INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and $$ \mathcal{N} $$ N finite module categories over a finite tensor category $$ \mathcal{C} $$ C , the category $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) of right exact module functors is a finite module category over the Drinfeld center $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map $$ \mathcal{C} $$ C -$$ \mathcal{C} $$ C -bimodule functors to objects of $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and $$ \mathcal{N} $$ N are exact $$ \mathcal{C} $$ C -modules and $$ \mathcal{C} $$ C is pivotal, then the $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C )-module $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is exact. We compute its relative Serre functor and show that if ℳ and $$ \mathcal{N} $$ N are even pivotal module categories, then $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ).

Finitary ideals of direct products in quantales

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

On the representation theory of the vertex algebra L−5/2(sl(4))

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

Modified Traces and the Nakayama Functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

Motzkin algebras and the An tensor categories of bimodules

We discuss the structure of the Motzkin algebra [Formula: see text] by introducing a sequence of idempotents and the basic construction. We show that [Formula: see text] admits a factor trace if and only if [Formula: see text] and the higher commutants of these factors depend on [Formula: see text]. Then a family of irreducible bimodules over these factors is constructed. A tensor category with [Formula: see text] fusion rule is obtained from these bimodules.

Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

L 2-Betti Numbers of C*-Tensor Categories Associated with Totally Disconnected Groups

We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of [ 7]. We apply this criterion to show that the categories constructed from totally disconnected groups in [ 6] have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda <\Gamma $, with $\Gamma $ acting on a type $\textrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $P\rtimes \Lambda \subset P\rtimes \Gamma $ to that of the Schlichting completion $G$ of $\Lambda <\Gamma $. If $\Lambda <\Gamma $ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $P\rtimes \Lambda \subset P\rtimes \Gamma $ are equal to those of $G$.

Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups Γθ, Γ0(2) and Γ0(2) of SL2(ℤ). Each subgroup corresponds to one of the spin structures on the torus. The pole structures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups. We focus on the first and second order holomorphic MLDEs without poles and use them to find a large class of ‘Fermionic Rational Conformal Field Theories’, which have non-negative integer coefficients in the q-series expansion of their characters. We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic Modular Tensor Category.