Semisimple Tensor Research Articles

On the dualizability of fusion 2-categories

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple tensor 2-categories, separable bimodule 2-categories, and their morphisms. Categorifying a result of Douglas, Schommer-Pries, Snyder [Mem. Amer. Math. Soc. 268 (2021)], we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of Décoppet [Comp. Math.] that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object of the above Morita 4-category. We explain how this can be extended to any field of characteristic zero. Finally, we discuss the field theoretic interpretation of our results.

Multiplicities and Dimensions in Enveloping Tensor Categories

Abstract In a previous paper, semisimple tensor categories were constructed from certain regular Mal’cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple objects. This yields also the Grothendieck ring. The main tool is the subquotient decomposition of the generating objects.

Compact semisimple 2-categories

Working over an arbitrary field, we define compact semisimple 2-categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then, we prove that, over an algebraically closed field or a real closed field, compact semisimple 2-categories are finite. Finally, we explain how a number of key results in the theory of finite semisimple 2-categories over an algebraically closed field of characteristic zero can be generalized to compact semisimple 2-categories.

Affinization of q-oscillator representations of $$U_q(\mathfrak {gl}_n)$$

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.

An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

Trivializing group actions on braided crossed tensor categories and graded braided tensor categories

For an abelian group $A$, we study a close connection between braided $A$-crossed tensor categories with a trivialization of the $A$-action and $A$-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a tensor category $\mathcal{C}$ is given by an element $O(T) \in H^2(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T) = 0$, the set of obstructions forms a torsor over $\operatorname{Hom}(G, \operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}}))$, where $\operatorname{Aut}_{\otimes}(\operatorname{Id}_{\mathcal{C}})$ is the abelian group of tensor natural automorphisms of the identity. The cohomological interpretation of trivializations, together with the homotopical classification of (faithfully graded) braided $A$-crossed tensor categories developed Etingof et al., allows us to provide a method for the construction of faithfully $A$-graded braided tensor categories. We work out two examples. First, we compute the obstruction to the existence of trivializations for the braided $A$-crossed tensor category associated with a pointed semisimple tensor category. In the second example, we compute explicit formulas for the braided $\mathbb{Z}/2\mathbb{Z}$-crossed structures over Tambara–Yamagami fusion categories and, consequently, a conceptual interpretation of the results by Siehler about the classification of braidings over Tambara–Yamagami categories.

Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

The $${\mathcal {B}}_{p}$$ -algebras are a family of vertex operator algebras parameterized by $$p\in {\mathbb {Z}}_{\ge 2}$$ . They are important examples of logarithmic CFTs and appear as chiral algebras of type $$(A_1, A_{2p-3})$$ Argyres–Douglas theories. The first member of this series, the $${\mathcal {B}}_2$$ -algebra, are the well-known symplectic bosons also often called the $$\beta \gamma $$ vertex operator algebra. We study categories related to the $${\mathcal {B}}_{p}$$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $${\mathfrak {sl}}_{2}$$ . These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

On the semisimple tensor product of MV-algebras

In this note we show how the MV-algebraic tensor product can be used to define natural adjunctions between the algebraic structures of Łukasiewicz logic with product. In order to do this, we define the semisimple tensor PMV-algebra of a semisimple MV-algebra.

MV-algebras, infinite dimensional polyhedra, and natural dualities

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.

Super duality and crystal bases for quantum ortho-symplectic superalgebras II

Let $$\mathcal {O}^\mathrm{int}_q(m|n)$$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type B, C, D introduced in Kwon (Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076 ). It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type B, C, D from a viewpoint of super duality. Continuing the previous work on type B and C (Kwon in Int Math Res Not, 2015. doi: 10.1093/imrn/rnv076 ), we classify the irreducible modules in $$\mathcal {O}^\mathrm{int}_q(m|n)$$ and prove the existence and uniqueness of their crystal bases in case of type D. A new combinatorial model of classical crystals of type D is introduced, whose super analog gives a realization of crystals for the highest weight modules in $$\mathcal {O}^\mathrm{int}_q(m|n)$$ .

Super Duality and Crystal Bases for Quantum Ortho-Symplectic Superalgebras

We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{m+n}$, $C_{m+n}$, $D_{m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\mc{O}^{int}_q(m|n)$ and show that an irreducible module in $\mc{O}^{int}_q(m|n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatorial object called ortho-symplectic tableaux.

Irreducible characters of Kac-Moody Lie superalgebras

Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategory, whose Littlewood-Richardson tensor product multiplicities coincide with those in the Kac-Moody algebra setting.

Polyhedral MV-algebras

A polyhedron in Rn is a finite union of simplexes in Rn. An MV-algebra is polyhedral if it is isomorphic to the MV-algebra of all continuous [0,1]-valued piecewise linear functions with integer coefficients, defined on some polyhedron P in Rn. We characterize polyhedral MV-algebras as finitely generated subalgebras of semisimple tensor products S⊗F with S simple and F finitely presented. We establish a duality between the category of polyhedral MV-algebras and the category of polyhedra with Z-maps. We prove that polyhedral MV-algebras are preserved under various kinds of operations, and have the amalgamation property. Strengthening the Hay–Wójcicki theorem, we prove that every polyhedral MV-algebra is strongly semisimple, in the sense of Dubuc–Poveda.

Gauge semi-simple extension of the Poincaré group

An approach to the cosmological term problem is proposed, using the gauge semi-simple tensor extension of the D-dimensional Poincaré group as a basis.

A construction of semisimple tensor categories

Let A be an Abelian category such that every object has only finitely many subobjects. From A we construct a semisimple tensor category T . We show that T interpolates the categories Rep ( Aut ( p ) , K ) where p runs through certain projective pro-objects of A . This extends a construction of Deligne for symmetric groups. To cite this article: F. Knop, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

On braided tensor categories of typeBCD

We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep(O(\infty) (formally), Rep(O(N), Rep(Sp(N) or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. Note that the paper is followed by a brief erratum, which corrects a mistake, which does not affect the main results of the paper.

Representations of Algebraic Quantum Groups and Reconstruction Theorems for Tensor Categories

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka–Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka–Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras.

Near-group categories

We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules, and an explicit construction of matrix solutions to the pentagon equations in the cases where we establish existence. Many of these also support (braided) commutative and tortile structures and we indicate when this is possible. Small examples are presented in detail.

Tannaka duals in semisimple tensor categories

Tannaka duals of finite-dimensional Hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories, which are shown to include the Tannaka dual of the dual Hopf algebras. The second orbifolds are then monoidally equivalent to the initial tensor categories in a canonical fashion.

Polygonal presentations of semisimple tensor categories

A polygonal description of semisimple tensor categories is presented and the rigidity as well as the associated involutions are analysed in terms of this.