On geometric models in representation theory

TL;DR

This survey develops and analyzes geometric surface models for the derived categories of graded gentle algebras, using ribbon graphs and dual dissections to encode indecomposables as graded curves and morphisms as curve intersections. It establishes a concrete correspondence between $\mathcal{D}^b(A)$ (and $\operatorname{per}(A)$) and topological data on a compact surface, including a complete derived invariant given by the graded marked surface $(S,M,\eta)$, and explains how Serre functors, mapping cones, and recollements are interpreted geometrically. The work further connects these models to Fukaya categories of surfaces with stops, explores exceptional sequences and braid group actions, and develops an deformation theory via Hochschild cohomology and $A_\infty$-deformations, including orbifold and orbifold-disk generalizations. Overall, the approach provides a unifying framework linking representation theory of tame algebras to symplectic and topological methods, with wide implications for derived equivalences, deformations, and categorical dynamics. The results yield practical tools for computing derived invariants, constructing tilting/silting objects, and understanding deformation paths through partial compactifications of associated surfaces.

Abstract

Geometric models have emerged as an important tool in the representation theory of algebras. Surface models associated to gentle algebras have been particularly fruitful in advancing our understanding of their module and derived categories. We give an overview of some of the theoretical advances that geometric surface models for the derived categories of graded gentle algebras and their connections to Fukaya categories of surfaces have made possible.

Figures (7)

• Figure 1: Two mutually Koszul dual (graded) gentle algebras, their ribbon graphs and the corresponding admissible dissections. Every path in the quiver induces a morphism between the corresponding indecomposable dg $A$-modules $e_i A = P_i$. For example, for the model on the left side, we have morphisms $P_1 \stackrel{a.}\longrightarrow P_2[|a|]$, $P_2 \stackrel{b.}\longrightarrow P_3[|b|]$ and $P_1 \stackrel{ba.}\longrightarrow P_3[|a|+|b|]$, where $|a|$ and $|b|$ denote the degrees of $a$ and $b$, respectively.
• Figure 2: Example of a bounded twisted complex $X_\alpha$ and an unbounded twisted complex $X_\beta$ with differentials $d_\alpha$ and $d_\beta$, respectively, and a morphism $f$ between them induced by the identity on the component $P_1 \oplus P_3 [|a|+|b|-1]$.
• Figure 3: Example of a recollement of derived categories of graded gentle algebras in terms of surface cuts where the algebra $A(\Delta)$ is given by $1 \mkern 5.5mu \mkern 5.5mu 2 \mkern 5.5mu \mkern 5.5mu 3 \mkern 5.5mu \mkern 5.5mu 4 \mkern 5.5mu \mkern 5.5mu 5 \mkern 5.5mu \mkern 5.5mu 6 \mkern 5.5mu \mkern 5.5mu 7 \mkern 5.5mu \mkern 5.5mu 8 \mkern 5.5mu \mkern 5.5mu 9$ and where $\Gamma = \{2,3,4,8\}$. Then $A_\Gamma \colon (1 \mkern 5.5mu \mkern 5.5mu 5 \mkern 5.5mu \mkern 5.5mu 6 \mkern 5.5mu \mkern 5.5mu 7) \times 9$ with $|b| = |a_1|+|a_2|+|a_3|+|a_4|-3$ and $A(\Gamma) \colon 2 \mkern 5.5mu \mkern 5.5mu 3 \mkern 5.5mu \mkern 5.5mu 4 \mkern 5.5mu \mkern 5.5mu 8$ with $|c| = |a_4|+|a_5|+|a_6|+|a_7|$.
• Figure 4: Formal and non-formal generators of $\operatorname{{\rm tw}} (\mathcal{A}) = \mathcal{W}(S,M,\eta)$.
• Figure 5: Geometric representation of the three types of non-Eulerian Hochschild cocycles generating $\operatorname{HH}^*(A)$ as a graded commutative algebra. For $a = a_m \dotsb a_1$ and $b = b_n \dotsb b_1$, the generator $x_{a,b}$ corresponds to a morphism $A^{\otimes m} \to A$ which maps $a_m \otimes \cdots \otimes a_1$ to $b$ (left), $x_{a^i, \mathrm s (a)} \colon A^{\otimes im} \to A$ maps $(a_{j+m-1} \otimes \dotsb \otimes a_j)^{ \otimes i}$ to $e_{\mathrm s (a_j)}$ for all $0 \leq j < m$ (middle) and, dually, $x_{\mathrm s (b), b^i} \colon \Bbbk Q_0 \to A$ maps $e_{\mathrm s (b_j)}$ to $(b_{j+n-1} \dotsb b_j)^i$ for all $0 \leq j < n$ (right), indices being labelled cyclically, and $i = 2$ unless $\mathrm w_\eta (C)$ is even or $\mathrm{char} (\Bbbk) = 2$ in which case $i = 1$.

Theorems & Definitions (1)

• Definition 2.1