We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. A dual version of Brauer's equivalence theorem is presented, allowing a characterization of symmetric coalgebras by comparing certain functors. We define an automorphism of the ring with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is used to give a new characterization to symmetric coalgebras and to describe Hopf algebras that are symmetric as coalgebras. As a corollary we obtain as a consequence the known characterization of Hopf algebras that are symmetric as algebras.
We investigate left and right co-Frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of Frobenius algebras. We prove that a coalgebra is left and right co-Frobenius if and only if C is isomorphic to Rat(C*) as right C*-modules and also that this is eqhivalent to the fact that the functors Hom_K(-,K) and Hom_{C*}(-,C*) from the category of right C-comodules to the category of all left C*-modules are isomorphic. This allows a definition of a left-right symmetric concept of co-Frobenius coalgebras that is perfectly dual to the one of Fro…
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 $S^4$-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.
"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or prod…
We generalize the results on existence and uniqueness of integrals from compact groups and Hopf algebras in a pure (co)algebraic setting, and find a series of new results on (quasi)-co-Frobenius and semiperfect coalgebras. For a coalgebra $C$, we introduce the generalized space of integrals $\int_M=\Hom^C(C,M)$ associated to a right $C$-comodule $M$ and study connections between "uniqueness of integrals" $\dim(\int_M)\leq \dim(M)$ and "existence of integrals" $\dim(\int_M)\geq \dim(M)$ for all $M$ and representation theoretic properties of $C$: (quasi)-co-Frobenius, semiperfect. We show that a…
We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We also discuss the coreflexive property for the considered classes of coalgebras. Finally, we investigate which of the co-Frobenius…
Entwined modules arose from the coalgebra-Galois theory. They are a generalisation of unified Doi-Hopf modules. In this paper, Frobenius properties and Maschke-type theorems, known for Doi-Hopf modules are extended to the case of entwined modules.
A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient…
We show that Turaev's group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier's version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over Hopf group-coalgebras using the center construction.
The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra an…
We prove that the shifted Hochschild chain complex $C\_*(A,A)[m]$ of a symmetric open Frobenius algebra $A$ of degree $m$ has a natural homotopy coBV-algebra structure. As a consequence $HH\_*(A,A)[m]$ and $HH^*(A,A^\vee)[-m]$ are respectively coBV and BV algebras. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural homotopy BV-algebra structure on $C\_*(A,A)[m]$ hence a BV-structure on $HH\_*(A,A)[m]$. Moreover we prove that the product and coproduct on $HH\_*(A,A)[m]$ satisfy the Frobenius compatibility condition i.e. $HH\_*(A,A)[m…
OCHA is the homotopy algebra of open-closed strings. It can be defined as a sequence of multilinear operations on a pair of DG spaces satisfying certain relations which include the $L_\infty$ relations in one space and the $A_\infty$ relations in the other. In this paper we show that the OCHA structure is intrinsic to the tensor product of the symmetric and tensor coalgebras. We also show how an OCHA can be obtained from $A_\infty$-extesions and define the {\it universal enveloping} $A_\infty$-algebra of an OCHA as an $A_\infty$-extension of the universal enveloping of its $L_\infty$ part by i…
We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusio…
Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $θ_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteri…
We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.
The `spider theorem' for a general Frobenius algebra $A$, classifies all maps $A^{\otimes m}\to A^{\otimes n}$ that are built from the operations and, in a graphical representation, represented by a {\it connected} diagram. Here the algebra can be noncommutative and the Frobenius form can be asymmetric. We view this theorem as reducing any connected diagram to a standard form with $j$ beads $B$, where $j$ is the number of bounded connected components of the original diagram. We study the associated F-dimension Hilbert series $\dim_x=\sum_{j=0}^\infty x^j\dim_j$, where $\dim_j=ε\circ B^j\circ 1…
We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra $H$. If $H$ is a group Hopf algebra, we study a more general Frobenius type property and uncover the structure of graded Frobenius algebras. Graded symmetric algebras are also investigated.
We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ i…
We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.
We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric grou…
We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, separable Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace. In order to put this result into perspective, we introduce the concept of the equivariantization of a 2-functor between bicategories endowed with a G-action. We prove that the equivalence between Froben…
An associative algebra is nothing but an odd quadratic codifferential on the tensor coalgebra of a vector space, and an A-infinity algebra is simply an arbitrary odd codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A-infinity algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an A-infinity algebra preserving the inner product. Similarly, a graded Lie algebra is simply a special case of an odd codifferential on the exterior coalgebra of a vector space, and an …
In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the q…
We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras, bimodules and intertwiners. The results are motivated by the 2-dimensional Cobordism Hypothesis for oriented manifolds, and can hence be interpreted in the realm of Topological Quantum Field Theory.
We define a dg-category of looped diagrams which we use to construct operations on the Hochschild complex of commutative Frobenius dg-algebras. We show that we recover the operations known for symmetric Frobenius dg-algebras constructed using Sullivan chord diagrams as well as all formal operations for commutative algebras (including Loday's lambda operations) and prove that there is a chain level version of a suspended Cactus operad inside the complex of looped diagrams. This recovers the suspended BV algebra structure on the Hochschild homology of commutative Frobenius algebras defined by Ab…
For a finite dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite dimensional Hopf kernel and with image the universal enveloping algebra. In this article we define and describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases the Frobenius-Lusztig kernel is associated to a different Lie algebra than the initial …
While semisimple artinian rings and semisimple coalgebras over a field can be described in terms of matrices (either matrix ring over division rings or comatrix coalgebras over the ground field), semisimple corings seem to have a more intrincated structure in general. It turns out that some well-known properties of semisimple rings or coalgebras, which are immediately deduced from the aforementioned structure, are not evident over a (left) semi-simple coring. For instance, it is not evident that the notion of semi-simple coring is left-right symmetric. To be precise, if every left comodule dec…
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. Thes…
Let $U:\mathcal{C}\rightarrow\mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R(\mathbb{1}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor…
In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.
We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.
We study the representations and their Frobenius-Schur indicators of two semisimple Hopf algebras related to the symmetric group $S_n$, namely the bismash products $H_n = k^{C_n}# kS_{n-1}$ and its dual $J_n = k^{S_{n-1}}# kC_n = (H_n)^*,$ where $k$ is an algebraically closed field of characteristic 0. Both algebras are constructed using the standard representation of $S_n$ as a factorizable group, that is $S_n = S_{n-1}C_n = C_nS_{n-1}$. We prove that for $H_n$, the indicators of all simple modules are +1. For the dual Hopf algebra $J_n = k^{S_{n_1}}# kC_n$, the indicator can have values eith…
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a noncommutative Frobenius algebra, hence justifying this choice of data in the Moore-Segal axioms. Our…
We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of Frobenius algebra and the latter is called a double construction of Connes cocycle which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras an…
We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an …
By computing Frobenius-Schur indicators of modules of certain weak Hopf algebras, we give a formula for the number of involutions in symmetric groups, which are contained in a given coset with respect to a given Young subgroup.
We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde fiber functor $F(X)$ of $X$ (defined by the second author) in terms of the usual and super Frobenius-Perron dimension of $X$. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique…
We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincar{é} duality hypothesis, such as Calabi-Yau algebras, derived Poincar{é} duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$. As an example, after proving that the chain complex of the Moore loop space of a manifold $M$ is a CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a BV-st…
A Morita class of symmetric special Frobenius algebras A in the modular tensor category of a chiral CFT determines a full CFT on oriented world sheets. For unoriented world sheets, A must in addition possess a reversion, i.e. an isomorphism from A^opp to A squaring to the twist. Any two reversions of an algebra A differ by an element of the group Aut(A) of algebra automorphisms of A. We establish a group homomorphism from Aut(A) to the Picard group of the bimodule category C_AA, with kernel consisting of the inner automorphisms, and we refine Morita equivalence to an equivalence relation betwe…
