The category of graded modules of a generalized Weyl algebra Robert Won University of California, San Diego AMS Central Sectional Meeting, East Lansing, MI, March 2015 March 14, 2015 1 / 24
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