The category of graded modules of a generalizedWeyl algebra

Robert WonUniversity of California, San Diego

AMS Central Sectional Meeting, East Lansing, MI, March 2015

March 14, 2015 1 / 24

Overview

1 BackgroundSierra (2009)Smith (2011)Generalized Weyl algebras

2 ResultsNon-congruent rootsMultiple rootCongruent roots

3 Future direction

March 14, 2015 2 / 24

The Weyl algebra

• Fix k = k̄, char k = 0• A noncommutative k-algebra, the first Weyl algebra A1(k) = A1

A1 = k〈x, y〉/(xy− yx− 1)

• Simple noetherian domain• A1 is Z-graded by deg x = 1, deg y = −1• Exists an outer automorphism ω, reversing the grading

ω(x) = y ω(y) = −x

Background March 14, 2015 3 / 24

Sierra (2009)

• Sue Sierra, Rings graded equivalent to the Weyl algebra• Examined the graded module category gr-A:

−3 −2 −1 0 1 2 3

• For each λ ∈ k \ Z, one simple module Mλ

• For each n ∈ Z, two simple modules, X〈n〉 and Y〈n〉

X Y

• For each n, exists a nonsplit extension of X〈n〉 by Y〈n〉 and anonsplit extension of Y〈n〉 by X〈n〉

Background March 14, 2015 4 / 24

Sierra (2009)• Computed Pic(gr-A)

• Shift functor S :

• Autoequivalence ω:

Theorem (Sierra)Let F ∈ Pic(gr-A). Then there exist a = ±1 and b ∈ Z such that

{F(X〈n〉),F(Y〈n〉)} ∼= {X〈an + b〉,Y〈an + b〉}.

So Pic(gr-A) ∼= Pic0(gr-A) o D∞.

Background March 14, 2015 5 / 24

Sierra (2009)

Theorem (Sierra)There exist ιn, autoequivalences of gr-A, permuting X〈n〉 and Y〈n〉 andfixing all other simple modules.

Also ιiιj = ιjιi and ι2n ∼= Idgr-A

Theorem (Sierra)

Pic0(gr-A) ∼= (Z/2Z)(Z) ∼= Zfin.

Background March 14, 2015 6 / 24

Smith (2011)

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

• Proves that gr-A ≡ Qcohχ• χ is a quotient stack “whose coarse moduli space is the affine line

Spec k[z], and whose stacky structure consists of stacky points BZ2supported at each integer point”

• gr-A ≡ gr(C,Zfin) ≡ Qcohχ

Background March 14, 2015 7 / 24

Smith (2011)

• Zfin the group of finite subsets of Z, operation XOR• Constructs a Zfin graded ring

C :=⊕

J∈Zfin

hom(A, ιJA) ∼=k[xn | n ∈ Z]

(x2n + n = x2

m + m | m,n ∈ Z)

∼= k[z |√

z− n]

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

Theorem (Smith)There is an equivalence of categories

gr-A ≡ gr(C,Zfin).

Background March 14, 2015 8 / 24

Generalized Weyl algebras

• Introduced by V. Bavula in Generalized Weyl algebras and theirrepresentations (1993)

• The generalized Weyl algebra A(f )

A(f ) ∼=k〈x, y, z〉(

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

A(z) ∼=k〈x, y〉

(xy− yx− 1)= A1.

Background March 14, 2015 9 / 24

The main object

Properties of A(f ):

• Noetherian domain• Krull dimension 1• Simple if and only if no congruent roots

• gl.dim. A(f ) =

1, f has neither multiple nor congruent roots2, f has congruent roots but no multiple roots∞, f has a multiple root

• A(f ) is Z-graded by letting deg x = 1, deg y = −1, deg z = 0

Background March 14, 2015 10 / 24

Questions and strategy

For these generalized Weyl algebras A(f ):• What does gr-A(f ) look like?• What is Pic(gr-A(f ))?• Can we construct a commutative (or otherwise nicer) Γ-graded

ring C such that gr(C,Γ) ≡ gr-A(f )?Strategy:

• First quadratic f• Distinct, non-congruent roots• Double root• Congruent roots

Results March 14, 2015 11 / 24

Distinct, non-congruent roots

Let f (z) = z(z + α) for some α ∈ k \ Z.

A = A(f ) ∼=k〈x, y, z〉(

xy = z(z + α) yx = (z− 1)(z + α− 1)xz = (z + 1)x yz = (z− 1)y

)• We still have A is simple, K.dim(A) = gl.dim(A) = 1• Still exists an outer automorphism ω reversing the grading

ω(x) = y ω(y) = x ω(z) = 1 + α− z

Results March 14, 2015 12 / 24

Distinct, non-congruent roots

Graded simple modules:

−2

α− 2

−1

α− 1

0

α

1

α+1

2

α+2

• One simple graded module Mλ for each λ ∈ k \ (Z ∪ Z + α)

• For each n ∈ Z two simple modules X0〈n〉 and Y0〈n〉• For each n ∈ Z two simple modules Xα〈n〉 and Yα〈n〉• A nonsplit extension of X0〈n〉 by Y0〈n〉 and vice versa• A nonsplit extension of Xα〈n〉 by Yα〈n〉 and vice versa

Results March 14, 2015 13 / 24

Distinct, non-congruent roots

Theorem (W)There exist numerically trivial autoequivalences, ι(n,∅)permuting X0〈n〉 and Y0〈n〉 and fixing all other simple modules.Similarly, there exist ι(∅,n) permuting Xα〈n〉 and Yα〈n〉.

Theorem (W)

Pic(gr-A) ∼= Pic0(gr-A) o D∞ ∼= (Z/2Z)(Z) o D∞

Results March 14, 2015 14 / 24

Distinct, non-congruent roots

• Define a Zfin × Zfin graded ring C:

C = k[an, bn | n ∈ Z]

modulo the relations

a2n + n = a2

m + m and a2n = b2

n + α for all 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.

Results March 14, 2015 15 / 24

Multiple root

Let f (z) = z2.

The graded simple modules

−3 −2 −1 0 1 2 3

• For each λ ∈ k \ Z, Mλ

• For each n ∈ Z, X〈n〉 and Y〈n〉

X Y

• Nonsplit extensions of X〈n〉 by Y〈n〉 and vice versa. Alsoself-extensions of X〈n〉 by X〈n〉 and Y〈n〉 and Y〈n〉.

Results March 14, 2015 16 / 24

Multiple root

Theorem (W)There exist numerically trivial autoequivalences, ιn permutingX〈n〉 and Y〈n〉 and fixing all other simple modules.

Theorem (W)

Pic(gr-A) ∼= Pic0(gr-A) o D∞ ∼= (Z/2Z)(Z) o D∞

Results March 14, 2015 17 / 24

Multiple root

• Define a Zfin graded ring B:

B =⊕

J∈Zfin

homA(A, ιJA).

• Commutative k-algebra, not a domain.• B has the presentation

B ∼=k[z][bn | n ∈ Z]

(b2n = (z + n)2 | n ∈ Z)

where deg bn = {n}.

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

Results March 14, 2015 18 / 24

Two congruent roots

Let f (z) = z(z + m) for some m ∈ N.

A = A(f ) ∼=k〈x, y, z〉(

xy = z(z + m) yx = (z− 1)(z + m− 1)xz = (z + 1)x yz = (z− 1)y

)Different from previous cases:

• A no longer simple• There exist new finite-dimensional simple modules• Now gl.dim(A) = 2

Results March 14, 2015 19 / 24

Two congruent roots

The graded simple modules• For each λ ∈ k \ Z, Mλ

• For each n ∈ Z, X〈n〉, Y〈n〉, and Z〈n〉

X YZ

• Z〈n〉 is finite dimensional• proj.dim Z〈n〉 = 2

Results March 14, 2015 20 / 24

Two congruent roots

Theorem (W)There exist numerically trivial autoequivalences, ιn permutingX〈n〉 and Y〈n〉 and fixing all other simple modules.

Theorem (W)

Pic(gr-A) ∼= Pic0(gr-A) o D∞ ∼= (Z/2Z)(Z) o D∞

Results March 14, 2015 21 / 24

Two congruent roots

• Let qgr-A denote the quotient category of gr-A modulo its fullsubcategory of finite dimensional modules.

Theorem (W)

qgr-A (z(z + m)) ≡ gr-A(z2).

Corollary (W)qgr-A (z(z + m)) ≡ gr(B,Zfin)

Results March 14, 2015 22 / 24

Summary

• For all f , there exist numerically trivial autoequivalencespermuting X and Y and fixing all other simples.

• For all f , Pic(gr-A(f )) ∼= (Z2)Z o D∞.• If f has distinct roots:

gr-A(f ) ≡ gr(C,Zfin × Zfin).

• If f has a multiple root:

gr-A(f ) ≡ gr(B,Zfin).

• If f has congruent roots:

qgr-A(f ) ≡ gr(B,Zfin).

Results March 14, 2015 23 / 24

Questions

• What are the properties of the commutative ring B?

• Can we piece together the picture for general (non-quadratic) f ?

• Other Z-graded domains of GK dim 2?

Future direction March 14, 2015 24 / 24