Lecture 11- Calogero-Moser system and Hamiltonian reduction

Lecture 11- Calogero-Moser system and Hamiltonian reduction...

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LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 11. Calogero-Moser system and Hamiltonian reduction 11.1. Calogero-Moser system. The Calogero-Moser system is the system of n distinct points of the same mass, say 1, on the line (we work over C so we consider the complex line) with coordinates x 1 , . . . , x n that interact with pairwise potentials of the form c ( x i x j ) 2 , where c is some nonzero constant. We can rescale and assume that c = 1. The total potential is V = i<j 1 ( x i x j ) 2 . We want to consider the corresponding Hamiltonian system but we first need to decide what will be the symplectic variety to “accommodate” the system. Our original configuration space is ( C n ) Reg := { ( x 1 , . . . , x n ) | x i ̸ = x j , i ̸ = j } . So we could take the variety X := T ( C n ) Reg and consider the Hamiltonian H = 1 2 n i =1 y 2 i i<j 1 ( x i x j ) 2 . However, there is a better choice. Our points are indistinguishable and so we can view ( x 1 , . . . , x n ) as an unordered n -tuple. So the configuration space is ( C n ) Reg / S n and we consider its cotangent bundle X := T ( C n ) Reg / S n . Also S n acts naturally on T ( C n ) Reg , the action is induced from ( C n ) Reg and hence preserves the symplectic form. As the following exercise shows T (( C n ) Reg / S n ) = ( T ( C n ) Reg ) / S n . So we can view H (that is an S n -invariant function) as a function on X . This is a Hamiltonian of our system. Below we will write C Reg instead X . We will see that it is still not the best possible phase space for our system: we can actually embed C Reg to some symplectic affine variety C (Calogero-Moser space) to “avoid collisions”. Exercise 11.1. Let X 0 be a smooth algebraic variety equipped with a free action of a finite group Γ . Show that T ( X 0 / Γ) is naturally identified with ( T X 0 ) / Γ (an isomorphism of symplectic varieties). Let us explain what kind of results regarding the Calogero-Moser (CM) system we want to get. First of all, we want to describe the trajectories, as explicitly as possible. Second, we want to produce 1st integrals subject to certain conditions. Namely, we want 1st integrals H 1 , . . . , H n with H 2 = H such that { H i , H j } = 0 for all i, j and d x H 1 , . . . , d x H n being lin- early independent for a general point x X . By some general results of Algebraic geometry (generic smoothness) the last condition is equivalent to H 1 , . . . , H n being algebraically inde- pendent. As the following exercise shows, here n is the maximal possible number for which such first integral may exist. Exercise 11.2. Let f 1 , . . . , f m be functions on a symplectic variety X such that { f i , f j } = 0 for all i, j . Show that the dimension of the span of d x f 1 , . . . , d x f m has dimension not exceeding 1 2 dim X . Systems admitting such collection of functions H 1 , . . . , H n (with Hamiltonian H being one of them) are called completely integrable . The reason is the Arnold-Liouville theorem that roughly states that (well, under some additional assumptions, and in C -setting) such 1
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2 IVAN LOSEV systems can be explicitly integrated. As we can explicitly integrate the system under con- sideration without using that theorem, we want to skip the details.
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