Category Of Vector Spaces Research Articles

On the cohomology of pure mapping class groups as FI-modules

In this paper we apply the theory of finitely generated FI-modules developed by Church et al. to certain sequences of rational cohomology groups. An FI-module over \({\mathbb {Q}}\) is a functor from the category of finite sets and injections to the category of vector spaces over \({\mathbb {Q}}\). Our main examples are the cohomology of the moduli space of \(n\)-pointed curves, the cohomology of the pure mapping class group of surfaces and some manifolds of higher dimension, and the cohomology of classifying spaces of some diffeomorphism groups. We introduce the notion of FI\([G]\)-module and use it to strengthen and give new context to results on representation stability discussed by the author in a previous paper. Moreover, we prove that the Betti numbers of these spaces and groups are polynomial and find bounds on their degree. Finally, we obtain rational homological stability of certain wreath products.

Higher genus mapping class group invariants from factorizable Hopf algebras

Lyubashenkoʼs construction associates representations of mapping class groups Mapg:n of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the category of bimodules over a finite-dimensional factorizable ribbon Hopf algebra H. For any such Hopf algebra we find an invariant of Mapg:n for all values of g and n. More generally, we obtain such invariants for any pair (H,ω), where ω is a ribbon automorphism of H.Our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple, so-called logarithmic conformal field theories.

RESOLUTIONS AND COHOMOLOGIES OF TORIC SHEAVES: THE AFFINE CASE

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤn-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

An axiomatic construction of an almost full embedding of the category of graphs into the category of [formula omitted]-objects

We construct embeddings G of the category of graphs into categories of R-modules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of GR[HomGraphs(X,Y)]⟶HomR(GX,GY) are isomorphisms. The symbol R[S] above denotes the free R-module with the basis S. This implies that, for any cotorsion-free ring R, the categories of R-modules are not less complicated than the category of graphs. A similar embedding of graphs into the category of vector spaces with four distinguished subspaces (over any field, e.g. F2={0,1}) is obtained.

Relative centers and tensor products of tensor and braided fusion categories

In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a bimodule category. In particular we investigate the canonical tensor category structure and braiding inherited by the relative tensor product when module categories are themselves tensor/braided. As a basic example we show that the category of representations of the fibre product of finite groups may be expressed in terms of the relative product and show how to multiply braided fusion categories arising from pre-metric groups. Also, we consider relative centers of braided fusion categories and look at the relative tensor product over Müger centers. We finish with an in-depth example hinting at a categorification of conjugacy classes for finite groups.

Tensor [formula omitted]-categories arising as bimodule categories of II1 factors

We prove that if C is a tensor C∗-category in a certain class, then there exists an uncountable family of pairwise non stably isomorphic II1 factors (Mi) such that the bimodule category of Mi is equivalent to C for all i. In particular, we prove that every finite tensor C∗-category is the bimodule category of a II1 factor. As an application we prove the existence of a II1 factor for which the set of indices of finite index irreducible subfactors is {1,5+132,12+313,4+13,11+3132,13+3132,19+5132,7+132}. We also give the first example of a II1 factor M such that Bimod(M) is explicitly calculated and has an uncountable number of isomorphism classes of irreducible objects.

Rigid C*-tensor categories of bimodules over interpolated free group factors

Given a countably generated rigid C*-tensor category ${\sf C}$C, we construct a planar algebra P• whose category of projections ${\sf Pro}$Pro is equivalent to ${\sf C}$C. From P•, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C*-tensor category ${\sf Bim}$Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show ${\sf Bim}$Bim is equivalent to ${\sf Pro}$Pro. We use these constructions to show ${\sf C}$C is equivalent to a category of bifinite bimodules over $L(\mathbb {F}_\infty )$L(F∞).

Monoidal categories in, and linking, geometry and algebra

This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, machinery such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might be, used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.

A diagrammatic categorification of the $q$-Schur algebra

In this paper we categorify the q-Schur algebra \mathbf{S}_q(n,d) as a quotient of Khovanov and Lauda’s diagrammatic 2-category \mathcal{U}(\mathfrak{sl}_n) (Khovanov and Lauda 2010). We also show that our 2-category contains Soergel’s (1992) monoidal category of bimodules of type A , which categorifies the Hecke algebra H_q(d) , as a full sub-2-category if d\leq n . For the latter result we use Elias and Khovanov’s diagrammatic presentation of Soergel’s monoidal category of type A ; see (Elias and Khovanov, 2009).

Braidings on the Category of Bimodules, Azumaya Algebras and Epimorphisms of Rings

Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ is separable, then $A$ admits a canonical R-matrix; in particular, any Azumaya algebra admits a canonical R-matrix. Working over a field, we find a remarkable new characterization of central simple algebras: these are precisely the finite dimensional algebras that admit a canonical R-matrix. Canonical R-matrices give rise to a new class of examples of simultaneous solutions for the quantum Yang-Baxter equation and the braid equation.

Tilting, deformations and representations of linear groups over Euclidean algebras

We study irreducible unitary representations of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of a Dynkin or Euclidean quiver. In particular, we describe such generic representations: we prove that there exists an open dense subset of the space of irreducible unitary representations isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.

On the tensor product of bimodule categories over Hopf algebras

Let H be a finite-dimensional Hopf algebra. We give a description of the tensor product of bimodule categories over \(\operatorname {Rep}(H)\). When the bimodule categories are invertible this description can be given explicitly. We present some consequences of this description in the case H is a pointed Hopf algebra.

Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms

For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule Fω and a bimodule morphism Zω from Lyubashenkoʼs Hopf algebra object K for the bimodule category to Fω. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to Fω. We further show that the bimodule Fω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.The bimodules K and Fω can both be characterized as coends of suitable bifunctors. The morphism Zω is obtained by applying a monodromy operation to the coproduct of Fω; a similar construction for the product of Fω exists as well.Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.

A coalgebraic perspective on linear weighted automata

Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence.Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence.Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages.

A Structure Theorem for $\displaystyle\mathbb{P}^{{1}}$ — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.

Character D-modules via Drinfeld center of Harish-Chandra bimodules

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method.

The braided monoidal structures on the category of vector spaces graded by the Klein group

AbstractLet k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

STANDORT Info

Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework to study the algebra of such functors, which I call formal plethories, in the case where $E_*$ is a Pr\"ufer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C . The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedlerʼs Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).

TENSOR FUNCTORS ON A CERTAIN CATEGORY CONSTRUCTED FROM SPHERICAL CATEGORIES

We construct functors from a certain algebraic category [Formula: see text], defined by Hopf algebra generators and relations, to the category of vector spaces, based on spherical categories. The category [Formula: see text] is proposed by Habiro to be isomorphic to the cobordism category of once-punctured surfaces. If the proposal is proved valid, the result of this paper would imply a construction of a TQFT functor based on a spherical category.