Finite Tensor Research Articles

Support varieties for finite tensor categories: the tensor product property

Support varieties for finite tensor categories: the tensor product property

Modular tensor categories arising from central extensions and related applications

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras Eq, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of Rep(Eq) (the representation category of Eq). By further studying the simplicity of Eq (whether it is a simple Hopf algebra or not), we conclude that(1)there exists a twist J of uq(sl2⊕3) such that uq(sl2⊕3)J is a simple Hopf algebra,(2)there is no relation between the simplicity of a Hopf algebra H and the primality of Rep(H),(3)there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.

Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials

Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials

Adjacency preserving maps between tensor spaces

Let r and s be positive integers such that r⩾3. Let U1,…,Ur be vector spaces over a field F and V1,…,Vs be vector spaces over a field K such that dim⁡Ui,dim⁡Vj⩾2 for all i,j. In this paper, we characterize maps ψ:⨂i=1rUi→⨂i=1sVi that preserve adjacency in both directions, which extends Hua's fundamental theorem of geometry of rectangular matrices. We also characterize related results concerning locally full maps preserving adjacency in both directions between tensor spaces, maps preserving adjacency in both directions between tensor spaces over a field all whose nonzero endomorphisms are automorphisms, and injective continuous adjacency preserving maps on finite dimensional tensor spaces over the real field.

Support varieties without the tensor product property

AbstractWe show that over a perfect field, every non‐semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties.

Homology of complexes over finite tensor categories

We generalize a recent result by J. F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many non-isomorphic and nontrivial bounded complexes of projective objects, and with small homology. We also prove a version for finite dimensional algebras with finitely generated cohomology.

On Tensor Product of c-Spaces

This paper is an extension of the research on (cartesian) product of c-spaces. This paper demonstrates that the finite (tensor)product of quotients of c-spaces can be represented as a quotient of its (tensor)product. Some properties of the tensor product of c-spaces have been investigated in this context. Properties of the space of c-continuous functions have been probed and the relevance of the standard c-structure on it has been established.

On the spectrum and support theory of a finite tensor category

On the spectrum and support theory of a finite tensor category

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.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

On unimodular module categories

Let C be a finite tensor category and M an exact left C-module category. We call M unimodular if the finite multitensor category RexC(M) of right exact C-module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Relative Serre functor for comodule algebras

Let C be a finite tensor category, and let M be an exact left C-module category. The relative Serre functor of M is an endofunctor S on M together with a natural isomorphism Hom_(M,N)⁎≅Hom_(N,S(M)) for M,N∈M, where Hom_ is the internal Hom functor of M. In this paper, we discuss the case where C and M are the category of finite-dimensional left modules over a finite-dimensional Hopf algebra H and a finite-dimensional left H-comodule algebra L, respectively. We give an explicit description of the relative Serre functor of M and its twisted module structure in terms of the Frobenius structure of L. We also study pivotal structures on M and give some concrete examples.

Separable equivalences, finitely generated cohomology and finite tensor categories

We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik. Moreover, for such categories we also determine the representation dimension and the Rouquier dimension of the stable category. Finally, we recover a number of results on the cohomology of stably equivalent and singularly equivalent algebras.

Exact factorizations and extensions of finite tensor categories

We extend [13] to the nonsemisimple case. We define and study exact factorizations B=A•C of a finite tensor category B into a product of two tensor subcategories A,C, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories [8]. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category [14] by another one. We also provide several examples to illustrate our results.

Robust Max Entrywise Error Bounds for Tensor Estimation From Sparse Observations via Similarity-Based Collaborative Filtering

Consider the task of estimating a 3-order $n \times n \times n$ tensor from noisy observations of randomly chosen entries in the sparse regime. We introduce a similarity based collaborative filtering algorithm for estimating a tensor from sparse observations and argue that it achieves sample complexity that nearly matches the conjectured computationally efficient lower bound on the sample complexity for the setting of low-rank tensors. Our algorithm uses the matrix obtained from the flattened tensor to compute similarity, and estimates the tensor entries using a nearest neighbor estimator. We prove that the algorithm recovers a finite rank tensor with maximum entry-wise error (MEE) and mean-squared-error (MSE) decaying to $0$ as long as each entry is observed independently with probability $p = \Omega(n^{-3/2 + \kappa})$ for any arbitrarily small $\kappa > 0$. More generally, we establish robustness of the estimator, showing that when arbitrary noise bounded by $\varepsilon \geq 0$ is added to each observation, the estimation error with respect to MEE and MSE degrades by $\text{poly}(\varepsilon)$. Consequently, even if the tensor may not have finite rank but can be approximated within $\varepsilon \geq 0$ by a finite rank tensor, then the estimation error converges to $\text{poly}(\varepsilon)$. Our analysis sheds insight into the conjectured sample complexity lower bound, showing that it matches the connectivity threshold of the graph used by our algorithm for estimating similarity between coordinates.

Homotopy invariants of braided commutative algebras and the Deligne conjecture for finite tensor categories

It is easy to find algebras T∈C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T∈Z(C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end ∫X∈CX⊗X∨. Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T, naturally come with the structure of a differential graded E2-algebra. This way, we obtain a rich source of differential graded E2-algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E2-structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E2-structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E2-structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E2-algebras, thereby refining a result of Menichi.

Nakayama functors for coalgebras and their applications to Frobenius tensor categories

We introduce Nakayama functors for coalgebras and investigate their basic properties. These functors are expressed by certain (co)ends as in the finite case discussed by Fuchs, Schaumann, and Schweigert. This observation allows us to define Nakayama functors for Frobenius tensor categories in an intrinsic way. As applications, we establish the categorical Radford S4-formula for Frobenius tensor categories and obtain some related results. These are generalizations of works of Etingof, Nikshych, and Ostrik on finite tensor categories and some known facts on co-Frobenius Hopf algebras.

Ribbon structures of the Drinfeld center of a finite tensor category

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double of a finite-dimensional Hopf algebra. Our result implies that $\mathcal{Z}(\mathcal{C})$ is a modular tensor category in the sense of Lyubashenko if $\mathcal{C}$ is a spherical finite tensor category in the sense of Douglas, Schommer-Pries and Snyder.

On the stable equivalences between finite tensor categories

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact k - \Bbbk \text {-} linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.

The adjoint algebra for 2-categories

For any 0-cell B in a 2-category B, we introduce the notion of adjoint algebra AdB. This is an algebra in the center of B. We prove that if C is a finite tensor category, this notion applied to the 2-category CMod of C-module categories coincides with the one introduced by Shimizu. As a consequence of this general approach, we obtain new results on the adjoint algebra for tensor categories.