Skew Calabi-Yau algebras and homological identities
Manuel L. Reyes
Bowdoin College
Joint international AMS-RMS meeting
Alba Iulia, Romania
June 30, 2013
(joint work with Daniel Rogalski and James J. Zhang)
Manuel Reyes (Bowdoin College)
Skew Calabi-Yau algebras
June 30, 2013
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Outline
1
Calabi-Yau algebras: basics and sources of examples
2
Calabi-Yau algebras from smash products
3
A curious example of a Calabi-Yau algebra
4
Skew Calabi-Yau algebras and homological identities
5
Problems and questions
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The Calabi-Yau property in noncommutative algebra
The Calabi-Yau property has its origin in geometry:
But it has now made its way into noncommutative algebra!
Geometry
Physics
triangulated
noncomm.
(string theory)
categories
algebras
There are (at least) two ways to think of these algebras:
(1) Noncommutative version of coordinate ring of Calabi-Yau variety.
(2) Graded case: noncommutative version of polynomial ring k[x, y , z].
I will emphasize (2).
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Preliminaries: the enveloping algebra
Let A be an algebra over a field k. We write − ⊗ − for − ⊗k −.
The enveloping algebra of A is Ae = A ⊗ Aop . A left Ae -module M is the
same as a k-central (A, A)-bimodule, via:
(a ⊗ b op ) · m = a · m · b.
Provides a convenient way to discuss homological algebra for bimodules:
Projective/injective bimodules ! Projective/injective Ae -modules
Resolutions of (A, A)-bimodules ! resolutions of Ae -modules
Note: A bimodule A MA that is left or right A-projective need not be
Ae -projective. Most important example: A AA . (It’s only Ae -projective if A
is a “separable” k-algebra.)
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Ginzburg’s definition
Here’s how Ginzburg generalized the Calabi-Yau condition to
noncommutative algebras:
Definition
(i) A is homologically smooth if A has a projective resolution in Ae -Mod of
finite length whose terms are finitely generated over Ae . (A is a “perfect
object” in Ae -Mod.)
(ii) [Ginzburg, 2006] A is Calabi-Yau of dimension d if it is homologically
smooth and if
(
0 if i 6= d,
ExtiAe (A, Ae ) ∼
=
A if i = d,
as Ae -modules.
This condition amounts to a “self-duality” under M ∨ = RHomAe (M, Ae )
in the derived bimodule category: A ∼
= A∨ [d].
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First examples of Calabi-Yau algebras
Commutative examples: coordinate rings of Calabi-Yau varieties
[Ginzburg].
From now on, we will consider only graded Calabi-Yau algebras: take the
projective Ae -resolution and Ext isomorphism to be in the graded category.
Graded commutative examples: k[x1 , . . . , xn ].
So graded Calabi-Yau algebras are “noncommutative polynomial rings.”
But so are the Artin-Schelter regular algebras. How do these compare?
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AS-regular vs. Calabi-Yau properties
Definition
A connected graded k-algebra A is Artin-Schelter Gorenstein if
A has left and right injective dimension d < ∞
there is an integer l (“AS
( index”) such that, as both left and right
0,
i 6= d,
modules, ExtiA (k, A) ∼
=
k(l) i = d.
An algebra A as above is AS-regular if it also has finite global dimension,
which will equal d.
(This amounts to a “duality” between the one-sided resolutions of k under
the dual functor HomA (−, A).)
Note the similarity with the CY condition! However:
Theorem: An AS-regular algebra A (with finite GK dimension) of
dimension 3 is Calabi-Yau if and only if it is of “type A” in Artin &
Schelter’s terminology [Berger & Taillefer].
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Further examples
Here are some more ways to find noncommutative examples:
Enveloping algebras: U(g) is Calabi-Yau if g is finite-dimensional and
N-graded.
E.g., g = {upper-triangular n × n matrices}, graded by “level of the
diagonal.”
Morita invariance: The CY property is Morita invariant.
In particular, A is Calabi-Yau if and only if Mn (A) is.
Polynomial rings: If A is Calabi-Yau then so is A[x].
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1
Calabi-Yau algebras: basics and sources of examples
2
Calabi-Yau algebras from smash products
3
A curious example of a Calabi-Yau algebra
4
Skew Calabi-Yau algebras and homological identities
5
Problems and questions
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Calabi-Yau algebras from skew group algebras
Here is a well known way to produce Calabi Yau algebras.
L
Consider k[x1 , . . . , xn ] as the symmetric algebra on V =
k · xi . Let
G ≤ SL(V ) be a finite subgroup. Then the skew group algebra
k[x1 , . . . , xn ] o G is Calabi-Yau. Recall, this has multiplication:
(a ⊗ g )(b ⊗ h) = ag (b) ⊗ gh.
Theorem: Given G as above, the algebra k[x1 , . . . , xn ] o G is Calabi-Yau
of dimension n [“well-known,” see Bocklandt/Schedler/Wemyss].
One might think of this as a ”factory” for producing CY algebras that are
not commutative.
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Calabi-Yau algebras from smash products
Thm: If G ≤ SL(V ) is a finite subgroup, then k[V ] o G is CY.
The idea of using an action on a CY algebra to produce a new CY algebra
has received much interest. Just a sample of those who’ve been working
on this: Bocklandt/Schedler/Wemyss, Farniati, LeMeur, Liu/Wu/Zhu,
Yu/Zhang, etc.
Q: If A is CY and a finite group G acts on A, what condition ensures that
A o G is CY?
One of the most general results is due to Liu/Wu/Zhu.
What is their analogue of the condition that G ≤ SL(V )?
That a certain homological determinant is trivial.
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Motivation: determinants the “coordinate-free” way
Suppose that a group G acts on V = Cd .
This induces a graded action on the exterior algebra
Λ(V ) = Λ0 (V ) ⊕ Λ1 (V ) ⊕ · · · ⊕ Λd (V ).
Here, Λ1 (V ) ∼
= V and Λ0 (V ) ∼
= Λd (V ) ∼
= C.
As the action is graded, G acts in particular on Λd (V ). Fix any generator
e ∈ Λd (V ). Then for each g ∈ G , g e is a scalar multiple of e. It turns out
that this is the determinant!
g (e) = det(g )e
And this can be (almost) recovered from homological algebra: over the
symmetric algebra A = k[V ], we have Ext•A (k, k) ∼
= Λ(V ∗ ).
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The homological determinant
There are a few ways to define the homological determinant, that agree in
the cases where they make sense.
The one that directly generalizes the above requires A to be AS-regular
[Kirkman/Kuzmanovich/Zhang].
An action of G on A induces an action on Ext•A (k, k).
The action is graded, so the highest-degree part ExtdA (k, k) is
G -invariant. It’s also one dimensional.
Fix generator e, then for he = η(h)e, we have hdet = η ◦ S : kG → k.
P
P
(Here S : kG → kG is the antipode:
ag g 7→
ag g −1 .)
There is another formulation using local cohomology, which works for more
general AS-Gorenstein algebras. Uses top-degree local cohomology instead.
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Examples of the homological determinant
In general, it’s not easy to compute the homological determinant. Here are
some special cases where we do know how to compute it:
Below, let g ∈ GrAut(A).
Examples
1
If A = k[x1 , . . . , xn ], then hdet(g ) = det(g |A1 ).
2
If A = k−1 [x, y ] = khx, y | xy = −yxi, then hdet(g ) = − det(g |A1 ).
3
If A = khx, y | x 2 y = yx 2 , xy 2 = y 2 xi, then hdet(g ) = det(g |A1 )2 .
We could use more techniques for computing hdet!
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Hopf algebra actions
At this level of generality, we might as well work with Hopf algebra actions
(“quantized group actions”).
Recall: A Hopf algebra H has a comultiplication ∆ : H → H ⊗ H, a counit
: H → k (respectively “dual” to multiplication m : H ⊗ H → H and unit
η : k → H), and an antipode S : kG → kG . Assume S bijective.
Ex: for a group G , P
the group algebra
H = kG is a Hopf P
algebra under
P
P the
comultiplication ∆( ag g ) = ag g ⊗ g and counit ( ag g ) = ag .
Sweedler notation: ∆(h) =
P
h1 ⊗ h2 .
An algebra A is a left H-module algebra if its a left H-module satisfying
X
h(ab) =
h1 (a)h2 (b).
Note: Group action of G on A ! left kG -module action on A.
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Homological determinants and smash products
Homological determinant: The definition of the homological
determinant extends to actions of Hopf algebras in a straightforward way:
if A is an AS-regular left H-module algebra, we get an induced algebra
homomorphism hdet : H → k, via H-action on ExtdA (k, k).
Smash products: Generalizing the skew group algebra A o G , we can
form the smash product A#H when A is a left H-module algebra.
This is defined to be the vector space A ⊗ H with multiplication given by
X
(a ⊗ g )(b ⊗ h) =
ag1 (b) ⊗ g2 h.
When H = kG is a group algebra, we have A#H ∼
= A o G.
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Smash products of Calabi-Yau algebras
Basic example: Given G a group of graded automorphisms of
A = k[x1 , . . . , xn ], the resulting map hdet : kG → k gives the usual
determinant of each group element.
In particular: G ≤ SLn (k) ⇐⇒ hdet is trivial.
For our context, this is the appropriate generalization of the condition
G ≤ SL(V )!
Theorem: [Liu/Wu/Zhu] Let A be an N-Koszul Calabi-Yau algebra and
H an involutory Calabi-Yau Hopf algebra, where A is a left H-module
algebra. Then A#H is Calabi-Yau if and only if the homological
determinant of the H-action on A is trivial.
Related result: Yu/Zhang have a related result when A is also a Hopf
algebra.
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1
Calabi-Yau algebras: basics and sources of examples
2
Calabi-Yau algebras from smash products
3
A curious example of a Calabi-Yau algebra
4
Skew Calabi-Yau algebras and homological identities
5
Problems and questions
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Calabi-Yau algebras from quivers
Next I want to present an example that complicates this nice picture. It
arises as a quotient of a path algebra, and gives us an excuse to discuss
more examples of CY algebras that are not connected.
A quiver Q is a directed graph (multiple arrows and loops are allowed).
The path algebra kQ is the k-vector space spanned by (possibly trivial)
paths in Q (read right-to-left), where the product of two paths is either
their concatenation (if defined) or zero (if concatenation is undefined).
Quick examples:
Q=
a1
< b
a2
b
/
k 0
kQ 0 ∼
=
k k
kQ ∼
= khx, y i
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Q0 =
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Calabi-Yau algebras and superpotentials
A popular theme in the study of CY algebras (coming from physics),
especially in dimension 3, is that they tend to have relations defined by
“superpotentials.”
Superpotential W : a linear combination of cycles in Q. (May be
identified, up to permutation, with an element of kQ/[kQ, kQ].)
If a ∈ ar(Q) is an arrow, then the “cyclic partial derivative” ∂a W is the
partial derivative with respect to a of all cyclic permutations of W .
The Jacobi algebra of W is kQ/(∂a W : a ∈ Q1 ).
CY-3 Ex: Q = Yh
x,y ,z
with W = xyz − xzy .
kQ
khx, y , zi
∼
∼
=
= k[x, y , z].
(∂a W : a ∈ ar(Q))
(yz − zy , zx − xz, xy − yx)
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Calabi-Yau algebras from superpotentials
(Note: not every Jacobi algebra is CY; only for “good” superpotentials.)
It’s been shown that many CY algebras come from superpotentials
[Bocklandt, Bocklandt/Schedler/Wemyss, Van den Bergh] but not all
CY-3 algebras come from superpotentials [Davidson].
Example (Bocklandt)
a1 ,a2
For Q = ai
!) and the superpotential W = a a a a + a a a a , the
1 3 2 4
3 1 4 2
a3 ,a4
resulting Jacobi algebra B = CQ/(∂a W ) is Calabi-Yau of dimension 3.
The explicit relations are:
∂ a1 W = 0
a3 a2 a4 = −a4 a2 a3
∂ a2 W = 0
a4 a1 a3 = −a2 a3 a1
∂ a3 W = 0
a2 a4 a1 = −a1 a4 a2
∂ a4 W = 0
a1 a3 a2 = −a2 a3 a1
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Why Bocklandt’s example is “strange”
a1 ,a2
Ex [Bocklandt]: For the quiver Q = ai
!) and the superpotential
a3 ,a4
W = a1 a3 a2 a4 + a3 a1 a4 a2 , the resulting Jacobi algebra
B = CQ/(∂ai W : i = 1, . . . , 4) is Calabi-Yau of dimension 3.
Fact: We noticed that this can be realized as a skew group algebra: Let
A = Chx, y | x 2 y = yx 2 , y 2 x = xy 2 i.
Let µ : A → A send x 7→ −x and y 7→ −y . Then for G = hµi = {1, µ}, it
turns out that B ∼
= A o G.
However, A is not Calabi-Yau! It’s AS-regular of dimension 3, but of the
“wrong” type (type A).
What’s going on here? To better understand this example, we must step
outside of the world of CY algebras.
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1
Calabi-Yau algebras: basics and sources of examples
2
Calabi-Yau algebras from smash products
3
A curious example of a Calabi-Yau algebra
4
Skew Calabi-Yau algebras and homological identities
5
Problems and questions
Manuel Reyes (Bowdoin College)
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Skew Calabi-Yau algebras
Let M be an (A, A)-bimodule and µ, ν : A → A two automorphisms. The
twist µ M ν is equal to M as a k-vector space, with
a · m · b = µ(a)mν(b).
Definition
A (graded) k-algebra A is (graded) skew Calabi-Yau of dimension d if it is
homologically smooth and there is a (graded) automorphism µ : A → A
and isomorphisms of (graded) bimodules
(
i 6= d,
i
e ∼ 0,
ExtAe (A, A ) = 1 µ
A , i = d.
(Also called twisted Calabi-Yau algebras.)
This µ = µA is called the Nakayama automorphism (generalized from
Frobenius algebras). It’s only unique up to an inner automorphism. So A
is CY ⇐⇒ µ is inner.
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Examples of skew Calabi-Yau algebras
Where do we find examples of skew CY algebras? There are two major
classes.
(1) A connected graded algebra A is graded skew Calabi-Yau if and only if
it is AS-regular [Yekutieli/Zhang, R./Rogalski/Zhang].
(Note: no assumption of finite GK dimension here!)
In this sense, graded skew CY algebras are a non-connected generalization
of AS-regular algebras.
Some particular examples:
1
Skew polynomial rings kqij [x1 , . . . , xn ]
2
Coordinate ring of quantum matrices
3
Sklyanin algebras
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Examples of skew Calabi-Yau algebras
(2) A noetherian Hopf algebra H is skew CY if and only if it is AS-regular
in the sense of Brown/Zhang [Brown/Zhang, Lu/Wu/Zhu]. Explicit
Nakayama automorphism for such H: S −2 ◦ ΞrR l .
(For an algebra map α : H → k, the right winding automorphism of α is
Rl
P
defined by Ξrα (h) = h1 α(h2 ). Also, : H → k is a map induced by the
“left homological integral.”)
Some examples:
1
U(g) for finite-dimensional g. (It’s CY if g is N-graded.)
2
O(G )
3
Lots of quantum groups (Uq (g), Oq (G ))
4
Affine noetherian PI Hopf algebras with finite global dimension
Question of Brown: are all noetherian Hopf algebras “AS-Gorenstein”?
Positive answer ⇒ all noetherian H with gl.dim(H) < ∞ are skew CY.
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Three homological identities
So what happens when we take smash products with skew CY algebras?
This is our first “homological identity.”
Theorem (R./Rogalski/Zhang)
Let H be a finite dimensional semisimple Hopf algebra acting on a
connected graded skew Calabi-Yau algebra A (compatible with grading).
Then A#H is skew Calabi-Yau as well, with Nakayama automorphism
µA#H = µA #(µH ◦ Ξlhdet ),
(HI1)
where hdet is the homological determinant of the action of H on A.
In particular, if hdet is trivial, then the winding automorphism of hdet is
the identity and
hdet trivial =⇒ µA#H = µA #µH .
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Constructing Calabi-Yau algebras
This formula indicates a way to construct new Calabi-Yau algebras.
Theorem (R./Rogalski/Zhang)
Let A be a noetherian AS-regular algebra with Nakayama automorphism
µ. Suppose that hdet(µ) = 1. Then B = A o hµi is Calabi-Yau.
Proof.
For simplicity, assume µ has finite order. Then H = khµi has µH = idH .
Also, hdet(µ) = 1 means that hdetH is trivial, so Ξlhdet = idH . So
µB = µA #(µH ◦ Ξlhdet ) = µA # idH = µ#1.
But this is an inner automorphism of B = A#H, implemented by
1 ⊗ µ ∈ A#H. So µB inner ⇒ B is CY!
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Bocklandt’s example revisited
Recall Bocklandt’s quiver algebra B = kQ/(∂a W ) ∼
= A o {1, µ} for
A = Chx, y | x 2 y = yx 2 , y 2 x = xy 2 i
and µ : A → A given by µ(x) = −x, µ(y ) = −y .
Set H = k{1, µ}. Then B ∼
= A#H is Calabi-Yau, while A is not
Calabi-Yau.
But A is skew Calabi-Yau! (It’s AS-regular.)
It turns out that µ = µA .
Kirkman and Kuzmanovich: for any σ ∈ GrAut(A),
hdet(σ) = det(σ|A1 )2 . Thus, hdet(µ) = 1 and hdet : H → k is trivial.
Thus the previous theorem recovers the fact that B is Calabi-Yau!
Again, the key idea is that while µA was not an inner automorphism of A,
it “becomes an inner automorphism” when we pass to B = A o hµi.
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hdet of the Nakayama automorphism
Here is a second “homological identity” that we established.
Theorem (R./Rogalski/Zhang)
Let A be a noetherian connected graded Koszul skew Calabi-Yau algebra.
Then
hdet(µA ) = 1.
We strongly believe that it should be possible to omit the Koszul property.
Here’s a little evidence:
Recall that, if hdet(µA ) = 1, then A o hµA i is Calabi-Yau.
But more recently, J. Goodman and U. Krähmer have shown that if A is
skew CY, not necessarily graded, then A o hµA i is CY. This is consistent
with hdet(µA ) = 1 in the connected graded case.
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1
Calabi-Yau algebras: basics and sources of examples
2
Calabi-Yau algebras from smash products
3
A curious example of a Calabi-Yau algebra
4
Skew Calabi-Yau algebras and homological identities
5
Problems and questions
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Food for thought
Problems
1
If A and H are skew CY with A a left H-module algebra, is A#H
skew Calabi-Yau? If so, what is µA#H (in terms of µA and µH )?
2
If A is connected graded skew Calabi-Yau (or even AS-Gorenstein), is
hdet(µA ) = 1?
3
What are some good, general techniques to compute the Nakayama
automorphism of an algebra or the homological determinant of an
action?
4
Big, big question, even for regular algebras: How can we understand
the ring-theoretic structure of these algebras?
There is much that is not yet understood. Skew CY algebras are very
natural in noncommutative algebra/geometry and deserve more attention!
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Selected references
R. Berger and R. Taillefer, Poincaré-Birkhoff-Witt deformations of
Calabi-Yau algebras, J. Noncommut. Geom. (2007).
R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl.
Algebra (2008).
R. Bocklandt, T. Schedler, M. Wemyss, Superpotentials and higher order
derivations, J. Pure Appl. Algebra (2010).
K.A. Brown and J.J. Zhang, Dualizing complexes and twisted Hochschild
(co)homology for Noetherian Hopf algebras, J. Algebra (2008).
B. Davidson, Superpotential algebras and manifolds, Adv. Math. (2012).
V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139 (2006).
J. Goodman and U. Krähmer, Untwisting a twisted Calabi-Yau algebra,
arXiv:1304.0749 (2013).
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More selected references
P. Jørgensen and J.J. Zhang, Gourmet’s guide to Gorensteinness, Adv.
Math. (2000).
E. Kirkman, J. Kuzmanovich, J.J. Zhang, Gorenstein subrings of invariants
under Hopf algebra actions, J. Algebra (2009).
P. LeMeur, Crossed products of Calabi-Yau algebras by finite groups,
arXiv:1006.1082 (2010).
L.-Y. Liu, Q.-S. Wu, and C. Zhu, Hopf action on Calabi-Yau algebras,
Contemp. Math. 562 (2012).
M. Reyes, D. Rogalski, J.J. Zhang, Skew Calabi-Yau algebras and
homological identities, arXiv:1302.0437 (2013).
M. Van den Bergh, Calabi-Yau algebras and superpotentials,
arXiv:1008.0599 (2010).
Manuel Reyes (Bowdoin College)
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Even more selected references
Q.-S. Wu and C. Zhu, Skew group algebras of Calabi-Yau algebras, J.
Algebra (2011).
A. Yekutieli and J.J. Zhang, Homological transcendence degree, Proc.
London Math. Soc. (2006).
X. Yu and Y. Zhang, The Calabi-Yau property of smash products, J.
Algebra (2012).
Thank you!
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