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Home , Weyl algebra

The graded module category of a generalized Weyl algebra Final Defense

Robert Won Advised by: Daniel Rogalski

May 2, 2016 1 / 39 Overview

1 Graded rings and things

2 Noncommutative is not commutative

3 Noncommutative projective schemes

4 Z-graded rings The first Weyl algebra Generalized Weyl algebras

5 Future direction

May 2, 2016 2 / 39 Graded k-algebras

• Throughout, k an algebraically closed field, char k = 0. • A k-algebra is a A with 1 which is a k- such that

λ(ab) = (λa)b = a(λb) for λ ∈ k and a, b ∈ A.

Examples • The ring, k[x]

• The ring of n × n matrices, Mn(k) • The complex numbers, C = R · 1 + R · i

Graded rings and things May 2, 2016 3 / 39 Graded k-algebras

• Γ an abelian (semi)group • A Γ-graded k-algebra A has k-space decomposition M A = Aγ γ∈Γ

such that AγAδ ⊆ Aγ+δ.

• If a ∈ Aγ, deg a = γ and a is homogeneous of degree γ.

Graded rings and things May 2, 2016 4 / 39 Graded k-algebras

Examples • Any ring A with trivial grading • k[x, y] graded by N − • k[x, x 1] graded by Z • If you like physics: graded by Z/2Z • Or combinatorics: symmetric and alternating graded by Z/2Z

Remark Usually, “graded ring” means N-graded.

Graded rings and things May 2, 2016 5 / 39 Graded modules and morphisms

• A graded right A-module M: M M = Mi i∈Z

such that Mi · Aj ⊆ Mi+j. • Finitely generated graded right A-modules with graded module homomorphisms form a category gr-A.

Example N-graded ring k[x, y] • Homogeneous ideals (x), (x, x + y), (x2 − xy) are graded submodules • Nonhomogeneous ideals (x + 3), (x2 + y) are not

Graded rings and things May 2, 2016 6 / 39 Graded modules and morphisms

• Idea: Study A by studying its module category, mod-A (analogous to studying a group by its representations). • If mod-A ≡ mod-B, we say A and B are Morita equivalent.

Remarks • For commutative rings, Morita equivalent ⇔ isomorphic

• Mn(A) is Morita equivalent to A • Morita invariant properties of A: simple, semisimple, noetherian, artinian, hereditary, etc.

Graded rings and things May 2, 2016 7 / 39 Proj of a graded k-algebra

• A a commutative N-graded k-algebra • The scheme Proj A

• As a set: homogeneous primes ideals of A not containing A≥0 • Topological space: the Zariski topology with closed sets

V(a) = {p ∈ Proj A | a ⊆ p}

for homogeneous ideals a of A • Structure sheaf: localize at homogeneous prime ideals

Noncommutative is not commutative May 2, 2016 8 / 39 Proj of a graded k-algebra

Interplay beteween algebra and geometry

algebra geometry graded k-algebra A space Proj A homogeneous prime ideals points I ⊆ J V(I) ⊇ V(J) factor rings subschemes homogeneous elements of A functions on Proj A

Noncommutative is not commutative May 2, 2016 9 / 39 Noncommutative is not commutative

Noncommutative rings are ubiquitous.

Examples • Ring A, nonabelian group G, the group algebra A[G]

• Mn(A), n × n matrices • Differential operators on k[t], generated by t· and d/dt is isomorphic to the first Weyl algebra

A1 = khx, yi/(xy − yx − 1).

If you like physics, position and momentum operators don’t commute in .

Noncommutative is not commutative May 2, 2016 10 / 39 Noncommutative is not commutative

• Localization is different. • Given A commutative and S ⊂ A multiplicatively closed,

−1 −1 −1 −1 a1s1 a2s2 = a1a2s1 s2

• If A noncommutative, can only form AS−1 if S is an Ore set.

Definition S is an Ore set if for any a ∈ A, s ∈ S

sA ∩ aS 6= ∅.

Noncommutative is not commutative May 2, 2016 11 / 39 Noncommutative is not commutative Even worse, not enough (prime) ideals.

The Weyl algebra

khx, yi/(xy − yx − 1) is a noncommutative analogue of k[x, y] but is simple.

The quantum The N-graded ring

khx, yi/(xy − qyx)

is a “noncommutative P1” but for qn 6= 1 has only three homogeneous ideals (namely (x), (y), and (x, y)).

Noncommutative is not commutative May 2, 2016 12 / 39 Sheaves to the rescue

• The Beatles (paraphrased):

“All you need is sheaves.”

• Idea: You can reconstruct the space from the sheaves.

Theorem (Rosenberg, Gabriel, Gabber, Brandenburg) Let X, Y be quasi-separated schemes. If qcoh(X) ≡ qcoh(Y) then X and Y are isomorphic.

Noncommutative is not commutative May 2, 2016 13 / 39 Sheaves to the rescue

• All you need is modules

Theorem Let X = Proj A for a commutative, f.g. k-algebra A generated in degree 1.

(1) Every coherent sheaf on X is isomorphic to Me for some f.g. graded A-module M.

(2) Me =∼ Ne as sheaves if and only if there is an isomorphism ∼ M≥n = N≥n.

• Let gr-A be the category of f.g. A-modules. Take the quotient category qgr-A = gr-A/fdim-A • The above says qgr-A ≡ coh(Proj A).

Noncommutative is not commutative May 2, 2016 14 / 39 Noncommutative projective schemes

A (not necessarily commutative) connected graded k-algebra A is

A = k ⊕ A1 ⊕ A2 ⊕ · · ·

such that dimk Ai < ∞ and A is a f.g. k-algebra.

Definition (Artin-Zhang, 1994)

The noncommutative projective scheme ProjNC A is the triple (qgr-A, A, S)

where A is the distinguished object and S is the shift functor.

Noncommutative projective schemes May 2, 2016 15 / 39 Noncommutative curves

• “The Hartshorne approach” • A a k-algebra, V ⊆ A a k-subspace generating A spanned by {1, a1,..., am}.

• V0 = k, Vn spanned by monomials of length ≤ n in the ai. • The Gelfand-Kirillov dimension of A

GKdim A = lim sup logn(dimk Vn) n→∞

• GKdim k[x1,..., xm] = m. • So noncommutative projective curves should have GKdim 2.

Noncommutative projective schemes May 2, 2016 16 / 39 Noncommutative curves

Theorem (Artin-Stafford, 1995) Let A be a f.g. connected graded generated in degree 1 with GKdim(A) = 2. Then there exists a projective curve X such that qgr-A ≡ coh(X)

Or “noncommutative projective curves are commutative”.

Noncommutative projective schemes May 2, 2016 17 / 39 Noncommutative surfaces

• The “right” definition of a noncommutative polynomial ring?

Definition A a f.g. connected graded k-algebra is Artin-Schelter regular if (1) gl.dimA = d < ∞ (2) GKdim A < ∞ and i k k (3) ExtA( , A) = δi,d .

• Interesting invariant theory for these rings. • Behaves homologically like a commutative polynomial ring.

• k[x1,..., xm] is AS regular of dimension m. • Noncommutative P2s should be qgr-A for A AS-regular of dimension 3.

Noncommutative projective schemes May 2, 2016 18 / 39 Noncommutative surfaces

Theorem (Artin-Tate-Van den Bergh, 1990) Let A be an AS regular ring of dimension 3 generated in degree 1. Either (a) qgr-A ≡ coh(X) for X = P2 or X = P1 × P1, or (b) A  B and qgr-B ≡ coh(E) for an elliptic curve E.

• Or “noncommutative P2s are either commutative or contain a commutative curve”. • Other noncommutative surfaces (noetherian connected graded domains of GKdim 3)? • Noncommutative P3 (AS-regular of dimension 4)?

Noncommutative projective schemes May 2, 2016 19 / 39 The first Weyl algebra

• Throughout, “graded” really meant N-graded • Recall the first Weyl algebra:

A1 = khx, yi/(xy − yx − 1)

• Simple noetherian domain of GK dim 2

• A1 is not N-graded

Z-graded rings May 2, 2016 20 / 39 The first Weyl algebra

A1 = khx, yi/(xy − yx − 1)

• Ring of differential operators on k[t] • x ←→ t· • y ←→ d/dt • A is Z-graded by deg x = 1, deg y = −1 ∼ • (A1)0 = k[xy] 6= k • Exists an outer automorphism ω, reversing the grading

ω(x) = y ω(y) = −x

Z-graded rings May 2, 2016 21 / 39 Sierra (2009)

• Sue Sierra, Rings graded equivalent to the Weyl algebra

• Classified all rings graded Morita equivalent to A1

• Examined the graded module category gr-A1:

−3 −2 −1 0 1 2 3

• For each λ ∈ k \ Z, one simple module Mλ • For each n ∈ Z, two simple modules, Xhni and Yhni

X Y

• For each n, exists a nonsplit extension of Xhni by Yhni and a nonsplit extension of Yhni by Xhni

Z-graded rings May 2, 2016 22 / 39 Sierra (2009)

• Computed Pic(gr-A1), the Picard group (group of autoequivalences) of gr-A1 • Shift functor, S:

• Autoequivalence ω:

Z-graded rings May 2, 2016 23 / 39 Sierra (2009)

Theorem (Sierra)

There exist ιn, autoequivalences of gr-A1, permuting Xhni and Yhni and fixing all other simple modules.

2 ∼ Also ιiιj = ιjιi and ιn = Idgr-A

Theorem (Sierra) ∼ (Z) ∼ Pic(gr-A1) = (Z/2Z) o D∞ = Zfin o D∞.

Z-graded rings May 2, 2016 24 / 39 Smith (2011)

• Paul Smith, A quotient stack related to the Weyl algebra

• Proves that Gr-A1 ≡ Qcoh(χ) • χ is a quotient stack “whose coarse moduli space is the affine line Spec k[z], and whose stacky structure consists of stacky points BZ2 supported at each integer point”

• Gr-A1 ≡ Gr(C, Zfin) ≡ Qcoh(χ)

Z-graded rings May 2, 2016 25 / 39 Smith (2011)

• Zfin the group of finite subsets of Z, operation XOR

• Constructs a Zfin-graded ring k M [xn | n ∈ Z] C := hom(A , ι A ) =∼ 1 J 1 (x2 + n = x2 + m | m, n ∈ ) J∈ n m Z Zfin√ ∼ = k[z][ z − n | n ∈ Z]

where deg xn = {n} • C is commutative, integrally closed, non-noetherian PID

Theorem (Smith) There is an equivalence of categories

Gr-A1 ≡ Gr(C, Zfin).

Z-graded rings May 2, 2016 26 / 39 Z-graded rings

Theorem (Artin-Stafford, 1995) Let A be a f.g. connected N-graded domain generated in degree 1 with GKdim(A) = 2. Then there exists a projective curve X such that qgr-A ≡ coh(X).

Theorem (Smith, 2011)

A1 is a f.g. Z-graded domain with GKdim(A1) = 2. There exists a C and quotient stack χ such that

gr-A1 ≡ gr(C, Zfin) ≡ coh(χ).

Z-graded rings May 2, 2016 27 / 39 Generalized Weyl algebras (GWAs)

• Introduced by V. Bavula • Generalized Weyl algebras and their representations (1993) • D a ring; σ ∈ Aut(D); a ∈ Z(D) • The generalized Weyl algebra D(σ, a) with base ring D

Dhx, yi D(σ, a) =  xy = a yx = σ(a)  dx = xσ(d), d ∈ D dy = yσ−1(d), d ∈ D

Theorem (Bell-Rogalski, 2015) Every simple Z-graded domain of GKdim 2 is graded Morita equivalent to a GWA.

Z-graded rings May 2, 2016 28 / 39 Generalized Weyl algebras

• For us, D = k[z]; σ(z) = z − 1; a = f (z)

khx, y, zi A(f ) =∼  xy = f (z) yx = f (z − 1) xz = (z + 1)x yz = (z − 1)y

• Two roots α, β of f (z) are congruent if α − β ∈ Z

Example (The first Weyl algebra) Take f (z) = z k k [z]hx, yi ∼ hx, yi D(σ, a) = = = A1.  xy = z yx = z − 1  (xy − yx − 1) zx = x(z − 1) zy = y(z + 1)

Z-graded rings May 2, 2016 29 / 39 Generalized Weyl algebras

Properties of A(f ): • Noetherian domain • Krull dimension 1 • Simple if and only if no congruent roots  1, f has neither multiple nor congruent roots  • gl.dim A(f ) = 2, f has congruent roots but no multiple roots  ∞, f has a multiple root • We can give A(f ) a Z grading by deg x = 1, deg y = −1, deg z = 0

Z-graded rings May 2, 2016 30 / 39 Questions

Focus on quadratic f (z) = z(z + α). For these GWAs A(f ): • What does gr-A(f ) look like? • What is Pic(gr-A(f ))? • Can we construct a commutative Γ-graded ring C such that gr(C, Γ) ≡ gr-A(f )?

Z-graded rings May 2, 2016 31 / 39 Results

If α ∈ k \ Z (distinct roots):

α − 2 α − 1 α α+1

−2 −1 0 1 2

If α = 0 (double root):

−3 −2 −1 0 1 2 3

+ If α ∈ N (congruent roots):

Z-graded rings May 2, 2016 32 / 39 Results

Theorem(s) (W)

In all cases, there exist numerically trivial autoequivalences ιn permuting “Xhni” and “Yhni” and fixing all other simple modules.

Corollary (W) ∼ For quadratic f , Pic(gr-A(f )) = Zfin o D∞.

Z-graded rings May 2, 2016 33 / 39 Results

k Let α ∈ \ Z. Define a Zfin × Zfin graded ring C: M  C = homA A, ι(J,J0)A 0 (J,J )∈Zfin×Zfin k ∼ [an, bn | n ∈ Z] = 2 2 2 2 (an + n = am + m, an = bn + α | m, n ∈ Z)

with deg an = ({n}, ∅) and deg bn = (∅, {n})

Theorem (W)

There is an equivalence of categories gr(C, Zfin × Zfin) ≡ gr-A.

Z-graded rings May 2, 2016 34 / 39 Results

Let α = 0. Define a Zfin-graded ring B:

M k[z][b | n ∈ ] = ( , ι ) ∼ n Z . B homA A JA = 2 2 (bn = (z + n) | n ∈ Z) J∈Zfin

Theorem (W) B is a reduced, non-noetherian, non-domain of Kdim 1 with uncountably many prime ideals.

Theorem (W)

There is an equivalence of categories gr(B, Zfin) ≡ gr-A.

Z-graded rings May 2, 2016 35 / 39 Results

+ • In congruent root (α ∈ N ), we have finite-dimensional modules. • Consider the quotient category qgr-A = gr-A/fdim-A.

Theorem (W)

There is an equivalence of categories gr(B, Zfin) ≡ qgr-A.

Z-graded rings May 2, 2016 36 / 39 The upshot

In all cases, • There exist numerically trivial autoequivalences permuting X and Y and fixing all other simples. ∼ • Pic(gr-A(f )) = Zfin o D∞.

• There exists a Zfin-graded commutative ring R such that

qgr-A(f ) ≡ gr(R, Zfin).

Z-graded rings May 2, 2016 37 / 39 Questions k • Zfin-grading on B gives an action of Spec Zfin on Spec B

 Spec B  χ = k Spec Zfin

• Other Z-graded domains of GK dimension 2? Other GWAs? • U(sl(2)), Down-up algebras, quantum Weyl algebra, Simple Z-graded domains • Construction of B: {Fγ | γ ∈ Γ} ⊆ Pic(gr-A) M B = Homqgr-A(A, FγA) γ∈Γ

• Opposite view: O a quasicoherent sheaf on χ:

M n HomQcoh(χ)(O, S O) n∈Z

Future direction May 2, 2016 38 / 39 Thank you!

May 2, 2016 39 / 39