--- title: 'CT2Rep: Classifying Simple Transitive 2-Representations' url: https://www.emergentmind.com/topics/ct2rep type: topic --- # CT2Rep: Classifying Simple Transitive 2-Representations 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 [1404.7589]. ## 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 ($\leq_J$). These induce a partition into left, right, and two-sided cells. Cell 2-representations arise by fixing a left cell $\mathcal{L}$ and forming the sub-2-representation of the principal 2-representation generated by all 1-morphisms in $\mathcal{L}$. Modding out by the maximal ideal that avoids $\mathcal{L}$ yields the cell 2-representation $\mathbf{C}_\mathcal{L}$. In the fiat case, cell 2-representations categorify Kazhdan–Lusztig cell modules and capture the combinatorics of the Hecke algebra [1011.3322]. 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 [1207.6236]. ## 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 [1404.7589][1703.10093]. Methodologically, much of the theory reduces to combinatorial and spectral considerations. For instance, when $\mathscr{C}$ 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 $A_2 = \Bbbk(1 \to 2)$ and the "dotted-arrow" algebra $A_3 = \Bbbk(1 \to 2 \to 3)/(\beta\alpha=0)$, the crucial step is to show that the only integer matrices satisfying $M^2 = cM$ (for $c = \dim A$), subject to irreducibility and cell combinatorics, correspond to those coming from the cell 2-representations [1601.00097]. 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 $\mathscr{C}_A$ of projective endofunctors of $A$-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 [1601.00097][1705.01149]. ### Soergel Bimodules In the fiat 2-category of Soergel bimodules for type $B_2$ or $I_2(5)$ (dihedral), all simple transitive 2-representations are cell 2-representations except in type $B_2$, 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 [1602.04314][1509.01441]. For small-quotient 2-categories derived from Soergel bimodules in even dihedral type $I_2(2k)$, two extra non-cell simple transitive 2-representations exist, constructed via an inductive limit and orbit category construction reminiscent of skew group algebras [1605.01373]. ### Star Algebras and Left Cell Subcategories In the left-cell 2-subcategories of projective functors for star algebras (generalizing the $A_2$ quiver), in the simplest case ($n=1$), all simple transitive 2-representations are cell, but for $n>1$, it is conjectured (and partially evidenced) that additional, non-cell simple transitive 2-representations exist, likely governed by combinatorial data of set partitions [1805.05724]. ## 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 $k$-vector space [1207.6236]. 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 [1612.06325]. This furnishes a 2-categorical analogue of Morita–Takeuchi theory and a 2-Artin–Wedderburn structure theorem for $J$-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 $p$-dg 2-categories (with $p$-differentials relevant for categorification at roots of unity) preserves these structural results under suitable regularity conditions [1706.07725]. 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 $n>1$). - 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 [1703.10093][1404.7589][1011.3322][1612.06325]. --- **References:** - [1601.00097] Simple transitive 2-representations for two non-fiat 2-categories of projective functors - [1011.3322] Cell 2-representations of finitary 2-categories - [1404.7589] Transitive 2-representations of finitary 2-categories - [1207.6236] Endomorphisms of cell 2-representations - [1706.07725] Cell 2-Representations and Categorification at Prime Roots of Unity - [1612.06325] Simple transitive 2-representations via (co)algebra 1-morphisms - [1605.01373] Simple transitive 2-representations of small quotients of Soergel bimodules - [1602.04314] Simple transitive 2-representations for some 2-subcategories of Soergel bimodules - [1509.01441] Simple transitive $2$-representations of Soergel bimodules in type $B_2$ - [1705.01149] Simple transitive $2$-representations of some $2$-categories of projective functors - [1805.05724] Simple transitive $2$-representations of left cell $2$-subcategories of projective functors for star algebras - [1703.10093] Classification problems in 2-representation theory Source: https://www.emergentmind.com/topics/ct2rep