛ ∧⎝ ∂ |µ | ∂ log xn|µkk | (k)−1 − ∂ ∂ log xnαα (k) ∂ |µ | ∂ log xn|µkk | (k) ⎞ ⎠. 42 V. V. Fock and A. B. Goncharov Remark. One can check that in the case when D is reduced, our function εij is related to the cluster function bij deﬁned in [BFZ3] for the corresponding double Bruhat cell as follows.
4) We would like to stress that the multiplication m is a projection with ﬁbers of nonzero dimension. 6 Cluster X -varieties related to the Hecke semigroup Let π : B → H be the canonical projection of semigroups. Considered as a projection of sets it has a canonical splitting s : H → B. Namely, for every H ∈ H there is a unique reduced element s(H ) in π −1 (H ), the reduced representative of H in B. So given an element H ∈ H there is a cluster variety Xs(H ) . Abusing notation, we will denote it by XH .
Let us elaborate on this point. 2 Amalgamation We introduce operations of amalgamation and defrosting of seeds. The amalgamation of a collection of seeds I(s), parametrised by a set S, is a new seed K = (K, K0 , εij , d). The set K is deﬁned by gluing some of the frozen vertices of the sets I (s). The frozen subset K0 is obtained by gluing the frozen subsets I0 (s). The rest of the data of K is also inherited from those of I(s). Defrosting simply shrinks the subset of the frozen vertices of K, without changing the set K.
Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)