Tensor Functor Research Articles

Rieffel Deformation of Tensor Functor and Braided Quantum Groups

In this paper we apply Rieffel deformation to C*- tensor product viewed as a functor on the category of C*-algebras with an abelian group action. In the case of the Rieffel deformation of a quantum group with the action by automorphisms the deformed tensor product enables us to view the deformed object as a braided quantum group. We construct a bicharacter for a braided quantum group and the dual braided quantum group. We employ our method to get a braided quantum Minkowski space. Its description in terms of the deformed space-time coordinates is provided.

Weak Hom-Hopf algebras and their (co)representations

In this paper we will introduce the notion of a weak Hom-Hopf algebra, generalizing both weak Hopf algebras and Hom-Hopf algebras. Then we study the category R e p ( H ) of Hom-modules with bijective Hom-structure maps over a weak Hom-Hopf algebra H and show that the tensor functor of a (weak) Hom-bialgebra is actually a (weak) bimonad on a Hom-type category. At last, we prove that if H is a quasitriangular weak Hom-bialgebra (resp. ribbon weak Hom-Hopf algebra), then R e p ( H ) is a braided monoidal category (resp. ribbon category).

On p-adic differential equations on semistable varieties II

This paper is a complement to the paper “On p-adic differential equations on semistable varieties” by Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber. In this paper, we prove that, with convenable hypothesis, this functor is a tensor functor whose essential image is closed under extensions and subquotients. As a consequence, we can find suitable Tannakian subcategories of log overconvergent isocrystals and of modules with regular integrable connection on which the algebraization functor is an equivalence of Tannakian categories.

Handlebody-knot invariants derived from unimodular Hopf algebras

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.

Fuzzy projective modules and tensor products in fuzzy module categories

Let R be a commutative ring. We write Hom( A; B) for the set of all fuzzy R-morphisms from A to B, where A and B are two fuzzy R-modules. We make Hom( A; B) into fuzzy R-module by redening a func- tion : Hom( A; B) ! (0; 1). We study the properties of the functor Hom( A; ) : FR-Mod! FR-Mod and get some unexpected results. In ad- dition, we prove that Hom( p; ) is exact if and only if P is a fuzzy projective R-module, when R is a commutative semiperfect ring. Finally, we investigate tensor product of two fuzzy R-modules and get some related properties. Also, we study the relationships between Hom functor and tensor functor.

On Normal Tensor Functors and Coset Decompositions for Fusion Categories

We introduce the notion of double cosets relative to two fusion subcategories of a fusion category. Given a tensor functor \(F : {\mathcal {C}} \to {\mathcal {D}}\) between fusion categories, we introduce an equivalence relation ≈F on the set \(\Lambda _{\mathcal {C}}\) of isomorphism classes of simple objects of \({\mathcal {C}}\), and when F is dominant, an equivalence relation ≈F on \(\Lambda _{\mathcal {D}}\). We show that the equivalence classes of ≈F are cosets. We also give a description of the image of F when F is a normal tensor functor, and we show that F is normal if and only if the images of ≈F equivalent elements of \(\Lambda _{\mathcal {C}}\) are colinear. We study the situation where the composition of two tensor functors F = F′ F″ is normal, and we give a criterion of normality for F″, with an application to equivariantizations. Lastly, we introduce the radical of a fusion subcategory and compare it to its commutator in the case of a normal subcategory.

Categorical duality for Yetter-Drinfeld algebras

We study tensor structures on (\operatorname{Rep} G) -module categories defined by actions of a compact quantum group G on unital C ^* -algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of G to the structure of a braided-commutative Yetter–Drinfeld algebra. This shows that the category of braided-commutative Yetter–Drinfeld G -C ^* -algebras is equivalent to the category of generating unitary tensor functors from \operatorname{Rep} G into C ^* -tensor categories. To illustrate this equivalence, we discuss coideals of quotient type in C(G) , Hopf–Galois extensions and noncommutative Poisson boundaries.

Central exact sequences of tensor categories, equivariantization and applications

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if ${\mathcal C}$ is a fusion category and $F: {\mathcal C} \to {\mathcal D}$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if ${\mathcal C}$ is weakly integral and $p$ is the smallest prime factor of FPdim ${\mathcal C}$.

Tensor functors between categories of quasi-coherent sheaves

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction f↦f⁎ implements an equivalence between the discrete category of morphisms Y→X and the category of cocontinuous tensor functors Qcoh(X)→Qcoh(Y). This is an improvement of a result by Lurie and may be interpreted as the statement that algebraic geometry is 2-affine. Moreover, we prove the analogous version of this result for Durovʼs notion of generalized schemes.

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors − ⊗ k V and V ⊗ k − are known to be left adjoint to some kind of Hom-functors as endofunctors of H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case. In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors − ⊗V and − ⊗V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors − ⊗V* and Hom k (V, −) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.

Modules M such that ${\rm {\bf Ext}}_{{\bf \textit{R}}}^{\bf{1}}{({\bf \textit{M}},-)}$ Commutes with Direct Limits

We will use Watts’s theorem together with Lenzing’s characterization of finitely presented modules via commuting properties of the induced tensor functor in order to study commuting properties of covariant Ext-functors.

A theory of induction and classification of tensor C*-categories

This paper addresses the problem of describing the structure of tensor C*-categories \mathcal{M} with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley–Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension ≥ 2 . The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from \mathcal{M} to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G -equivariant Hermitian bundles over compact homogeneous G -spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of \mathcal{M} into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if \mathcal{M} is generated by a pseudoreal object of dimension 2.

Pullback of parabolic bundles and covers of ℙ1\{0,1,∞}

We work over an algebraically closed ground field of characteristic zero. A $G$-cover of ${\mathbb P}^1$ ramified at three points allows one to assign to each finite dimensional representation $V$ of $G$ a vector bundle $\oplus \mathscr{O}(s_i)$ on ${\mathbb P}^1$ with parabolic structure at the ramification points. This produces a tensor functor from representation of $G$ to vector bundles with parabolic structure that characterises the original cover. This work attempts to describe this tensor functor in terms of group theoretic data. More precisely, we construct a pullback functor on vector bundles with parabolic structure and describe the parabolic pullback of the previously described tensor functor.

Exact Sequences of Tensor Categories

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adjoint there exists a canonical commutative algebra A in the center of C such that F is tensor equivalent to the free module functor C -> mod_C A, where mod_C A denotes the category of A-modules in C endowed with a monoidal structure defined using the half-braiding of A. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.

Schur-Finiteness and Endomorphisms Universally of Trace Zero Via Certain Trace Relations

We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a ℚ-linear ⊗-category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the field of combinatorics. Our main tool is Berele–Regev's theory of Hook Schur functions. We use their generalization of the classic Schur–Weyl duality to the “super” case, together with their factorization formula.

Discriminant algebras of finite rank algebras and quadratic trace modules

Based on the construction of the discriminant algebra of an even-ranked quadratic form and Rost’s method of shifting quadratic algebras, we give an explicit rational construction of the discriminant algebra of finite-rank algebras and, more generally, of quadratic trace modules, over arbitrary commutative rings. The discriminant algebra is a tensor functor with values in quadratic algebras, and a symmetric tensor functor with values in quadratic algebras with parity. The automorphism group of a separable quadratic trace module is a smooth, but in general not reductive, group scheme admitting a Dickson type homomorphism into the constant group scheme Z2.

On fusion categories

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

An analogue of Radford's S4 formula for finite tensor categories

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ:V**→D⊗**V⊗D−1. This provides a categorical generalization of Radford's S 4 formula for finite-dimensional Hopf algebras (presented in 1976), which was proved by Nikshych (2002) for weak Hopf algebras and by Hausser and Nill (1999) for quasi-Hopf algebras, and was conjectured in general by Etingof and Ostrik (2003). When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of Radford (1992). We also show that a factorizable braided tensor category is unimodular (i.e., D=1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories defined by Etingof and Ostrik (2003).

Regular representations and Huang–Lepowsky tensor functors for vertex operator algebras

This is the second paper in a series devoted to studies of regular representations for vertex operator algebras. In this paper, given a module W for a vertex operator algebra V , we construct, from the dual space W ∗ , a family of canonical (weak) V ⊗ V -modules called D Q(z) (W) parameterized by a nonzero complex number z . We prove that for V -modules W , W 1 , and W 2 , a Q ( z )-intertwining map of type W′ W 1 W 2 in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V -homomorphism from W 1 ⊗ W 2 to D Q(z) (W) and that a Q ( z )-tensor product of V -modules W 1 and W 2 in the sense of Huang and Lepowsky amounts to a universal from W 1 ⊗ W 2 to the functor F Q(z) , where F Q(z) is a functor from the category of V -modules to the category of weak V ⊗ V -modules defined by F Q(z) (W)= D Q(z) (W′) for a V -module W . Furthermore, Huang–Lepowsky's P ( z )- and Q ( z )-tensor functors for the category of V -modules are extended to functors T P ( z ) and T Q ( z ) from the category of V ⊗ V -modules to the category of V -modules. It is proved that functors F P(z) and F Q(z) are right adjoints of T P ( z ) and T Q ( z ) , respectively.

REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION

The notion of left (respectively right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.