Lecture 1- Kleinian Singularities

Lecture 1- Kleinian Singularities - LECTURES ON SYMPLECTIC...

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LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 1. Kleinian singularities Kleinian singularities are remarkable singular affine surfaces (varieties of dimension 2). They arise as quotients of C 2 by finite subgroups of SL 2 ( C ). Our main interest is not in these singularities themselves but in their (not necessarily commutative) deformations. This lecture is organized as follows. First, 1.1, we present Kleinian singularities as surfaces in C 3 . Next, we recall the classification of finite subgroups of SL 2 ( C ) in 1.2. In 1.3 we present the simplest version of the so called McKay correspondence that relates finite subgroups of SL 2 ( C ) to Dynkin diagrams. Then we relate the singularities and the subgroups as promised in the previous paragraph. One advantage of this realization, is that the singularities acquire a natural grading . We discuss graded algebras in 1.5. After that we start to proceed to our next topic and discuss the general notion of a deformation. Finally, we sketch a purely algebro-geometric way to connect the Kleinian singularities to Dynkin diagrams, 1.7. For more information on Kleinian singularities (and, in particular, their relation to simple Lie algebras) see [Sl], Section 6, in particular. 1.1. Singularities. There are some remarkable singular affine algebraic varieties of dimen- sion 2. They have many names (Kleinian singularities, rational double points, du Val singu- larities) and also many nice properties (e.g., these are only normal Gorenstein singularities in dimension 2). They can be described very explicitly, as surfaces in C 3 given by a single equation on the variables x 1 , x 2 , x 3 . They split into two families and three exceptional types. Here are the equations ( A r ) x r +1 1 + x 2 x 3 = 0, r > 1. ( D r ) x r - 1 1 + x 1 x 2 2 + x 2 3 , r > 4. ( E 6 ) x 4 1 + x 3 2 + x 2 3 = 0. ( E 7 ) x 3 1 x 2 + x 3 2 + x 2 3 = 0. ( E 8 ) x 5 1 + x 3 2 + x 2 3 = 0. Of course, A r , D r , E 6 , E 7 , E 8 are precisely the simply laced Dynkin diagrams. In a way, this and three subsequent lectures are to explain relationship between the singularities and the diagrams. 1.2. Finite subgroups of SL 2 ( C ) . It turns out that the Kleinian singularities can be real- ized as quotients of C 2 by finite subgroups of SL 2 ( C ). Let us recall their classification. Any finite subgroup in SL 2 ( C ) admits an invariant hermitian product on C 2 and so is conjugate to a subgroup in SU 2 . Recall that there is a covering SU 2 SU 2 / E } = SO 3 ( R ) given by the adjoint representation of SU 2 . So the first step in classifying finite subgroups of SU 2 is to classify those in SO 3 ( R ). Inside SO 3 ( R ) we have the following finite subgroups: (1) The cyclic group of order n , its generator is a rotation by the angle of 2 π/n . 1
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2 IVAN LOSEV (2) The dihedral group of order 2 n with n > 2 realized as the group of rotation sym- metries of a regular n -gon on the plane inside of the 3D space. Of course, a regular 2-gon is just a segment.
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