The mathematical β-tilting theory idea (the mathematical β-t theory) originated from the conceptual paintings of Adachi and his colleagues in 2014 [1], and it rapidly emerged as a primary focus of investigation within representation theory(R-theory) of finite-dimensional algebras (F-D algebras ). Integrating the idea of tilt, this framework presents an efficient combinatorial and isomorphic tool for reading rotation training, silt complexes, and cluster- tilting (CT) objects within modular classes .This study provides a systematic and self-contained introduction to β-t theory , especially designed for readers with a standard background in representation theory. It aims to enable researchers to grasp the fundamental concepts without the need for extensive reference to external sources, while maintaining full commitment to high mathematical rigor. In addition, the study unifies silting theory and cluster-tilting theory (CT theory) within a common perspective, gathering in one place the basic definitions, important bijections, and mutation techniques that form the core of this field [1]. In addition to the classical foundations, the research criticizes major developments published between 2023 and 2026, including new properties of β-tilting ( β-t) finiteness for Borel–Schur algebras and group algebras of generalized symmetric groups, and their applications to Frobenius–Perron dimensions and generalized preprojective algebras [2–8]. Particular attention is given to recent developments in higher torsion classes, βd-tilting theory ( βd-t-theory) and duplicated algebras. The study concludes by highlighting several key open problems such as the classification of minimal β-tilting infinite algebras ( β-t-i algebras ) and the explicit description of the vital bijections for important families of algebras which continue to motivate current research [9].This work aims to be an accessible entry point for beginners and a comprehensive reference for active researchers in representation theory ( R- theory ) and related fields [1,2].

Abstract We prove that the Drinfeld center $\mathcal {Z}(\operatorname {Vec}^{\omega }_{A_5})$ of the pointed category associated with the alternating group $A_5$ is the unique example of a perfect weakly group-theoretical modular category of Frobenius–Perron dimension less than $14400$ .

Fusion categories over non-algebraically closed fields

In this paper, we study the group extension of a Tambara-Yamagami category which has Frobenius-Perron dimension $2pq,$ where $p,q$ are prime numbers. We prove that there are two possible category types when $p\ne q,$ and five possible category types when $p=q.$

Abstract We apply the recently introduced notion, due to Dyckerhoff, Kapranov, and Schechtman, of $N$-spherical functors of stable infinity categories, which generalise spherical functors, to the setting of monoidal categories. We call an object $N$-bounded if the corresponding regular endofunctor on the derived category is $N$-spherical. Besides giving new examples of $N$-spherical functors, the notion of $N$-bounded objects gives surprising connections with Jones-Wenzl idempotents, Frobenius-Perron dimensions, and central conjectures in the field of symmetric tensor categories in positive characteristic.

Let M be an indecomposable representation of type-mathbbDquiver, s be an integer, and [s] be the suspension functor. In this paper, we study the brick sets of the indecomposable representation of a class of type-mathbbDquivers in the representation category and bounded derived category, and give the Frobenius-Perron dimensions of M and M[s] of this type-mathbbDquiver.

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example.

Abstract The integral group rings for finite groups are precisely those fusion rings whose basis elements have Frobenius–Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here, we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius–Perron dimensions take exactly one value distinct from 1. Our goal is a set of results to assist in characterizing when such fusion rings are categorifiable. As proof of concept, we complete the classification of categorifiable near‐group fusion rings for an infinite collection of finite abelian groups, a task that to‐date has only been completed for three such groups.

Let $p, q$ be odd primes, and let $d$ be an odd square-free integer such that $(pq,d)=1$. We show that slightly degenerate fusion categories of Frobenius-Perron dimensions $2p^2q^2d$, $2p^2q^3d$ and $2p^3q^3d$ are group-theoretical.

A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic $p$ version of this theorem. Namely, we show that a pre-Tannakian category over an algebraically closed field of characteristic $p>0$ admits a fiber functor into the Verlinde category $\mathrm{Ver}_p$ (i.e., is the representation category of an affine group scheme in $\mathrm{Ver}_p$) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderategrowth has a fiber functor to $\mathrm{Ver}_p$ (so in particular Deligne's theorem holds on the nose for semisimple pre-Tannakian categories in characteristics $2$,$3$). This settles a conjecture of the third author from 2015. In particular, this result applies to semisimplifications of categories of modular representations of finite groups (or, more generally, affine group schemes), which gives new applications to classical modular representation theory. For example, it allows us to characterize, for a modular representation $V$, the possible growth rates of the number of indecomposable summands in $V^{\otimes n}$of dimension prime to $p$.

The Frobenius-Perron theory of an endofunctor of a k \Bbbk -linear category (recently introduced in Chen et al. [Algebra Number Theory 13 (2019), pp. 2005–2055]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} -modules of infinite rank and compute it for the Z \mathbb {Z} -modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group.

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

Fusion bialgebras and Fourier analysis: Analytic obstructions for unitary categorification

Support varieties for finite tensor categories: Complexity, realization, and connectedness

Abstract We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)} ); a similar construction of the underlying additive functor appeared independently in [K. Coulembier, Tannakian categories in positive characteristic, preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik, On symmetric fusion categories in positive characteristic, Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if 𝒞 {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when 𝒞 {\mathcal{C}} is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor 𝒞 → Ver p {\mathcal{C}\to{\rm Ver}_{p}} . We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F, and use it to show that for categories with finitely many simple objects F does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory 𝒞 ex {\mathcal{C}_{\rm ex}} inside any STC 𝒞 {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by F. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Ver p {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in 𝒞 ex {\mathcal{C}_{\rm ex}} . Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra 𝒌 ⁢ [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d {R=1\otimes 1+d\otimes d} ), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki, Exact sequences of tensor categories with respect to a module category, Adv. Math. 308 2017, 1187–1208].

The goal of this paper is to classify fusion categories $${\mathcal {C}}$$ which are $$\otimes $$-generated by an object X of Frobenius–Perron dimension less than 2, with the additional mild assumption that the adjoint subcategory of $${\mathcal {C}}$$ is $$\otimes $$-generated by the object $$X\otimes X^*$$. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint ADE type. Our main tools in this classification are the results of Etingof et al. (Quantum Topol 1(3);209–273, 2010. https://doi.org/10.4171/QT/6), classifying cyclic extensions of a given category in terms of data computed from the Brauer–Picard group, and Drinfeld centre of that category, and the results of Edie-Michell (Int. J. Math. 29(5):1850036, 2018. https://doi.org/10.1142/S0129167X18500362) which compute the Brauer–Picard group and Drinfeld centres of the categories of adjoint ADE type. Our classification includes the expected categories, constructed from cyclic groups and the categories of ADE type. More interestingly we have categories in our classification that are non-trivial de-equivariantizations of these expected categories. Most interesting of all, our classification includes three infinite families constructed from the exceptional quantum subgroups $${\mathcal {E}}_4$$ of $${\mathcal {C}}( \mathfrak {sl}_4, 4)$$, and $${\mathcal {E}}_{16,6}$$ of $${\mathcal {C}}( \mathfrak {sl}_2, 16)\boxtimes {\mathcal {C}}( \mathfrak {sl}_3,6)$$.

We introduce the Frobenius–Perron dimension of an endofunctor of a k -linear category and provide some applications.

We introduce the notion of the Frobenius-Perron dimension of an integral \(\mathbb {Z}_{+}\)-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension p.

Symmetric tensor categories in characteristic 2