## 3-Calabi-Yau algebras

Last month I spoke at the workshop Subfactors, higher geometry, higher twists and almost Calabi-Yau algebras. The workshop was part of the six-month program Operator algebras: subfactors and their applications at the Newton Institute. I talked about the paper The classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations written with Izuru Mori. The slides for my talk are here (Talk-3-CY_algebras).

I was rather surprised to be asked to speak at the workshop because I don’t know much about subfactors. I’ve been an interested spectator ever since Vaughan Jones proved his remarkable Index Theorem in 1984. But my own work has not interacted with subfactors at all. Still, I was keen to go because of my general interest in subfactors and because it was a chance to return to Cambridge, one of the most beautiful cities in the world.

I lived in Cambridge for about 4 months in late 1975 and early 1976. I had long hair and a beard then. (I had just finished my undergraduate degree at the University of Canterbury in New Zealand.) I had a job at the Pye Electronics factory. Pretty low-level stuff but my appearance made potential employers wary. After a couple of months packing things in cardboard boxes with styrofoam bubbles I shaved off my beard, cut my hair, borrowed 50 pounds from Barclays Bank, bought a suit and briefcase, and found a job in London at Datastream working as something like a quant, before the word “quant” was invented. While living in London I visited Cambridge several times with my girlfriend, Gillian, who was doing her Ph.D. in pharmacology at University College. We punted on the Cam, picnicked, and drank champagne under the willows, on sunny days. Good memories. I spent another 3 months in Cambridge in late 2006, again at the Newton Institute. So, it was nice to be invited to return even though I felt I didn’t have much to add to the workshop. I was tempted to decline but decided to go once the slate of speakers was drawn up: at least half the speakers were not experts on subfactors, and I was quite interested in the topics they planned to speak on. So, I went. And I was glad I did. I found the workshop very stimulating. Lots of interesting talks. I was delighted to hear and meet Akhil Matthews whose blog I have been reading since he was in high school!

What is a subfactor? First, what is a factor? A von Neumann algebra, of the $C^*$-flavor not of the von Neumann regular algebra flavor, is called a factor if its center is ${\mathbb C}$. A subfactor is a von Neumann subalgebra of a factor that is itself a factor. It’s best to read about these things on Wikipedia. If $N$ is a subfactor of a factor $M$ Jones Index Theorem concerns the number $[M:N]$, the index of $N$ in $M$. Roughly speaking, $[M:N]$ measures the size of $M$ relative to $N$. The Index Theorem says the index can be any real number $\ge 4$ or $4 \cos^2(\pi/n)$ for an integer $n \ge 3$, and all these values occur. If you are interested, I suggest you read the survey article On the origin and development of subfactors and quantum topology by Vaughan Jones. That article was written in 1995. A more up-to-date account can be found in the very beautiful and stimulating article The classification of sub factors of index at most 5. If nothing else, I suggest you read the introduction; the parallel with Galois theory provides a helpful framework.

At the workshop, Iyama gave a very beautiful talk about pre-projective algebras aimed at non-experts. It was a “perfect” talk: the “perfect” examples and the most efficient and elegant perspectives on the main themes. Gus Lehrer spoke about some of his old work on Temperley-Lieb algebras and categories. For 30 years, people have asked me whether I had me Gus. I suppose they thought that all antipodean algebraists knew each other. Heck, I had only met Vaughan Jones once before even though we lived in the same town for several years when we were in high school, him as a student at Auckland Grammar, me at King’s College. (I should boast that Vaughan’s Fields Medal means that New Zealand has more Fields Medallists per capita than any other country.) Auckland Grammar and King’s College have been keen rivals for the best part of a century, especially in sporting events, the most important of which is the annual rugby match between their First XV’s. Probably, sometime in the early 1970’s Vaughan and I stood in the rain on opposite sidelines supporting our respective schools.

A couple of months before the workshop I talked about my work with Izuru in our weekly Algebra & Algebraic Geometry seminar. It wasn’t a good talk. I had also spoken about our work in Shanghai in late 2015. That wasn’t a good talk either. I think I am finally beginning to understand what we proved, and I think the talk in Cambridge reflected this.

The organizers scheduled me to speak on my birthday. They had also scheduled the conference dinner for that day too 🙂 The dinner was at Christ’s College. The size and magnificence of the college is breathtaking. Founded in 1437. Alumni include Darwin and John Milton…

## I’m back

I’m back. After a five-year hiatus. Perhaps only briefly. I’m going to post the slides from a some talks I’ve given over the last few years. That won’t require too much work.

## Are these algebras 4-dimensional Sklyanin algebras?

In their paper “Noncommutative Homological Mirror Functor”, Cho, Hong, and Lau, produce a 2-parameter family of non-commutative algebras and conjecture that all of them are 4-dimensional Sklyanin algebras. Their family is parametrized by the points on ${\mathbb P}^1 \times {\mathbb P}^1$. There is a 2-parameter family of 4-dimensional Sklyanin algebras; the parameters consist of an elliptic curve $E$ and a point $\tau \in E$.

4-dimensional Sklyanin algebras are a very interesting class of non-commutative algebras that are deformations of the polynomial ring on 4 variables. They are, like the polynomial ring, graded algebras, i.e., there is a notion of degree that behaves like the more familiar notion of degree for polynomials: in particular, ${\rm deg}(ab)={\rm deg}(a)+{\rm deg}(b)$. By “deformation” I mean that the 4-dimensional Sklyanin algebras belong to a flat family of graded algebras, one of which is the polynomial ring on 4 variables. Flatness simply means that the dimension of the degree-$n$ component of the algebra is the same for all algebras in the family.

What makes them so interesting is the presence of the elliptic curve in the background. The pair $(E,\tau)$ “controls” the properties of the corresponding Sklyanin algebra $A(E,\tau)$. For example, $A(E,\tau)$ is a finite module over its center if and only if $\tau$ is a torsion point. To make this precise I need to fix a group law on $E$; if I do that I can rephrase the previous sentence as follows:  $A(E,\tau)$ is isomorphic to a subalgebra of a matrix algebra over a commutative ring if and if and only if the translation automorphism $E \to E, \,\, p \mapsto p+\tau$, has finite order.

Sklyanin algebras appear naturally. In rough translation, the opening of Sklyanin’s 1982 paper in which 4-dimensional Sklyanin algebras first appeared reads as follows.
“One of the strongest methods of investigating the exactly solvable models of quantum and statistical physics is the quantum inverse problem method (QIPM). The problem of enumerating the discrete quantum systems that can be solved by the QIPM reduces to the problem of enumerating the operator-valued functions $L(u):V \to V$ that satisfy
the relation $latex R(u)L_{12}(u+v) L_{23}(v) \; = \; L_{23}(v) L_{12}(u+v)R(u)$ for a fixed solution $R(u):V \otimes V \to V \otimes V$ to the quantum Yang-Baxter equation. […] The present paper is devoted to a study of the above equation in the case when $R(u)$ is the simplest solution to the quantum Yang-Baxter equation, namely that found by R. Baxter. During our investigation it turned out that it is necessary to bring into the picture new algebraic structures, namely, the quadratic algebras of Poisson brackets and the quadratic generalization of the universal enveloping algebra of a Lie algebra. The theory of these mathematical objects is surprisingly reminiscent of the theory of Lie algebras, the difference being that it is more complicated. In our opinion, it deserves the greatest attention of mathematicians.”

Sklyanin algebras also appeared in the work of Connes and Dubois-Violette on non-commutative 3-spheres. So, it would be very nice if they were to appear in an entirely new setting, namely that of Cho, Hong, and Lau. Alex Chirvasitu and I settled their conjecture and posted a paper about this a couple of months ago. Alas, mostly in the negative, but their conjecture fails in a somewhat interesting way. I gave a talk about their work today at the AMS Sectional Meeting in Pullman, WA. Here’s a link to the talk.

## Non-noetherian subalgebras of finitely generated commutative algebras

First example

I first encountered a non-noetherian subalgebra of a finitely generated commutative algebra in the early 1980’s. Let $R=k[x,y]$ be the commutative polynomial ring in two variables over a field $k$. The subalgebra $S:=k+xR$ is not noetherian. It is a pleasant exercise to show that the ideal ${\mathfrak m} := xR$ is not a finitely generated ideal of $S$. As an ideal of $S$ it is equal to $xS+xyS+xy^2S+\cdots$.

A non-commutative contrast

If $A$ is the first Weyl algebra over a field $k$ of characteristic zero with “standard” generators $\partial$ and $t$, then $k+At$ is both left and right noetherian and is a finitely generated $k$-algebra. This was proved in a beautiful paper by Chris Robson, Idealizers and Hereditary Noetherian Prime Rings, J. Algebra, 22 (1972) 45-81; MR0299639 (45 #8687). Google Scholar lists 124 citations to this paper. (Math Reviews lists 26 citations, but the Google Scholar list is more complete.) Robson’s paper contains much more than this example but I find the example a striking illustration of how much the commutative and non-commutative worlds differ. Part of the point, in relation to the previous paragraph, is that the first Weyl algebra is often seen as a non-commutative analogue of the polynomial ring on two variables. In fact, there is a flat family of algebras with the special fiber being the polynomial ring and the other fibers isomorphic to the first Weyl algebra.

I don’t need to tell those who already know this example any more. Those who haven’t seen it before deserve a little more explanation.

The first Weyl algebra

The first Weyl algebra, which I will denote by $A$, over a field $k$, is the free algebra $k\langle t,\partial\rangle$ modulo the relation $\partial t-t\partial =1$. This is a standard example in non-commutative ring theory. If the characteristic of $k$ is zero, which I will assume from now on, then $A$ is isomorphic to the ring of differential operators on the polynomial ring in one variable: if that polynomial ring is $k[t]$, then $t \in A$ acts on $k[t]$ as multiplication by $t$ and $\partial \in A$ acts on $k[t]$ as the derivative $d/dt$. That these two operators on $k[t]$ satisfy the relation $\partial t-t\partial =1$ is the fact, well-known to our calculus students, that $(tf)'=tf'+f$.

The first Weyl algebra is considered a non-commutative analogue of the polynomial ring on two variables. It has the following properties:

1. $\{t^i\partial^j \; | \; i,j \ge 0\}$ is a $k$-basis for $A$;
2. $A$ has a filtration $F^nA= {\rm span}\{t^i\partial^j \; | \; i+j \le n\}$ such that the associated graded algebra is isomorphic to the polynomial ring on two variables;
3. $A$ is left and right noetherian;
4. $A$ is a domain, i.e., every product of non-zero elements is non-zero;
5. $A$ has projective dimension 2 as a left module over $A \otimes A^{op}$;
6. $A$ is a simple ring: its only two-sided ideals are $A$ and $\{0\}$;
7. $A$ has global homological dimension 1.

The first 5 of these properties, with appropriate changes of notation, hold for the polynomial ring on two variables.

A talk by Lance Small in 1982

In an algebra seminar talk at USC on 10/18/1982, Lance Small provided the ring

$B:=\begin{pmatrix} R & Rx \\ R & k+Rx \end{pmatrix}$

as an example of a finitely generated (!) prime subalgebra of matrices over a commutative ring (he was interested in the fact that it satisfies a polynomial identity) whose center, which is isomorphic to $k+Rx$, is not finitely generated. I show $B$ is finitely generated below. It is probably appropriate to mention the Artin-Tate Lemma at this point because it implies that $B$ is not a finitely generated module over its center.

The Artin-Tate Lemma. Let $k$ be a field and $A$ a finitely generated $k$-algebra. If $A$ is finitely generated as a module over a central subalgebra $Z$, then $Z$ is finitely generated. Hence $A$ is left and right noetherian.

In his talk, Lance also mentioned that the ring

$\begin{pmatrix} A & At \\ A & k+At \end{pmatrix}$

is noetherian and that $k+At$ is a finitely generated algebra.

Is the commutative example $k+xR$ more than a curiosity?

I considered the fact that $k+xR$ is not noetherian a curiosity that I didn’t need to pay much attention to. Same with Lance Small’s example, the ring $B$. Good to know, but then time to get on with other business. My attitude changed last summer when Charlie Beil gave a talk in Seattle about his paper Non-noetherian Geometry and Toric Superpotential Algebras.

Charlie showed that the algebra $B$ that appeared in Lance Small’s talk appears in a very natural way. Natural in the sense that similar constructions produce very beautiful, very useful, rings that have been intensively studied over the past several years for reasons both internal to mathematics (non-commutative resolutions of affine Gorenstein singularities) and arising from quiver gauge theories (in string theory).

There are several points of view on the context in which to view Charlie’s paper; one is the use of superpotential algebras and associated brane tilings to construct non-commutative crepant resolutions of certain affine varieties; another is that the superpotential algebras lead to quiver gauge theories that are relevant to string theory. These other papers produce non-commutative algebras with good properties that are finitely generated modules over their centers and these centers are finitely generated commutative rings that are coordinate rings of certain affine singular 3-folds. However, for these constructions to work the brane tiling must satisfy a certain “consistency” condition. I don’t want to say more about that here—if you want to know more google on consistent brane tiling.

The starting point for Charlie’s work is that brane tilings that fail to be consistent, or are non-cancellative in Charlie’s terminology, still produce gauge theories of physical interest. The simplest example of a  non-cancellative brane tiling corresponds to the algebra $B=kQ/(yba-bay)$ where $Q$ is the following quiver:

A standard way to view a quotient, let’s call it $B$, of a path algebra of a quiver is to use the idempotents (trivial paths) $e_i$ at the vertices $i \in Q_0$ to realize $B$ as a tiled matrix algebra with “tiles” $e_i B e_j$, $i,j \in Q_0$. When $Q$ has only two vertices, labelled 1 and 2 say, we have

$B=\begin{pmatrix} e_1Be_1 & e_1Be_2 \\ e_2 B e_1 & e_2Be_2 \end{pmatrix}.$

The “tile” $e_i B e_j$ is spanned by the paths that begin at $j$ and end at $i$; I adopt the convention that $pq$ means first traverse the path $q$, then $p$. The consistency conditions, which I have not stated, imply that all $e_iBe_i$ are isomorphic to a single finitely generated commutative algebra.

Now consider Charlie’s example above. Paths that begin and end at vertex 1 are words in $y$ and $ba$. The relation $yba-bay=0$ tells us that, as elements of $B$, $y$ commutes with $ba$. It follows that $e_1Be_1$ is the polynomial ring $k[y,ba]$. Non-trivial paths that begin and end at vertex 2 are of the form $apb$ where $p$ is a path that begins and ends at $1$. Hence

$B=\begin{pmatrix} k[y,ba] & k[y,ba]b \\ ak[y,ba] & k+ak[y,ba]b \end{pmatrix}=\begin{pmatrix} k[x,y] & k[x,y]b \\ ak[x,y] & k+ak[x,y]b \end{pmatrix}$

where the second equality comes from writing $x =ba$. A simple calculation shows that there is an isomorphism

$B \cong \begin{pmatrix} R & Rx \\ R & k+Rx \end{pmatrix}$

given by the map

$\Phi\begin{pmatrix} p & qb \\ ar & \lambda+asb \end{pmatrix}=\begin{pmatrix} p & qx \\ r & \lambda+sx \end{pmatrix}$.

What is the appropriate geometric picture for $B$ and the non-noetherian scheme $Spec(k+xR)$?

This is the question Charlie addresses in his paper. There is a rather well-developed theory for non-commutative rings that are finite, i.e., finitely generated, modules over their centers.  The ring $B$ is almost a finite module over its center so some variation on that theory should apply to $B$ in a useful way.

The first Weyl algebra has a non-noetherian subalgebra

Bavula has shown that the first Weyl algebra over a field of characteristic zero contains a non-noetherian subalgebra. Its construction is a little tricky so I’ll just refer you to his paper if you want to learn more about it.

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## Drinking and homological algebra?

Popular wisdom says one should not drink and derive but some of you might enjoy a little wine with your derived functors so, with that thought in mind, I suggest you try a glass of this from Tor Vineyards in California.

## Wine for Buchweitz

I have found the perfect wine for Ragnar Buchweitz who is giving two talks on Maximal Cohen-Macaulay modules and Orlov’s Theorem. One might say that the wine is complex and after having one bottle one often needs another, and another, and another. Syzygy is produced in Washington State, and it is good. As are Ragnar’s talks.

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## Two-sided ideals in a universal localization

Let $Q$ be a quiver with the property that only finitely many arrows emerge from each vertex—this condition is commonly referred to by saying the graph is row-finite but I find that terminology unsatisfactory; row-finite needs too much explanation (the incidence matrix and the convention about the entries in it).

Pinar Colak proved that every two-sided ideal in the Leavitt path algebra $L(Q)$ is generated by its intersection with the path algebra $kQ$. See Theorem 8 in his paper. He doesn’t phrase it like this because he doesn’t know that the canonical map $kQ \to L(Q)$ is injective; Proposition 4.1 in Xiao Wu Chen’s paper proves this map is injective.

Colak’s result is very precise. He proves that a two-sided ideal in $L(Q)$ is generated by elements of the form $e_i + \sum_{m=0}^n \lambda_m c^m$ where $e_i$ is the idempotent at a vertex $i$, $c$ is a cycle at $i$, and the $\lambda_i$ are scalars in the field $k$.

My question.

As I said in a previous post, $L(Q)$ is a universal localization of $kQ$ so Colak’s result prompted me to ask George Bergman the following question: if $f:R \to S$ is a  universal localization is every two-sided ideal $I$ in $S$ is generated by its preimage in $R$, i.e., by $f(f^{-1}(I))$? The answer is a big fat NO. The simplest example George provided is

$R= \begin{pmatrix} k & k+kx \\ 0 & k \end{pmatrix} \subset S=\begin{pmatrix} k[x] & k[x] \\ k[x] & k[x] \end{pmatrix}$

The ring $S$ is the universal localization of $R$ at the map $g:Re_{11} \to Re_{22}$ which is right multiplication by  $e_{12}$. The ring $S$ has many ideals that have zero intersection with  $R$; for example, any ideal of $S$ that is generated by an ideal of  the central copy of $k[x]$  whose generator has degree  $> 1$.

To see that $S$ is the universal localization of $R$ at $g$, first observe that $S \otimes g:S \otimes_R Re_{11} \to S\otimes_R Re_{22}$ is the map $Se_{11} \to Se_{22}$ given by right multiplication by $e_{12}$ and this is an isomorphism with inverse left multiplication by $e_{21}$. If $R_g$ denotes the universal localization its universal property says that the inclusion $R \to S$ must factor through $R_g$. Hence there is a homomorphism $R_g \to S$ with image the subalgebra of $S$ generated by $R$ and $e_{21}$. But that subalgebra is $S$ so $R_g \to S$ is surjective. I need to do a little more to show that $S$ is the universal localization; I think that should be done by using the fact that $R$ is the path algebra of the Kronecker quiver $Q$ which is

$\bullet \, {{\longrightarrow}\atop{\longrightarrow}} \, \bullet.$

Let’s label the arrows $a$ and $b$ and adjoin a new arrow $b^*$ in the opposite direction and impose the relation $bb^*=e_1$ and $b^*b=e_2$ where $e_1$ and $e_2$ are the idempotents at the vertices. Let’s write $S'$ for the path algebra modulo these relations. I think it is clear that $S'$ has the appropriate universal property to be the universal localization of $kQ =R$ at $b$. A simple calculation will show that $S' \cong S$.

Contrast with inverting elements in a ring.

If $\Sigma$ is a multiplicatively closed set of non-zero-divisors in a commutative ring $A$, then every ideal in the localization $A[\Sigma^{-1}]$ is generated by its intersection with $A$. To prove this we use the fact that elements in $A[\Sigma^{-1}]$ are of the form $ax^{-1}$ where $a \in A$ and $x \in \Sigma$; an ideal $I$ in $A[\Sigma^{-1}]$ that contains $ax^{-1}$ contains $a$ which is an element of $I \cap R$ so $ax^{-1}$ is in the ideal of $A[\Sigma^{-1}]$ that is generated by $I \cap R$.

This argument extends to one-sided ideals in localizations of non-commutative rings at Ore sets but there is an extra twist to the argument because one must apply the Ore condition to see that an element $ax^{-1}$ in a left ideal $I$ of the localization is equal to an element of the form $y^{-1}b$ in order to see that $ax^{-1}$ is in the left ideal of the localization that is generated by $b$, etc. I’m omitting the details.