Geometry, noncommutative algebra and representations

Outline

Geometry, noncommutative algebra and 1

Geometry and Commutative Algebra

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Singularities and Resolutions

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Noncommutative Algebra and Deformations

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Representation Theory

representations Iain Gordon http://www.maths.ed.ac.uk/˜igordon/ University of Edinburgh

16th December 2006

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Geometry, noncommutative algebra and representations

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Let k be a field. (Affine) algebraic geometry studies solutions of

What’s the point of studying polynomial equations?

systems of polynomial equations with coefficients in k .

We can collect together the information included a system of

For example, two elliptic curves:

polynomial equations, say f1 (x1 , . . . , xn ), · · · , fr (x1 , . . . , xn ), into the ideal of the polynomial ring k [x1 , . . . , xn ] generated by f1 , . . . , fr . 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 [x1 , . . . , xn ]} This is a fundamental idea across a lot of mathematics:

Obviously, this is sensitive to the choice of k . 3

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analysis, algebra, geometry, number theory (to name four!) 4

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This idea sets up a geometry–algebra dictionary. Of course, different types of geometry (smooth, analytic, ...) lead to

The last property shows that the points and the topology of X

different types of algebra; we’re focusing on algebraic

can be defined algebraically in terms of the spectrum of the ring

geometry.

of functions C[X ], written SpecC[X ]. 1−1

{closed algebraic sets X } ←→ {(radical) ideals I(X )} Algebraic geometry is flexible enough to allow techniques of

1−1

functions on X ←→ C[X ] := k [x1 , . . . , xn ]/I

complex geometry when k = C and to be of use to number

(if k is algebraically closed)

theory when k is a finite field.

1−1

(a1 , . . . , an ) ∈ X ←→ maximal ideals (x1 −a1 , . . . , xn −an ) ⊂ C[X ]

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We also need to introduce projective space Pn (k ) = {(x0 , · · · , xn ) ∈ k n+1 \ {0}}/ ∼

It’s useful to notice that projective spaces are covered by copies of affine space, k n : given i between 0 and n

where (x0 , . . . , xn ) ∼ (y0 , . . . , yn ) if (x0 , . . . , xn ) = λ(y0 , . . . , yn ) for some λ ∈ k \ {0}.

∼

k n −→ {(x0 : · · · : xn ) : xi 6= 0} ⊂ Pn (k )

Replace k n by Pn (k ), then apply the previous constructions to get closed projective sets. Why?

So we can define local properties on affine varieties and then

When k = C the points of Pn (k ) are compact in the usual

apply them to projective varieties.

complex topology.

In general we call these spaces varieties.

Counting (i.e. intersections) arguments work properly. Many invariants are finite. 7

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We want to define when a variety is smooth (an analogue of a

This definition depends only on partial derivatives, so is local.

manifold).

Hence if X is any variety and p ∈ X , then we can define the

Let X ⊂ k n be an affine variety defined by the system of

tangent space to X at p. There is a “typical value” for dim TX ,p ,

polynomials f1 , . . . , fr . If p = (a1 , . . . , an ) is a point of X then the

which we call d.

affine subspace of k n given by the system of linear equations in

A point p ∈ X is called singular if dim TX ,p > d.

T1 , . . . , Tn n X ∂fi (p)(Tj − aj ) = 0, ∂Tj

A variety X is singular if it contains a singular point. i = 1, . . . , r ,

Otherwise X is non-singular or smooth (of dimension d).

j=1

In algebra this corresponds to finite homological dimension.

is called the tangent space to X at p, denoted TX ,p .

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Here is an example. Let G = {±1}. This acts on V = C2 by multiplication and hence on C[V ] = C[x1 , x2 ], the coordinate ring of V . We ask for polynomial functions which are fixed by all Smooth varieties over C =⇒ techniques from complex

elements of the group:

geometry applicable.

C[V ]G = C[x12 , x1 x2 , x22 ] = C[a, b, c]/(ac − b2 ).

Thus we would like to replace singular varieties by non-singular

It’s easy to check that this has a singularity at the origin.

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ones.

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If we do this geometrically we should find a mapping π

When k = C a theorem of Hironaka proves such resolutions of

Y −→ X

singularities always exist, but there may be very many different

which is as efficient as possible for Y being smooth. This

ones. There is however a notion which measures how “large" a

means we should replace only singular points - π −1 (Xsm ) ∼ = Xsm

resolution is: this is called the discrepancy. When the

- and we should do it with projective fibres π −1 (x).

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?

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Noncommutative Algebra and Deformations C[V ] ∗ G is easy to describe. Recall C[V ] = C[x1 , x2 ] and G = {±1} = {id, σ}. Then

If we try to replace the singularities algebraically we should find π

a mapping C[X ] −→ R which is as efficient as possible for a

C[V ] ∗ G = C[x1 , x2 ]id ⊕ C[x1 , x2 ]σ

ring R with finite homological dimension. with multiplication In our example there’s always a canonical choice: the skew

p(x1 , x2 )g · q(x1 , x2 )h = p(x1 , x2 ) · g q(x1 , x2 )gh.

group ring C[V ] ∗ G.

It has finite homological dimension. Sitting inside it is C[V ]G ⊆ C[V ]id. 15

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Something amazing has happened: C[V ] ∗ G is Are these related?

noncommutative! For instance

The problem is that C[V ] ∗ G sees “everything” so it’s

σ

σ · x1 = x1 · σ = −x1 · σ

vanilla ice-cream. It’s hard to get any information. On the other hand it’s difficult to construct crepant

In fact

resolutions. Z (C[V ] ∗ G) := {z ∈ C[V ] ∗ G : zr = rz for all r ∈ C[V ] ∗ G} =

To get round the first problem, we use deformation theory.

G

C[V ]

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What is a deformation? Take a ring R with multiplication written r · s.

Sometimes the power series appearing may be a polynomial;

Form the polynomial ring R[[t]] consisting of power series

then we could set t = 1.

X

ri t i .

An example is C[x1 , x2 ] = Chx1 , x2 : x1 x2 − x2 x1 = 0i: it has

i≥0

a deformation with t = 1 called the first Weyl algebra

(Observe that R ⊂ R[[t]] is the subspace of constant power

Chx1 , x2 : x1 x2 − x2 x1 = 1i

series.) A deformation of R is a ring structure ? on R[[t]] which is

Such behaviour is pretty typical: deformations are less

t-linear and such that

commutative.

r ? s = r · s + e1 (r , s)t + e2 (r , s)t 2 + · · · 19

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Noncommutative Algebra and Deformations Does the variety corresponding to C[V ]G admit a crepant

Let’s go to the situation we’re going to study from now on.

resolution of singularities?

G finite group

This variety is V /G is the space of orbits of G acting on V

(V , ω) a complex symplectic vector space: ω : V × V −→ C

({g · v : g ∈ G}).

bilinear form which is

It’s an open problem to describe C[V ]G , and even in known

anticommutative: ω(v , w) = −ω(w, v )

special cases the description of this ring is hard to work with.

non-degenerate: ω(v , V ) = 0 ⇐⇒ v = 0

Some physicists say “The resolution is the deformation”:

G acts linearly on V preserving ω, i.e. G ,→ GL(V ) and

so we should deform!

ω(g v , g w) = ω(v , w), i.e. G ,→ Sp(V ).

Deform C[V ] ∗ G (a simple object to describe) and hope that C[V ]G is deformed simultaneously! 21

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Noncommutative Algebra and Deformations Deformations of C[V ] ∗ G were constructed earlier.

“The resolution is the deformation” was made precise in a

Definition (Etingof-Ginzburg, 2001)

recent paper.

Let V have basis {x1 , . . . , xn }. Then deformations of C[V ] ∗ G

Theorem (Ginzburg–Kaledin, 2004)

have the form

Suppose V /G admits a crepant resolution. Then there is a Chx1 , . . . , xn i ∗ G hxi xj − xj xi − κt,c (xi , xj ) : 1 ≤ i, j ≤ ni P where κt,c (xi , xj ) = tω(xi , xj ) + s∈S c(s)ωs (y , x)s.

commutative deformation of C[V ]G (the polynomial functions on

Ht,c =

V /G) which is smooth. So to show V /G doesn’t have a crepant resolution we need

S = {g ∈ G : rank (IdV − g) = 2}

only show that all commutative deformations of C[V ]G are

t ∈ C and c : S −→ C satisfies c(gsg −1 ) = c(s) for all

singular.

g ∈ G, s ∈ S. 23

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These algebras are called symplectic reflection algebras. They

From one point of view, representation theory is the study of the

have become increasingly important in representation theory

action of rings on modules, i.e. on vector spaces or maybe just

since their discovery with applications to algebra, geometry,

abelian groups. A fundamental problem is to describe the

combinatorics, differential equations, ...

irreducible representations, the atoms of representation theory. For example, you may have seen group representations: that’s

We concentrate on t = 0 (t 6= 0 behaves very differently). Let

just CG-modules. Or maybe representations of Lie algebras: Zc = {z ∈ H0,c : zr = rz for all r ∈ H0,c }.

that’s U(g)-modules.

We already know that Z0 = Z (C[V ] ∗ G) = C[V ]G : the algebras

In the first case there are only finitely many irreducible

Zc are the deformations of C[V ]G we are looking for.

representations, in the second case infinitely many.

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It turns out all irreducible H0,c -representations are finite

example of H0,c -representations.

dimensional vector spaces over C. There’s a great theorem proved over a period of 25 years which then applies here.

How do we describe the irreducible H0,c -representations?

Theorem (Artin–Procesi, LeBruyn, Brown–Goodearl)

Schur’s Lemma! If M is an irreducible representation of H0,c

There is an upper bound on the dimension of the irreducible

then this lemma says Zc acts by scalar multiplication, or

H0,c -representations. Furthermore

equivalently that there is a unique maximal ideal of Zc

χ−1 ((Max Zc )sm ) = {Irred. reps of maximal dimension}.

corresponding to M.

If Max Zc is singular H0,c must have “small” representations.

χ : {Irred. reps. of H0,c } −→ Max Zc

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We will describe the structure of Zc by describing the exotic

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Summary

Using a mixture of representation theory and the combinatorics

The dictionary: algebraic geometry ←→ commutative ring

surrounding Lie groups it is possible to define and work with a

theory.

class of representations for H0,c called baby Verma modules

Quantisation: commutative ring theory is too narrow;

which have particularly nice properties.

noncommutative structures are often required.

Theorem (Gordon, 2003)

Representations: noncommutative structures are rigid;

Description of groups G for which all Max Zc are singular.

representations provide very rich information.

Corollary Description of groups G < Sp(V ) for which the orbit space V /G

This leads to noncommutative geometry, and in this case deep

admits a crepant resolution.

links to algebraic geometry, differential equations, combinatorics,...and lots of beautiful representation theory.

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