CT2Rep: Classifying Simple Transitive 2-Representations

• CT2Rep is a framework for classifying simple transitive 2-representations, defining canonical cell 2-representations as basic building blocks in higher representation theory.
• It employs combinatorial, spectral, and matrix-elimination methods to verify equivalence conditions and structure the classification results in finitary and fiat 2-categories.
• The theory has significant implications for categorification, connecting deep algebraic invariants with applications in Lie theory, Soergel bimodules, and quiver algebras.

CT2Rep

CT2Rep refers to the classification of simple transitive 2-representations in the theory of finitary and fiat 2-categories. The main focus is on understanding when all simple transitive 2-representations of a given 2-category are equivalent to certain canonical 2-representations known as cell 2-representations. This theory provides a categorical structure that generalizes classical representation theory to higher categories, with deep connections to Lie theory, quiver algebras, Soergel bimodules, and categorification phenomena.

1. Finitary 2-Categories and Their 2-Representations

A finitary 2-category $\mathscr{C}$ over an algebraically closed field $\Bbbk$ consists of a finite set of objects, and each hom-category $\mathscr{C}(i,j)$ is a finitary idempotent-complete $\Bbbk$-linear category (having finitely many indecomposables and finite-dimensional Hom spaces), with biadditive horizontal composition. The identity 1-morphisms are required to be indecomposable.

A finitary 2-representation is a strict 2-functor $\mathbf{M}:\mathscr{C}\to\mathbf{Cat}^{f}_\Bbbk$, where $\mathbf{Cat}^{f}_\Bbbk$ is the 2-category of finitary $\Bbbk$-linear categories and additive functors. Simple transitive 2-representations are analogues of simple modules in this context and form the fundamental building blocks for 2-representation theory (Mazorchuk et al., 2014).

2. Cell Structure and Cell 2-Representations

The combinatorics of indecomposable 1-morphisms in $\mathscr{C}$ is captured by three preorders: left ($\leq_L$), right ($\leq_R$), and two-sided ($\Bbbk$0). These induce a partition into left, right, and two-sided cells.

Cell 2-representations arise by fixing a left cell $\Bbbk$1 and forming the sub-2-representation of the principal 2-representation generated by all 1-morphisms in $\Bbbk$2. Modding out by the maximal ideal that avoids $\Bbbk$3 yields the cell 2-representation $\Bbbk$4. In the fiat case, cell 2-representations categorify Kazhdan–Lusztig cell modules and capture the combinatorics of the Hecke algebra (Mazorchuk et al., 2010).

Cell 2-representations are always simple transitive. In fiat categories with strongly regular two-sided cells, cell 2-representations associated to different left cells inside the same two-sided cell are equivalent (Mazorchuk et al., 2012).

3. Classification Theorems and Matrix Techniques

A central problem (CT2Rep) is to classify all simple transitive 2-representations up to equivalence. The key results (type-I property) assert that under strong regularity and certain numerical conditions, every simple transitive 2-representation is equivalent to a cell 2-representation (Mazorchuk et al., 2014, Mazorchuk, 2017).

Methodologically, much of the theory reduces to combinatorial and spectral considerations. For instance, when $\Bbbk$5 is the 2-category of projective endofunctors for an explicit finite-dimensional algebra, transitivity and simplicity impose matrix equations on the Grothendieck group representations of certain generating 1-morphisms. For specific cases, such as the path algebra $\Bbbk$6 and the "dotted-arrow" algebra $\Bbbk$7, the crucial step is to show that the only integer matrices satisfying $\Bbbk$8 (for $\Bbbk$9), subject to irreducibility and cell combinatorics, correspond to those coming from the cell 2-representations (Mazorchuk et al., 2016).

This matrix-elimination method has proved particularly effective for small examples, leading to a full classification of simple transitive 2-representations for these algebras.

4. Notable Examples and Non-Cell Simples

Projective Functor 2-Categories

For the 2-category $\mathscr{C}(i,j)$0 of projective endofunctors of $\mathscr{C}(i,j)$1-mod (for self-injective or radical-square-zero algebras), every simple transitive 2-representation is a cell 2-representation. Explicit classification is achieved by comparing the action matrices of generating 1-morphisms and deducing that any simple transitive 2-representation must coincide with a cell 2-representation up to equivalence (Mazorchuk et al., 2016, Zimmermann, 2017).

Soergel Bimodules

In the fiat 2-category of Soergel bimodules for type $\mathscr{C}(i,j)$2 or $\mathscr{C}(i,j)$3 (dihedral), all simple transitive 2-representations are cell 2-representations except in type $\mathscr{C}(i,j)$4, where an additional “exotic” rank-1 simple transitive 2-representation emerges. This construction exploits orbit categories and infinite inflation, reflecting the subtlety and richness possible in 2-representation theory even in relatively small settings (Mackaay et al., 2016, Zimmermann, 2015).

For small-quotient 2-categories derived from Soergel bimodules in even dihedral type $\mathscr{C}(i,j)$5, two extra non-cell simple transitive 2-representations exist, constructed via an inductive limit and orbit category construction reminiscent of skew group algebras (Kildetoft et al., 2016).

Star Algebras and Left Cell Subcategories

In the left-cell 2-subcategories of projective functors for star algebras (generalizing the $\mathscr{C}(i,j)$6 quiver), in the simplest case ($\mathscr{C}(i,j)$7), all simple transitive 2-representations are cell, but for $\mathscr{C}(i,j)$8, it is conjectured (and partially evidenced) that additional, non-cell simple transitive 2-representations exist, likely governed by combinatorial data of set partitions (Zimmermann, 2018).

5. Structure, Endomorphisms, and Universal Properties

Cell 2-representations have endomorphism categories isomorphic to direct sums of the identity (a 2-Schur’s lemma); for a strongly regular two-sided cell, the only endomorphism 2-natural transformations are direct sums of the identity, and all intertwiners form a $\mathscr{C}(i,j)$9-vector space (Mazorchuk et al., 2012). This mimics classical module theory and situates cell 2-representations as "atomic" objects in 2-representation theory.

A universal property holds: cell 2-representations are characterized by a cyclic generator, and their abelianization provides the abelian cell 2-representation, further aligning them with simple modules.

Simple transitive 2-representations can, in the context of fiat 2-categories, be realized as categories of injective comodules over a simple coalgebra 1-morphism in the injective abelianization, or dually in terms of projective modules over a simple algebra 1-morphism in the projective abelianization (Mackaay et al., 2016). This furnishes a 2-categorical analogue of Morita–Takeuchi theory and a 2-Artin–Wedderburn structure theorem for $\Bbbk$0-simple fiat 2-categories.

6. Impact, Open Problems, and Outlook

The classification of simple transitive 2-representations (CT2Rep) underpins the uniqueness of categorifications of simple Lie algebra modules, the structure of 2-group actions, and the foundational theory for diagrammatic categorification (KLR algebras, Soergel bimodules, etc.). The extension to $\Bbbk$1-dg 2-categories (with $\Bbbk$2-differentials relevant for categorification at roots of unity) preserves these structural results under suitable regularity conditions (Laugwitz et al., 2017).

Open problems include:

• Extending classification results to 2-categories with non-strongly regular or non-fiat structure, where non-cell simple transitive 2-representations may exist.
• Understanding the full spectrum of non-cell simples in special configurations (e.g., even dihedral types, star algebras with $\Bbbk$3).
• The development of a comprehensive Ext-theory for 2-representations and homological invariants of 2-categories.
• Systematic study of discrete extensions, orbit-category constructions, and the role of coalgebra 1-morphisms.

Ongoing research aims at the complete description of all finitary 2-representations, connections to higher representation theory (Kac–Moody 2-categories, categorified quantum groups), and practical algorithms for explicit categorical constructions (Mazorchuk, 2017, Mazorchuk et al., 2014, Mazorchuk et al., 2010, Mackaay et al., 2016).

---

References:

• (Mazorchuk et al., 2016) Simple transitive 2-representations for two non-fiat 2-categories of projective functors
• (Mazorchuk et al., 2010) Cell 2-representations of finitary 2-categories
• (Mazorchuk et al., 2014) Transitive 2-representations of finitary 2-categories
• (Mazorchuk et al., 2012) Endomorphisms of cell 2-representations
• (Laugwitz et al., 2017) Cell 2-Representations and Categorification at Prime Roots of Unity
• (Mackaay et al., 2016) Simple transitive 2-representations via (co)algebra 1-morphisms
• (Kildetoft et al., 2016) Simple transitive 2-representations of small quotients of Soergel bimodules
• (Mackaay et al., 2016) Simple transitive 2-representations for some 2-subcategories of Soergel bimodules
• (Zimmermann, 2015) Simple transitive $\Bbbk$4-representations of Soergel bimodules in type $\Bbbk$5
• (Zimmermann, 2017) Simple transitive $\Bbbk$6-representations of some $\Bbbk$7-categories of projective functors
• (Zimmermann, 2018) Simple transitive $\Bbbk$8-representations of left cell $\Bbbk$9-subcategories of projective functors for star algebras
• (Mazorchuk, 2017) Classification problems in 2-representation theory