Lecture 10- Moment maps in the algebraic setting

Lecture 10- Moment maps in the algebraic setting - LECTURES...

Info icon This preview shows pages 1–2. Sign up to view the full content.

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 10. Moment maps in algebraic setting 10.1. Symplectic algebraic varieties. An affine algebraic variety X is said to be Poisson if C [ X ] is equipped with a Poisson bracket. Exercise 10.1. Let A be a commutative algebra and B be a localization of A . Let A be equipped with a bracket. Show that there is a unique bracket on B such that the natural homomorphism A B respects the bracket. Thanks to this exercise, the sheaf O X of regular functions on X acquires a bracket (i.e., we have brackets on all algebras of sections and the restriction homomorphisms are compatible with the bracket). We say that an arbitrary (=not necessarily affine) variety X is Poisson if the sheaf O X comes equipped with a Poisson bracket. Recall that on a variety X such that O X is equipped with a bracket we have a bivector (=a bivector field) P Γ( X reg , 2 TX reg ). This gives rise to a map v x : T x X T x X for x X reg , α 7→ P x ( α, · ). We say that P is nondegenerate in x if this map is an isomorphism. In this case, we can use this map to get a 2-form ω x 2 T x X : ω x ( v x ( α ) , v x ( β )) = P x ( α, β ) = α, v x ( β ) = −⟨ v x ( α ) , β . Now suppose X is smooth. Suppose that P is non-degenerate (=non-degenerate at all points). So we have a non-degenerate form ω on X . The condition that P is Poisson is equivalent to = 0. A non-degenerate closed form ω is called symplectic (and X is called a symplectic variety ). The most important for us class of symplectic varieties is cotangent bundles. Let X 0 be a smooth algebraic variety, set X := T X 0 . A symplectic form ω on X is introduced as follows. First, let us introduce a canonical 1-form α . We need to say how α x pairs with a tangent vector for any x X . A point X can be thought as a pair ( x 0 , β ), where x 0 X 0 and β T x 0 X 0 . Consider the projection π : X X 0 (defined by π ( x ) = x 0 ). For x = ( x 0 , β ) we define α x by α x , v = β, d x π ( v ) . We can write α in “coordinates”. If we worked in the C - or analytic setting, we could use the usual coordinates. However, we cannot do this because we want to show that α is an algebraic form. So we will use an algebro-geometric substitute for coordinate charts: ´ etale coordinates. Namely, we can introduce ´ etale coordinates in a neighborhood of each point x 0 X 0 . Let us choose functions x 1 , . . . , x n with a property that d x 0 x 1 , . . . d x 0 x n form a basis in T x 0 X 0 . Then dx 1 , . . . , dx n are linearly independent at any point from some neighborhood X 0 0 of x 0 . So the map φ : X 0 0 C n given by ( x 1 , . . . , x n ) is ´ etale and we call x 1 , . . . , x n ´ etale coordinates. Then we can get ´ etale coordinates y 1 , . . . , y n on T X 0 0 as follows: by definition y i ( x 0 , β ) is the coefficient of d x 0 x i in β , i.e., β = n i =1 y i ( x 0 , β ) d x 0 x i (and we view x 1 , . . . , x n as functions on T X 0 0 via pull-back). Then, on T X 0 0 , α is given by n i =1 y i dx i .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 2
This is the end of the preview. Sign up to access the rest of the document.
  • Fall '12
  • IvanLosev
  • Algebra, Hamiltonian mechanics, x0, Symplectic manifold, Symplectic geometry

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern