Algebraic Geometry and Number Theory: In Honor of Vladimir by victor ginzburg

By victor ginzburg

The most inventive mathematicians of our occasions, Vladimir Drinfeld obtained 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 renowned mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, largely mirror the diversity of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity idea.

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Thus the obtained set J (D) has a natural decoration π : J (D) −→ , extending those of the subsets J (αi ). 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).

Cluster automorphisms form a group called the mapping class group. The groups of cluster automorphisms of equivalent seeds are isomorphic. 3. Cluster transformations preserve the Poisson structure. In particular a cluster X -manifold has a canonical Poisson structure and the automorphism group of this manifold acts on it by Poisson transformations. 4. Cluster transformations are given by rational functions with positive integral coefficients. 5. If εij = εj i = 0, then µi µj µj µi = id. ∼ 6. If εij = −εj i = −1, then µi µj µi µj µi = id.

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 ⎠ . 2n + 1 We claim that the operator W[M] is a multiple of a unitary operator. Indeed, ⎛ ⎞ (W [M] ∗ −1 ) ⎜ = exp ⎝− M−1 2 n=0 1 ⎟ [M] ⎠W , 2n + 1 whence the result.

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