Introduction to algebras of partial triangulations

Laurent Demonet

arXiv:1701.07564·math.RT·January 27, 2017

Introduction to algebras of partial triangulations

Laurent Demonet

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TL;DR

This paper introduces algebras derived from partial triangulations of marked surfaces, exploring their properties, relations to classical Jacobian and Brauer graph algebras, and their derived equivalences.

Contribution

It provides a gentle introduction to these algebras, highlighting their structure, properties, and connections to existing algebra classes, with results proven in prior work.

Findings

01

Algebras of partial triangulations have finite rank.

02

They include Jacobian and Brauer graph algebras as special cases.

03

Representation theory and derived equivalences are analyzed.

Abstract

The aim of this note is to give a gentle introduction to algebras of partial triangulations of marked surfaces, following the structure of a talk given during the 49th symposium on ring theory and representation theory, held in Osaka. This class of algebras, which always have finite rank, contains classical Jacobian algebras of triangulations of marked surfaces and Brauer graph algebras. We discuss representation theoretical properties and derived equivalences. All results are proven in arXiv:1602.01592, under slightly milder hypotheses.

Equations20

A \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

A \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

M \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

M \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces \ignorespaces

a^{m_{D}}

a^{m_{D}}

ab

c d

ec

(ab)^{m_{M}}

(ab)^{m_{M}}

a^{2}

Δ_{τ} ≅ e_{τ} Δ_{σ} e_{τ}

Δ_{τ} ≅ e_{τ} Δ_{σ} e_{τ}

W_{σ} := T \in E \sum α_{T} β_{T} γ_{T} - M \in M \sum \frac{λ _{M}}{m _{M}} ω_{u, M}^{m_{M}}

W_{σ} := T \in E \sum α_{T} β_{T} γ_{T} - M \in M \sum \frac{λ _{M}}{m _{M}} ω_{u, M}^{m_{M}}

M \in M \cap \partial Σ \sum \frac{d _{M} ( d _{M} - 1 )}{2} + M \in M \cap (Σ ∖ \partial Σ) \sum m_{M} d_{M}^{2} + f

M \in M \cap \partial Σ \sum \frac{d _{M} ( d _{M} - 1 )}{2} + M \in M \cap (Σ ∖ \partial Σ) \sum m_{M} d_{M}^{2} + f

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Taxonomy

TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology

Full text

INTRODUCTION TO ALGEBRAS OF PARTIAL TRIANGULATIONS

Laurent Demonet

Graduate School of Mathematics

Nagoya University

Furocho, Chikusaku

Nagoya 464-8602 JAPAN

[email protected]

Abstract.

The aim of this note is to give a gentle introduction to algebras of partial triangulations of marked surfaces, following the structure of a talk given during the 49th symposium on ring theory and representation theory, held in Osaka. This class of algebras, which always have finite rank, contains classical Jacobian algebras of triangulations of marked surfaces and Brauer graph algebras. We discuss representation theoretical properties and derived equivalences. All results are proven in [2], under slightly milder hypotheses.

The paper is in a final form and no version of it will be submitted for publication elsewhere.

1. The algebra of a partial triangulation

Let $k$ be a unital ring and $\Sigma$ be a connected compact oriented surface with or without boundary. We fix a non-empty finite set $\mathbb{M}\subset\Sigma$ of marked points (some of them may be on the boundary $\partial\Sigma$ ). For each $M\in\mathbb{M}$ , we fix an invertible coefficient $\lambda_{M}\in k^{\times}$ and a multiplicity $m_{M}\in\mathbb{Z}_{>0}$ . For simplicity, we suppose here that if $\Sigma$ is a sphere then $\#\mathbb{M}\geq 5$ and if $\Sigma$ is a disc then $\#\mathbb{M}\geq 3$ .

Definition 1.

An arc on $(\Sigma,\mathbb{M})$ is a continuous map $u:[0,1]\to\Sigma$ satisfying:

•

The restriction of $u$ to $(0,1)$ is an embedding into $\Sigma\setminus\mathbb{M}$ ;

•

Extremities [math] and $1$ are mapped to $\mathbb{M}$ .

We consider arcs up to homotopy relative to their endpoints in $\Sigma\setminus\mathbb{M}$ . Moreover, we exclude arcs that are homotopic to a marked point or to a boundary component, that is the closure of a connected component of $\partial\Sigma\setminus\mathbb{M}$ .

In this note, for simplicity, we exclude arcs that are loops enclosing a unique marked point $M\in\mathbb{M}$ with $m_{M}\leq 2$ . We do not make this restriction in [2].

Definition 2.

We say that two arcs $u$ and $v$ are compatible if, up to homotopy, they are non-crossing. Then, a partial triangulation of $(\Sigma,\mathbb{M})$ is a set $\sigma$ of arcs of $(\Sigma,\mathbb{M})$ that are pairwise compatible. If $\sigma$ is a maximal partial triangulation and each connected component of $\partial\Sigma$ contains at least a marked point, $\sigma$ is called a triangulation.

In order to define the algebra $\Delta_{\sigma}$ of a partial triangulation $\sigma$ , we first need to construct a quiver $Q_{\sigma}$ .

Definition 3.

The quiver $Q_{\sigma}$ has set of vertices $\sigma$ , and arrows are winding in $\Sigma$ between successive arcs around marked points counter-clockwisely. We call bouncing path of $Q_{\sigma}$ any path of length $2$ consisting of two arrows that are not successive around the same endpoint.

We give two examples to illustrate this definition:

Example 4.

In the left example $\Sigma$ is a disc with four marked points $A$ , $B$ , $C$ and $D$ ( $A$ and $B$ are on the boundary). In the right one, $\Sigma$ is a torus with two marked points $M$ and $N$ . The partial triangulations are depicted with thick lines.

[TABLE]

Bouncing paths of the first example are $ab$ , $cd$ , $de$ and $ec$ . Bouncing paths of the second example are $a^{2}$ and $b^{2}$ .

To define the algebra $\Delta_{\sigma}$ , we need two more combinatorial concepts:

Definition 5.

For each edge $u\in\sigma$ and endpoint $M$ of $u$ , we denote by $\omega_{u,M}$ the path of $Q_{\sigma}$ going from $u$ to $u$ winding once around $M$ if $M\notin\partial\Sigma$ and $\omega_{u,M}=0$ if $M\in\partial\Sigma$ .

Definition 6.

A small triangle of $\sigma$ is a triple $(u,v,w)$ of arcs of $\sigma$ together with three marked points $M$ , $N$ and $P$ such that

•

$u$ is incident to $M$ and $N$ ;

•

$v$ is incident to $N$ and $P$ ;

•

$w$ is incident to $P$ and $M$ ;

•

The union of $u$ , $v$ and $w$ encloses clockwisely (in the order $u$ , $v$ , $w$ ) a disc without marked point inside.

We now define the algebra of the partial triangulation $\sigma$ by $\Delta_{\sigma}:=kQ_{\sigma}/I_{\sigma}$ where $I_{\sigma}$ is the ideal generated by the following relations:

(1)

For each $u\in\sigma$ that joins $M\in\mathbb{M}$ to $N\in\mathbb{M}$ , we require $\lambda_{M}\omega_{u,M}^{m_{M}}=\lambda_{N}\omega_{u,N}^{m_{N}}$ . 2. (2)

For each $u\in\sigma$ with endpoint $M$ , we require $\omega_{u,M}^{m_{M}}q=0$ for any arrow $q$ of $Q_{\sigma}$ . 3. (3)

Suppose that $\sigma$ contains a small triangle $M,N,P$ as in the following picture:

[TABLE]

where $\alpha$ , $\beta$ and $\gamma$ are arrows, $\omega$ is a the only possible simple path winding around $M$ if $M\notin\partial\Sigma$ and $\omega=0$ if $M\in\partial\Sigma$ . We require the relation $\alpha\beta=\lambda_{M}(\omega\gamma)^{m_{M}-1}\omega$ . 4. (4)

For any bouncing path $\alpha\beta$ that does not appear in case 3, we require $\alpha\beta=0$ .

Example 7.

Relations for the left partial triangulation of Example 4 are:

[TABLE]

and other relations are redundant.

Relations for the right partial triangulation are:

[TABLE]

It turns out that algebras of partial triangulations are particularly well behaved. A first result about them is that this definition is compatible with the naive notion of a sub-partial triangulation:

Theorem 8.

Let $\tau\subset\sigma$ . Then we have

[TABLE]

where $e_{\tau}$ is the sum of the primitive idempotents of $\Delta_{\sigma}$ corresponding to the arcs of $\tau$ .

Notice that we have naturally $kQ_{\tau}\subset kQ_{\sigma}$ . However, relations as defined in this note do not go through this inclusion. We have to take a more complicated variant of these relations, giving an isomorphic algebra, to obtain Theorem 8.

2. Brauer graph algebras and Jacobian algebras of surfaces

We explain here that the class of algebras of partial triangulations contains two important classes of algebras.

Theorem 9.

If $\sigma$ contains no small triangle, neither arc incident to $\partial\Sigma$ , then $\Delta_{\sigma}$ is the Brauer graph algebra of $\sigma$ considered as a ribbon graph. Moreover, any Brauer graph algebra is the algebra of a partial triangulation of a surface without boundary.

We will not recall what a Brauer graph algebra is. For more details, see for example [4] or [5]. However, this definition is very close to the definition of the algebra of a partial triangulation and Theorem 9 is mostly straightforward.

Theorem 10.

If all $m_{M}$ are invertible in $k$ and $\sigma$ is a triangulation, then $\Delta_{\sigma}$ is the Jacobian algebra of a quiver with potential $(Q_{\sigma},W_{\sigma})$ . To define $W_{\sigma}$ , consider the set $E$ of small triangles $T$ of $\sigma$ up to rotation and for each of them denote by $\alpha_{T}$ , $\beta_{T}$ and $\gamma_{T}$ the three arrows as in the figure defining relation (3) earlier. Then, for each $M\in\mathbb{M}$ take arbitrarily an arc $u_{M}$ incident to $M$ . Then

[TABLE]

(as usual for potentials, terms are only well defined up to cyclic permutations).

Notice that if $m_{M}=1$ for all $M\in\mathbb{M}$ , we recover the usual Jacobian algebra of a surface as defined in [3]. Recall that the Jacobian algebra of $(Q_{\sigma},W_{\sigma})$ is the quotient of the completed path algebra $\widehat{kQ_{\sigma}}$ by the cyclic derivatives of $W_{\sigma}$ . So we get here an improvement as $\Delta_{\sigma}$ is directly defined from $kQ_{\sigma}$ without completion.

3. Algebraic properties of $\Delta_{\sigma}$

We have the following result about $\Delta_{\sigma}$ :

Theorem 11.

The $k$ -algebra $\Delta_{\sigma}$ is a free $k$ -module of rank

[TABLE]

where, for $M\in\mathbb{M}$ , $d_{M}$ is the degree of $M$ in the graph $\sigma$ (without counting boundary components), and $f$ is the number of arcs in $\sigma$ with both endpoints on boundaries.

Example 12.

The algebra of the left partial triangulation of Example 4 has rank $5+4m_{C}+m_{D}$ , and the right one has rank $4m_{M}$ .

More precisely, there is a $k$ -basis of $\Delta_{\sigma}$ consisting of all strict and non-idempotent prefixes of all $\omega_{u,M}^{m_{M}}$ , together with primitive idempotents and elements $\lambda_{M}\omega_{u,M}^{m_{M}}=\lambda_{N}\omega_{u,N}^{m_{N}}$ .

The following property generalizes a known result for Brauer graph algebras and Jacobian algebras of surfaces without boundary:

Theorem 13.

If $\sigma$ has no arc incident to the boundary, then $\Delta_{\sigma}$ is a symmetric $k$ -algebra (i.e. $\operatorname{Hom}_{k}(\Delta_{\sigma},k)\cong\Delta_{\sigma}$ as $\Delta_{\sigma}$ -bimodules).

4. Representation type of $\Delta_{\sigma}$

The next theorem permits to expect that the classification of $\Delta_{\sigma}$ -modules is possible:

Theorem 14.

If $k$ is an algebraically closed field, then $\Delta_{\sigma}$ is of tame representation type.

The proof of this result relies on a deformation theorem by Crawley-Boevey [1]. Indeed, the relations defining $\Delta_{\sigma}$ can be deformed to the relations of a Brauer graph algebra in a suited manner. Moreover, Brauer graph algebras are of tame representation type. Notice that unfortunately, these techniques do not permit to deduce directly the classification of $\Delta_{\sigma}$ -modules even though modules over Brauer graph algebras are known.

5. Flip of partial triangulations and derived equivalences

Finally, we give a flip leading to derived equivalences. For an arc $u$ in $\sigma$ such that arcs marked by $+$ in the following diagrams are also in $\sigma$ (in particular, they are not in $\partial\Sigma$ ), we define $\mu_{u}(\sigma)$ by replacing $u$ by $u^{*}$ defined in the following way:

[TABLE]

We also define coefficients $\mu_{u}(\lambda)$ . For a marked point $M\in\mathbb{M}$ , $\mu_{u}(\lambda)_{M}=\lambda_{M}$ except in the following cases:

•

In case (F1), if $M$ is the topmost vertex of the figure then $\mu_{u}(\lambda)_{M}=-\lambda_{M}$ .

•

In case (F1), if $M$ is the rightmost vertex of the figure then $\mu_{u}(\lambda)_{M}=(-1)^{m_{M}}\lambda_{M}$ .

•

In case (F2), if $M$ is the unique marked point enclosed by $u$ then $\mu_{u}(\lambda)_{M}=-\lambda_{M}$ .

Then we get the following result:

Theorem 15.

There is a derived equivalence between $\Delta_{\sigma}$ and $\Delta_{\mu_{u}(\sigma)}^{\mu_{u}(\lambda)}$ where the second algebra is computed with respect to the coefficients $\mu_{u}(\lambda)$ .

Example 16.

We consider the two following partial triangulations of a disc with three marked points, none of them are in $\partial\Sigma$ :

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P}$

They are related by a flip so the following algebras, obtained for $\lambda_{M}=\lambda_{N}=\lambda_{P}=1$ and $m_{M}=m_{N}=m_{P}=m$ are derived equivalent:

[TABLE]

Bibliography5

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] William Crawley-Boevey, Tameness of biserial algebras , Arch. Math. (Basel), 65(5) (1995), 399–407.
2[2] Laurent Demonet, Algebras of partial triangulations , submitted to Adv. Math., ar Xiv: 1602.01592.
3[3] Daniel Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces , Proc. Lond. Math. Soc. (3), 98(3) (2009), 797–839.
4[4] Klaus W. Roggenkamp, Biserial algebras and graphs , in Algebras and modules, II (Geiranger, 1996), volume 24 of CMS Conf. Proc., Amer. Math. Soc., Providence, RI, (1998), 481–496.
5[5] Burkhard Wald and Josef Waschbüsch, Tame biserial algebras , J. Algebra, 95(2) (1985), 480–500.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

INTRODUCTION TO ALGEBRAS OF PARTIAL TRIANGULATIONS

Abstract.

1. The algebra of a partial triangulation

Definition 1**.**

Definition 2**.**

Definition 3**.**

Example 4**.**

Definition 5**.**

Definition 6**.**

Example 7**.**

Theorem 8**.**

2. Brauer graph algebras and Jacobian algebras of surfaces

Theorem 9**.**

Theorem 10**.**

3. Algebraic properties of Δσ\Delta_{\sigma}Δσ​

Theorem 11**.**

Example 12**.**

Theorem 13**.**

4. Representation type of Δσ\Delta_{\sigma}Δσ​

Theorem 14**.**

5. Flip of partial triangulations and derived equivalences

Theorem 15**.**

Example 16**.**

Definition 1.

Definition 2.

Definition 3.

Example 4.

Definition 5.

Definition 6.

Example 7.

Theorem 8.

Theorem 9.

Theorem 10.

3. Algebraic properties of $\Delta_{\sigma}$

Theorem 11.

Example 12.

Theorem 13.

4. Representation type of $\Delta_{\sigma}$

Theorem 14.

Theorem 15.

Example 16.