Outline. 1 Geometry and Commutative Algebra. 2 Singularities and Resolutions. 3 Noncommutative Algebra and Deformations. 4 Representation Theory

Transcription

1 Outline Geometry, noncommutative algebra and representations Iain Gordon igordon/ University of Edinburgh 16th December Iain Gordon Geometry, noncommutative algebra and representations 2 Iain Gordon Geometry, noncommutative algebra and representations Let k be a field. (Affine) algebraic geometry studies solutions of systems of polynomial equations with coefficients in k. For example, two elliptic curves: What s the point of studying polynomial equations? We can collect together the information included a system of polynomial equations, say f 1 (x 1,..., x n ),, f r (x 1,..., x n ), into the ideal of the polynomial ring k[x 1,..., x n ] generated by f 1,..., f r. The subsets of k n consisting of commons zero of polynomials in an ideal are called closed algebraic sets. {closed algebraic sets in k n } {ideals of k[x 1,..., x n ]} Obviously, this is sensitive to the choice of k. This is a fundamental idea across a lot of mathematics: analysis, algebra, geometry, number theory (to name four!) 3 Iain Gordon Geometry, noncommutative algebra and representations 4 Iain Gordon Geometry, noncommutative algebra and representations

2 This idea sets up a geometry algebra dictionary. Of course, different types of geometry (smooth, analytic,...) lead to different types of algebra; we re focusing on algebraic geometry. {closed algebraic sets X} 1 1 {(radical) ideals I(X)} functions on X 1 1 C[X] := k[x 1,..., x n ]/I (if k is algebraically closed) (a 1,..., a n ) X 1 1 maximal ideals (x 1 a 1,..., x n a n ) C[X] The last property shows that the points and the topology of X can be defined algebraically in terms of the spectrum of the ring of functions C[X], written SpecC[X]. Algebraic geometry is flexible enough to allow techniques of complex geometry when k = C and to be of use to number theory when k is a finite field. 5 Iain Gordon Geometry, noncommutative algebra and representations 6 Iain Gordon Geometry, noncommutative algebra and representations We also need to introduce projective space P n (k) = {(x 0,, x n ) k n+1 \ {0}}/ where (x 0,..., x n ) (y 0,..., y n ) if (x 0,..., x n ) = λ(y 0,..., y n ) for some λ k \ {0}. Replace k n by P n (k), then apply the previous constructions to get closed projective sets. Why? When k = C the points of P n (k) are compact in the usual complex topology. It s useful to notice that projective spaces are covered by copies of affine space, k n : given i between 0 and n k n {(x0 : : x n ) : x i 0} P n (k) So we can define local properties on affine varieties and then apply them to projective varieties. In general we call these spaces varieties. Counting (i.e. intersections) arguments work properly. Many invariants are finite. 7 Iain Gordon Geometry, noncommutative algebra and representations 8 Iain Gordon Geometry, noncommutative algebra and representations

3 We want to define when a variety is smooth (an analogue of a manifold). Let X k n be an affine variety defined by the system of polynomials f 1,..., f r. If p = (a 1,..., a n ) is a point of X then the affine subspace of k n given by the system of linear equations in T 1,..., T n n j=1 f i T j (p)(t j a j ) = 0, i = 1,..., r, is called the tangent space to X at p, denoted T X,p. This definition depends only on partial derivatives, so is local. Hence if X is any variety and p X, then we can define the tangent space to X at p. There is a typical value for dim T X,p, which we call d. A point p X is called singular if dim T X,p > d. A variety X is singular if it contains a singular point. Otherwise X is non-singular or smooth (of dimension d). In algebra this corresponds to finite homological dimension. 9 Iain Gordon Geometry, noncommutative algebra and representations 10 Iain Gordon Geometry, noncommutative algebra and representations Here is an example. Let G = {±1}. This acts on V = C 2 by multiplication and hence on C[V ] = C[x 1, x 2 ], the coordinate ring of V. We ask for polynomial functions which are fixed by all elements of the group: C[V ] G = C[x1 2, x 1x 2, x2 2 ] = C[a, b, c]/(ac b2 ). It s easy to check that this has a singularity at the origin. Smooth varieties over C = techniques from complex geometry applicable. Thus we would like to replace singular varieties by non-singular ones. 11 Iain Gordon Geometry, noncommutative algebra and representations 12 Iain Gordon Geometry, noncommutative algebra and representations

4 If we do this geometrically we should find a mapping Y π X which is as efficient as possible for Y being smooth. This means we should replace only singular points - π 1 (X sm ) = X sm - and we should do it with projective fibres π 1 (x). When k = C a theorem of Hironaka proves such resolutions of singularities always exist, but there may be very many different ones. There is however a notion which measures how large" a resolution is: this is called the discrepancy. When the discrepancy is zero, we say we have a crepant resolution. It s important to know: Given a variety X, does it have a crepant resolution? 13 Iain Gordon Geometry, noncommutative algebra and representations 14 Iain Gordon Geometry, noncommutative algebra and representations C[V ] G is easy to describe. Recall C[V ] = C[x 1, x 2 ] and If we try to replace the singularities algebraically we should find a mapping C[X] π R which is as efficient as possible for a ring R with finite homological dimension. In our example there s always a canonical choice: the skew group ring C[V ] G. G = {±1} = {id, σ}. Then C[V ] G = C[x 1, x 2 ]id C[x 1, x 2 ]σ with multiplication p(x 1, x 2 )g q(x 1, x 2 )h = p(x 1, x 2 ) gq(x 1, x 2 )gh. It has finite homological dimension. Sitting inside it is C[V ] G C[V ]id. 15 Iain Gordon Geometry, noncommutative algebra and representations 16 Iain Gordon Geometry, noncommutative algebra and representations

5 Something amazing has happened: C[V ] G is noncommutative! For instance σ x 1 = σ x 1 σ = x 1 σ In fact Z (C[V ] G) := {z C[V ] G : zr = rz for all r C[V ] G} = C[V ] G Are these related? The problem is that C[V ] G sees everything so it s vanilla ice-cream. It s hard to get any information. On the other hand it s difficult to construct crepant resolutions. To get round the first problem, we use deformation theory. 17 Iain Gordon Geometry, noncommutative algebra and representations 18 Iain Gordon Geometry, noncommutative algebra and representations What is a deformation? Take a ring R with multiplication written r s. Form the polynomial ring R[[t]] consisting of power series r i t i. i 0 (Observe that R R[[t]] is the subspace of constant power series.) A deformation of R is a ring structure on R[[t]] which is t-linear and such that Sometimes the power series appearing may be a polynomial; then we could set t = 1. An example is C[x 1, x 2 ] = C x 1, x 2 : x 1 x 2 x 2 x 1 = 0 : it has a deformation with t = 1 called the first Weyl algebra C x 1, x 2 : x 1 x 2 x 2 x 1 = 1 Such behaviour is pretty typical: deformations are less commutative. r s = r s + e 1 (r, s)t + e 2 (r, s)t Iain Gordon Geometry, noncommutative algebra and representations 20 Iain Gordon Geometry, noncommutative algebra and representations

6 Let s go to the situation we re going to study from now on. G finite group (V, ω) a complex symplectic vector space: ω : V V C bilinear form which is anticommutative: ω(v, w) = ω(w, v) non-degenerate: ω(v, V ) = 0 v = 0 G acts linearly on V preserving ω, i.e. G GL(V ) and ω( g v, g w) = ω(v, w), i.e. G Sp(V ). 21 Iain Gordon Geometry, noncommutative algebra and representations Does the variety corresponding to C[V ] G admit a crepant resolution of singularities? This variety is V /G is the space of orbits of G acting on V ({g v : g G}). It s an open problem to describe C[V ] G, and even in known special cases the description of this ring is hard to work with. Some physicists say The resolution is the deformation : so we should deform! Deform C[V ] G (a simple object to describe) and hope that C[V ] G is deformed simultaneously! 22 Iain Gordon Geometry, noncommutative algebra and representations The resolution is the deformation was made precise in a recent paper. Theorem (Ginzburg Kaledin, 2004) Suppose V /G admits a crepant resolution. Then there is a commutative deformation of C[V ] G (the polynomial functions on V /G) which is smooth. So to show V /G doesn t have a crepant resolution we need only show that all commutative deformations of C[V ] G are singular. 23 Iain Gordon Geometry, noncommutative algebra and representations Deformations of C[V ] G were constructed earlier. Definition (Etingof-Ginzburg, 2001) Let V have basis {x 1,..., x n }. Then deformations of C[V ] G have the form H t,c = C x 1,..., x n G x i x j x j x i κ t,c (x i, x j ) : 1 i, j n where κ t,c (x i, x j ) = tω(x i, x j ) + s S c(s)ω s(y, x)s. S = {g G : rank(id V g) = 2} t C and c : S C satisfies c(gsg 1 ) = c(s) for all g G, s S. 24 Iain Gordon Geometry, noncommutative algebra and representations

7 These algebras are called symplectic reflection algebras. They have become increasingly important in representation theory since their discovery with applications to algebra, geometry, combinatorics, differential equations,... We concentrate on t = 0 (t 0 behaves very differently). Let Z c = {z H 0,c : zr = rz for all r H 0,c }. We already know that Z 0 = Z (C[V ] G) = C[V ] G : the algebras Z c are the deformations of C[V ] G we are looking for. From one point of view, representation theory is the study of the action of rings on modules, i.e. on vector spaces or maybe just abelian groups. A fundamental problem is to describe the irreducible representations, the atoms of representation theory. For example, you may have seen group representations: that s just CG-modules. Or maybe representations of Lie algebras: that s U(g)-modules. In the first case there are only finitely many irreducible representations, in the second case infinitely many. 25 Iain Gordon Geometry, noncommutative algebra and representations 26 Iain Gordon Geometry, noncommutative algebra and representations We will describe the structure of Z c by describing the exotic example of H 0,c -representations. How do we describe the irreducible H 0,c -representations? Schur s Lemma! If M is an irreducible representation of H 0,c then this lemma says Z c acts by scalar multiplication, or equivalently that there is a unique maximal ideal of Z c corresponding to M. χ : {Irred. reps. of H 0,c } Max Z c It turns out all irreducible H 0,c -representations are finite dimensional vector spaces over C. There s a great theorem proved over a period of 25 years which then applies here. Theorem (Artin Procesi, LeBruyn, Brown Goodearl) There is an upper bound on the dimension of the irreducible H 0,c -representations. Furthermore χ 1 ((Max Z c ) sm ) = {Irred. reps of maximal dimension}. If Max Z c is singular H 0,c must have small representations. 27 Iain Gordon Geometry, noncommutative algebra and representations 28 Iain Gordon Geometry, noncommutative algebra and representations

8 Summary Using a mixture of representation theory and the combinatorics surrounding Lie groups it is possible to define and work with a class of representations for H 0,c called baby Verma modules which have particularly nice properties. Theorem (Gordon, 2003) Description of groups G for which all Max Z c are singular. Corollary Description of groups G < Sp(V ) for which the orbit space V /G admits a crepant resolution. The dictionary: algebraic geometry commutative ring theory. Quantisation: commutative ring theory is too narrow; noncommutative structures are often required. Representations: noncommutative structures are rigid; representations provide very rich information. This leads to noncommutative geometry, and in this case deep links to algebraic geometry, differential equations, combinatorics,...and lots of beautiful representation theory. 29 Iain Gordon Geometry, noncommutative algebra and representations 30 Iain Gordon Geometry, noncommutative algebra and representations