# Linking representation theory, singularity theory and non-commutative algebraic geometry (12w5067)

Arriving in Banff, Alberta Sunday, May 6 and departing Friday May 11, 2012

## Organizers

José Antonio de la Peña (Centro de Investigación en Matemáticas)

Vlastimil Dlab (Carleton University)

Osamu Iyama (Nagoya University)

Helmut Lenzing (University of Paderborn)

## Objectives

The proposed Workshop will deal with links between three main subjects: Representation Theory of Finite Dimensional Algebras, Singularity Theory and Non-commutative Algebraic Geometry. The main aim is to strengthen existing links and to uncover new links between those subjects. We are confident, by experience from previous workshops, to achieve such aims by giving the Workshop a proper focus defined by the following three test problems.

1. "Towers of abelian or triangulated categories" is an outcome of the analysis of the "Sequences of finite dimensional algebras", finding their expression (building law) in the limit of the accompanying module categories resp. derived categories. Here is a brief explanation how such towers arise. Assume $(A_n)$ is an infinite sequence of finite dimensional algebras, starting (for simplicity) with $A_1=k$ the base field and such that $A_{n+1}$ arises from $A_n$ as a one-point extension, typically by a "nice" $A_n$-module $E_n$.) Here, "nice" can be given different meanings; we mention two typical choices:(a) "nice"=indecomposable, (b) "nice"=exceptional, that is, $E_n$ has trivial endomorphism ring and no self-extensions. In case (b) one obtains so-called accessible algebras, introduced recently by Lenzing-de la Pe~{n}a.

Algebras, as above, all have finite global dimension. What is relevant here is that the module category $mod(A_n)$ sits nicely in $mod(A_{n+1})$ as an exact extension-closed subcategory, and that the bounded derived category $D^b(A_n)$ sits nicely in its successor $D^b(A_{n+1})$ as a triangulated subcategory, yielding towers of module categories and triangulated categories, respectively. We can interpret the direct limit (union) of the categories belonging to such an infinite tower as (a substitute for) a building law for $(A_n)$. These abelian (resp. triangulated) categories are new challenging mathematical objects. It is proposed to have a close look at them.

In particular, the above and related constructions provide a link between singularities and representation theory, making the limit categories particular interesting objects also for experts in singularity theory. We give one example for illustration. If $A_{2n-1}$ is given by the linear quiver $1 --> 2 -- > ... --> 2(n-1)$, together with all zero relations for compositions of three consecutive arrows, then the bounded derived category $D^b(A_n)$ is the (graded) singularity category of the triangle singularity $x^2+x^3+x^n$. From the preceding Workshop these categories are known to form an ADE-chain, extending the Dynkin diagrams, here the $E_n$-series beyond the range of classical Dynkin diagrams. The limit category in this case would thus - in a sense to be clarified - contain the information on this ADE-sequence.

Similar constructions can be done for abelian categories. Related to the above example is (in a sense not explained here) the tower of hereditary abelian categories $coh(mathbf{X}_n)$ of weighted projective lines of weight type $(2,3,n)$. The limit category thus gets an interpretation as the "category of coherent sheaves on the weighted projective line of weight type $(2,3,infty)$, which for $k$ the field of complex numbers, is conjecturally related to the modular group playing a dominant role in the theory of classical automorphic forms.

We note that the towers of finite dimensional semi-simple algebras have been studied in Functional Analysis already in the 90-ties and wish to refer to the investigations of F. M. Goodman, P. de la Harpe, and V. F. R. Jones in their monograph on Coxeter graphs and towers of algebras. The work of Dlab-Ringel extended their work in terms of the theory of finite dimensional hereditary algebras with radical square zero and clarified, in particular, the "mysteries" of "discrete" nature of the set of all possible values of the Jones index. Towers of algebras also occur implicitly in recent work by Angeleri-de la Pe~{n}a (2009). Depending on a more careful analysis during the next two-years, we intend to include an investigation of such towers related to theory of C*-algebras also in the proposed Workshop program.

To summarize: The challenge consists in investigating the limit (or union) of such towers of abelian and triangulated categories, related to suitably chosen sequences (towers) of finite dimensional algebras, and to exploit its bridge function between representation theory, singularity theory, theory of C*-algebras and other areas of mathematics.

2. The problem to understand the "Non-commutative projective geometry attached to preprojective algebras" is another outcome

of the preceding Workshop on "Test problems for the theory of finite dimensional algebras". The problem is around for about 25 years, so many properties are known but, still, a proper understanding is lacking and many fundamental questions remain open until today. Assume $A=kQ$ is the finite dimensional hereditary path algebra $A=kQ$ of an acyclic, finite, connected quiver $Q$ over a base field $k$. Then $A$ has an attached (graded) preprojective algebra $Pi=Pi(A)$ which can be defined combinatorially in terms of $Q$ (due to I.M. Gelfand and V.A. Ponomarev) or also as an orbit algebra of the (inverse) Auslander-Reiten translation. In this case the component of degree $n$ is given as $Hom(A,tau^{-n}A)$. For the present purpose, it is more useful to use the homological characterization of $Pi$ as the positively graded tensor algebra of the (A,A)-bimodule $M=Ext^1(DA,A)$, where $D$ refers to the formation of the $k$-dual. Thus the homogeneous component of degree $n$ equals the $n$th tensor power of $M$. The attached geometric object is a virtual 'space' $mathbf{X}$, whose category of coherent sheaves $coh(mathbf{X})$ arises from $Pi$ by Serre construction in forming the category of tails of graded finitely presented modules (it is thus allowed to change finitely many components). And, conversely, $Pi$ serves as the homogeneous coordinate algebra of the 'space' $mathbf{X}$.

If $A$ has tame representation type, then $mathbf{X}$ is a weighted projective line, and hence the geometry of $mathbf{X}$ is well understood. (In terms of differential geometry $mathbf{X}$ is then a 2-orbifold with cone points, if $k$ is the base field of complex numbers.)

Many aspects of $coh(mathbf{X})$ are also well understood for a wild quiver $Q$: we have Serre duality hence Auslander-Reiten theory, we have derived equivalence to the category $mod(A)$ of finite dimensional $A$-modules and know the shape of all Auslander-Reiten components. Many aspects in the wild situation however remain mysterious since then $coh(mathbf{X})$ has no simple objects and, hence, no natural concept of points for $mathbf{X}$ is in reach. In particular, we still miss a geometric interpretation (in terms of the hypothetical $mathbf{X}$) of the set of Auslander-Reiten components for the regular part, that is, of Kerner's exotic space $Omega(A)$. This aspect of the problem is related to old work by Lenzing and Baer-Geigle-Lenzing from the period 1985-1990 and to current work by Mori, Minamoto and others.

It is a particular challenge that this 'space' enters naturally in many different mathematical contexts, explaining also its relevance. Below, we list a selection of such interrelations, some of them to be considered as a result of (the preparation for) our BIRS-Workshop in September 2010 on "Test problems for the theory of finite dimensional algebras".

(a) As shown by Lenzing-de la Pe{~na}, the set of AR-components for the category of vector bundles on a weighted projective line of wild weight type $(p_1,ldots,p_t)$ is in natural bijection to the set of regular AR-components for a wild hereditary path algebra $kQ$ such that the graph underlying $Q$ is the star-shaped graph $[p_1,ldots,p_t]$ whose branch-lengths are given by the weight sequence of $mathbf{X}$.

(b) Fuchsian singularities and triangle singularities $x_1^{p_1}+x_2^{p_2}+x_3^{p_3}$ whose exponents satisfy $1/p_1+1/p_2+1/p_3<1$ yield triangulated categories whose sets of AR-components are in natural bijection to those from a category of vector bundles on a weighted projective line of wild type, hence to a `Kerner space' $Omega{A}$ for a suitable $A$. This is related to recent work of Kajiura-Saito-Takahashi, resp. Lenzing-de la Pe{~na} for Fuchsian singularities and Kussin-Lenzing-Meltzer for triangle singularities.

(c) Related to the previous examples, the derived categories for the algebras $A_{2(n-1)}$, discussed above, has a set of AR-components which is in natural bijection to the set of AR-components for the weighted projective line of weight type $(2,3,n)$, provided $ngeq7$. This is due to Kussin-Lenzing-Meltzer. The algebras $A_n$ and their relations to other problems have been further investigated recently by Happel-Seidel, Ladkani, Ladkani-Chen and Ringel.

(d) Ringel and Schmidmeier have investigated in 2008 ff. the invariant subspace problem for nilpotent operators (of nilpotency degree $n$) in a graded and an ungraded version. The resulting categories are Frobenius. In the graded case, due to results Kussin-Lenzing-Meltzer (2010) the attached triangulated category is triangle-equivalent to a stable category of vector bundles over a weighted projective line of type $(2,3,n)$ . For $ngeq 7$ we thus obtain a natural bijection between the set of all AR-components for the invariant subspace problem and the `space' $Omega{A}$ for a suitable $A$.

3. "Non-commutative (exotic) spaces arising from higher preprojective algebras".

Auslander-Reiten theory, which is central for most aspects of the finite dimensional representation theory, has (geometrically) a somewhat one-dimensional flavor, since the fundamental Auslander-Reiten duality formula, responsible for the existence of almost-split sequences, involves only first extensions, and thus resembles Serre duality for a 1-dimensional smooth projective curve. (Algebraically, for some reason not to be discussed here, one likes more to attach dimension two to this situation.) For a very long time it thus has been an open question whether appropriate higher analogues of the theory, that is, higher Auslander-Reiten theories exist where higher extensions are taken into account.

In the hands of O. Iyama and his collaborators this has been achieved in recent years; the theory is in rapid progress with many relations to Commutative Algebra (Cohen-Macaulay modules), Singularity theory and cluster categories (again in a higher version). What is relevant here is that, in particular, tilting modules, Auslander algebras and preprojective algebras now have such higher analogues. Here, in order to waive the restrictions of global dimension one or path algebras, we are interested in the higher preprojective algebras. Such (higher) preprojective algebras also exist for finite dimensional algebras $A$ of any global dimension $d$ by defining the graded $(d+1)$-preprojective algebra $Pi=Pi_{d+1}(A)$ as the tensor algebra of the $(A,A)$-bimodule $Ext^d(DA,A)$. The investigation of the properties of higher preprojective algebras is a matter of much current research. We propose to investigate the ring-theoretic properties of $Pi_{d+1}(A)$ wich are relevant for a study of the virtual non-commutative space $mathbf{X}$, whose projective coordinate ring equals $Pi$.

An important case is when $Ext^i(DA,A)$ vanishes for any $i

We note that in the "higher preprojective context" we expect a lot of new results, but still of a shape that is less specific than for the classical case of preprojective algebras for quivers. Particular questions to be addressed here are the following:

(a) When has $Pi$ a big center? To which extent does this correspond to tameness of $A$? Correspondingly, does a small center of $Pi$ relate in some way to wild representation type? (Such a connection is known to hold for the classical preprojective algebra $Pi=Pi_2(A)$ for $A$ hereditary by results of Baer-Geigle-Lenzing. Additionally, Iyama has preliminary results for the case of higher preprojective algebras.)

(b) How do the properties of $coh(mathbf{X})$ react on a big, resp. small center of $Pi$?

(c) When does the bounded derived category of $coh(mathbf{X})$ have a tilting object, Serre duality or almost-split triangles?

(c) What are the general properties of the correspondences $A<->Pi<->mathbf{X}$?

(d) Will we encounter Kerner's exotic spaces also in this context?

1. "Towers of abelian or triangulated categories" is an outcome of the analysis of the "Sequences of finite dimensional algebras", finding their expression (building law) in the limit of the accompanying module categories resp. derived categories. Here is a brief explanation how such towers arise. Assume $(A_n)$ is an infinite sequence of finite dimensional algebras, starting (for simplicity) with $A_1=k$ the base field and such that $A_{n+1}$ arises from $A_n$ as a one-point extension, typically by a "nice" $A_n$-module $E_n$.) Here, "nice" can be given different meanings; we mention two typical choices:(a) "nice"=indecomposable, (b) "nice"=exceptional, that is, $E_n$ has trivial endomorphism ring and no self-extensions. In case (b) one obtains so-called accessible algebras, introduced recently by Lenzing-de la Pe~{n}a.

Algebras, as above, all have finite global dimension. What is relevant here is that the module category $mod(A_n)$ sits nicely in $mod(A_{n+1})$ as an exact extension-closed subcategory, and that the bounded derived category $D^b(A_n)$ sits nicely in its successor $D^b(A_{n+1})$ as a triangulated subcategory, yielding towers of module categories and triangulated categories, respectively. We can interpret the direct limit (union) of the categories belonging to such an infinite tower as (a substitute for) a building law for $(A_n)$. These abelian (resp. triangulated) categories are new challenging mathematical objects. It is proposed to have a close look at them.

In particular, the above and related constructions provide a link between singularities and representation theory, making the limit categories particular interesting objects also for experts in singularity theory. We give one example for illustration. If $A_{2n-1}$ is given by the linear quiver $1 --> 2 -- > ... --> 2(n-1)$, together with all zero relations for compositions of three consecutive arrows, then the bounded derived category $D^b(A_n)$ is the (graded) singularity category of the triangle singularity $x^2+x^3+x^n$. From the preceding Workshop these categories are known to form an ADE-chain, extending the Dynkin diagrams, here the $E_n$-series beyond the range of classical Dynkin diagrams. The limit category in this case would thus - in a sense to be clarified - contain the information on this ADE-sequence.

Similar constructions can be done for abelian categories. Related to the above example is (in a sense not explained here) the tower of hereditary abelian categories $coh(mathbf{X}_n)$ of weighted projective lines of weight type $(2,3,n)$. The limit category thus gets an interpretation as the "category of coherent sheaves on the weighted projective line of weight type $(2,3,infty)$, which for $k$ the field of complex numbers, is conjecturally related to the modular group playing a dominant role in the theory of classical automorphic forms.

We note that the towers of finite dimensional semi-simple algebras have been studied in Functional Analysis already in the 90-ties and wish to refer to the investigations of F. M. Goodman, P. de la Harpe, and V. F. R. Jones in their monograph on Coxeter graphs and towers of algebras. The work of Dlab-Ringel extended their work in terms of the theory of finite dimensional hereditary algebras with radical square zero and clarified, in particular, the "mysteries" of "discrete" nature of the set of all possible values of the Jones index. Towers of algebras also occur implicitly in recent work by Angeleri-de la Pe~{n}a (2009). Depending on a more careful analysis during the next two-years, we intend to include an investigation of such towers related to theory of C*-algebras also in the proposed Workshop program.

To summarize: The challenge consists in investigating the limit (or union) of such towers of abelian and triangulated categories, related to suitably chosen sequences (towers) of finite dimensional algebras, and to exploit its bridge function between representation theory, singularity theory, theory of C*-algebras and other areas of mathematics.

2. The problem to understand the "Non-commutative projective geometry attached to preprojective algebras" is another outcome

of the preceding Workshop on "Test problems for the theory of finite dimensional algebras". The problem is around for about 25 years, so many properties are known but, still, a proper understanding is lacking and many fundamental questions remain open until today. Assume $A=kQ$ is the finite dimensional hereditary path algebra $A=kQ$ of an acyclic, finite, connected quiver $Q$ over a base field $k$. Then $A$ has an attached (graded) preprojective algebra $Pi=Pi(A)$ which can be defined combinatorially in terms of $Q$ (due to I.M. Gelfand and V.A. Ponomarev) or also as an orbit algebra of the (inverse) Auslander-Reiten translation. In this case the component of degree $n$ is given as $Hom(A,tau^{-n}A)$. For the present purpose, it is more useful to use the homological characterization of $Pi$ as the positively graded tensor algebra of the (A,A)-bimodule $M=Ext^1(DA,A)$, where $D$ refers to the formation of the $k$-dual. Thus the homogeneous component of degree $n$ equals the $n$th tensor power of $M$. The attached geometric object is a virtual 'space' $mathbf{X}$, whose category of coherent sheaves $coh(mathbf{X})$ arises from $Pi$ by Serre construction in forming the category of tails of graded finitely presented modules (it is thus allowed to change finitely many components). And, conversely, $Pi$ serves as the homogeneous coordinate algebra of the 'space' $mathbf{X}$.

If $A$ has tame representation type, then $mathbf{X}$ is a weighted projective line, and hence the geometry of $mathbf{X}$ is well understood. (In terms of differential geometry $mathbf{X}$ is then a 2-orbifold with cone points, if $k$ is the base field of complex numbers.)

Many aspects of $coh(mathbf{X})$ are also well understood for a wild quiver $Q$: we have Serre duality hence Auslander-Reiten theory, we have derived equivalence to the category $mod(A)$ of finite dimensional $A$-modules and know the shape of all Auslander-Reiten components. Many aspects in the wild situation however remain mysterious since then $coh(mathbf{X})$ has no simple objects and, hence, no natural concept of points for $mathbf{X}$ is in reach. In particular, we still miss a geometric interpretation (in terms of the hypothetical $mathbf{X}$) of the set of Auslander-Reiten components for the regular part, that is, of Kerner's exotic space $Omega(A)$. This aspect of the problem is related to old work by Lenzing and Baer-Geigle-Lenzing from the period 1985-1990 and to current work by Mori, Minamoto and others.

It is a particular challenge that this 'space' enters naturally in many different mathematical contexts, explaining also its relevance. Below, we list a selection of such interrelations, some of them to be considered as a result of (the preparation for) our BIRS-Workshop in September 2010 on "Test problems for the theory of finite dimensional algebras".

(a) As shown by Lenzing-de la Pe{~na}, the set of AR-components for the category of vector bundles on a weighted projective line of wild weight type $(p_1,ldots,p_t)$ is in natural bijection to the set of regular AR-components for a wild hereditary path algebra $kQ$ such that the graph underlying $Q$ is the star-shaped graph $[p_1,ldots,p_t]$ whose branch-lengths are given by the weight sequence of $mathbf{X}$.

(b) Fuchsian singularities and triangle singularities $x_1^{p_1}+x_2^{p_2}+x_3^{p_3}$ whose exponents satisfy $1/p_1+1/p_2+1/p_3<1$ yield triangulated categories whose sets of AR-components are in natural bijection to those from a category of vector bundles on a weighted projective line of wild type, hence to a `Kerner space' $Omega{A}$ for a suitable $A$. This is related to recent work of Kajiura-Saito-Takahashi, resp. Lenzing-de la Pe{~na} for Fuchsian singularities and Kussin-Lenzing-Meltzer for triangle singularities.

(c) Related to the previous examples, the derived categories for the algebras $A_{2(n-1)}$, discussed above, has a set of AR-components which is in natural bijection to the set of AR-components for the weighted projective line of weight type $(2,3,n)$, provided $ngeq7$. This is due to Kussin-Lenzing-Meltzer. The algebras $A_n$ and their relations to other problems have been further investigated recently by Happel-Seidel, Ladkani, Ladkani-Chen and Ringel.

(d) Ringel and Schmidmeier have investigated in 2008 ff. the invariant subspace problem for nilpotent operators (of nilpotency degree $n$) in a graded and an ungraded version. The resulting categories are Frobenius. In the graded case, due to results Kussin-Lenzing-Meltzer (2010) the attached triangulated category is triangle-equivalent to a stable category of vector bundles over a weighted projective line of type $(2,3,n)$ . For $ngeq 7$ we thus obtain a natural bijection between the set of all AR-components for the invariant subspace problem and the `space' $Omega{A}$ for a suitable $A$.

3. "Non-commutative (exotic) spaces arising from higher preprojective algebras".

Auslander-Reiten theory, which is central for most aspects of the finite dimensional representation theory, has (geometrically) a somewhat one-dimensional flavor, since the fundamental Auslander-Reiten duality formula, responsible for the existence of almost-split sequences, involves only first extensions, and thus resembles Serre duality for a 1-dimensional smooth projective curve. (Algebraically, for some reason not to be discussed here, one likes more to attach dimension two to this situation.) For a very long time it thus has been an open question whether appropriate higher analogues of the theory, that is, higher Auslander-Reiten theories exist where higher extensions are taken into account.

In the hands of O. Iyama and his collaborators this has been achieved in recent years; the theory is in rapid progress with many relations to Commutative Algebra (Cohen-Macaulay modules), Singularity theory and cluster categories (again in a higher version). What is relevant here is that, in particular, tilting modules, Auslander algebras and preprojective algebras now have such higher analogues. Here, in order to waive the restrictions of global dimension one or path algebras, we are interested in the higher preprojective algebras. Such (higher) preprojective algebras also exist for finite dimensional algebras $A$ of any global dimension $d$ by defining the graded $(d+1)$-preprojective algebra $Pi=Pi_{d+1}(A)$ as the tensor algebra of the $(A,A)$-bimodule $Ext^d(DA,A)$. The investigation of the properties of higher preprojective algebras is a matter of much current research. We propose to investigate the ring-theoretic properties of $Pi_{d+1}(A)$ wich are relevant for a study of the virtual non-commutative space $mathbf{X}$, whose projective coordinate ring equals $Pi$.

An important case is when $Ext^i(DA,A)$ vanishes for any $i

We note that in the "higher preprojective context" we expect a lot of new results, but still of a shape that is less specific than for the classical case of preprojective algebras for quivers. Particular questions to be addressed here are the following:

(a) When has $Pi$ a big center? To which extent does this correspond to tameness of $A$? Correspondingly, does a small center of $Pi$ relate in some way to wild representation type? (Such a connection is known to hold for the classical preprojective algebra $Pi=Pi_2(A)$ for $A$ hereditary by results of Baer-Geigle-Lenzing. Additionally, Iyama has preliminary results for the case of higher preprojective algebras.)

(b) How do the properties of $coh(mathbf{X})$ react on a big, resp. small center of $Pi$?

(c) When does the bounded derived category of $coh(mathbf{X})$ have a tilting object, Serre duality or almost-split triangles?

(c) What are the general properties of the correspondences $A<->Pi<->mathbf{X}$?

(d) Will we encounter Kerner's exotic spaces also in this context?