Frobenius Algebras Research Articles

Quaternion rings and octonion rings

In this paper, for rings R, we introduce complex rings ℂ(R), quaternion rings ℍ(R), and octonion rings O(R), which are extension rings of R; R ⊂ ℂ(R) ⊂ ℍ(R) ⊂ O(R). Our main purpose of this paper is to show that if R is a Frobenius algebra, then these extension rings are Frobenius algebras and if R is a quasi-Frobenius ring, then ℂ(R) and ℍ(R) are quasi-Frobenius rings and, when Char(R) = 2, O(R) is also a quasi-Frobenius ring.

Symmetric algebras in categories of corepresentations and smash products

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H⁎ with respect to the Frobenius and symmetric properties.

Representations of Frobenius-type triangular matrix algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.

A Review To Some Properties of Semiperfect and Quasi- Frobenius Rings

In [15] Ikeda showed that a ring R is left and right mininjective and left and right Artinian if and only if it is quasi-Frobenius. In [33] Utumi proved that any left and right continuous and left and right Artinian ring is quasi-Frobenius. In [7], Camillo and Yousif generalized Ikeda’s and Utumi’s result as a left and right continuous ring with ascending chain condition on left annihilators is quasi- Frobenius ring. It is natural to ask whether Faith’s or Camillo and Yousif ’s results can be extended to semiperfect CS-rings. However, it is a result of G´omez Pardo and Guil Asensio [29] that every right Kasch, right CS-ring has a finitely generated essential right socle, but it is unknown weather a right Kasch right CS-ring is always semiperfect [31]. In this paper, we review and prove some properties on Kasch and quasi-Frobenius rings which is related to these results.

Interacting Hopf algebras

We introduce the theory IHR of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IHR are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid–comonoid pairs. This construction is instrumental in showing that IHR is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R.

On the Hochschild homology of open Frobenius algebras

We prove that the shifted Hochschild homology HH_*(A,A)[m] of a symmetric open Frobenius algebra A of degree m has a natural coBV-algebra structure which is defined at the chain level. As a consequence HH^*(A,A^\vee)[-m] is a BV algebra. The underlying coalgebra and algebra structure may not be resp. counital and unital. We also introduce a natural BV-algebra structure on HH_*(A,A)[m] which is also defined at the chain level. Hence there is 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] is an open Frobenius algebras. If A is commutative, we also introduce a natural BV structure on the shifted relative Hochschild homology \widetilde{HH}_*(A)[m-1] . We conjecture that the product of this BV structure is identical to the Goresky–Hingston [10] product on the cohomology of free loop spaces when A is a commutative cochain algebra model for M .

Homotopy composition of cospans

It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra. In this paper, we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.

BV differential on Hochschild cohomology of Frobenius algebras

For a finite-dimensional Frobenius k-algebra R with a Nakayama automorphism ν, we define an algebra HH⁎(R)ν↑. If the order of ν is not divisible by the characteristic of k, this algebra is isomorphic to the Hochschild cohomology algebra of R. We prove that this algebra is a BV algebra. Moreover, we construct a BV differential on it in a canonical way. We use this fact to calculate the Gerstenhaber algebra structure and a BV structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type Dn.

Frobenius algebras derived from the Kauffman bracket skein algebra

The Kauffman bracket skein algebra of a compact oriented surface when the variable [Formula: see text] in the Kauffman bracket is set equal to [Formula: see text], where [Formula: see text] is an odd counting number, is a central extension of the ring of [Formula: see text]-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

Codes over local rings of order 16 and binary codes

We study codes over the commutative local Frobenius rings of order 16 with maximal ideals of size 8. We define a weight preserving Gray map and study the images of these codes as binary codes. We study self-dual codes and determine when they exist.

Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras

We investigate conditions that are sufficient to make the Ext-algebra of an object in a (triangulated) category into a Frobenius algebra, and compute the corresponding Nakayama automorphism. As an application, we prove the conjecture that $\mathrm {hdet}(\mu _A) = 1$ for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra $A$.

Boson condensation in topologically ordered quantum liquids

Boson condensation in topological quantum field theories (TQFT) has been previously investigated through the formalism of Frobenius algebras and the use of vertex lifting coefficients. While general, this formalism is physically opaque and computationally arduous: analyses of TQFT condensation are practically performed on a case by case basis and for very simple theories only, mostly not using the Frobenius algebra formalism. In this paper we provide a new way of treating boson condensation that is computationally efficient. With a minimal set of physical assumptions, such as commutativity of lifting and the definition of confined particles, we can prove a number of theorems linking Boson condensation in TQFT with chiral algebra extensions, and with the factorization of completely positive matrices over the nonnegative integers. We present numerically efficient ways of obtaining a condensed theory fusion algebra and S matrices; and we then use our formalism to prove several theorems for the S and T matrices of simple current condensation and of theories which upon condensation result in a low number of confined particles. We also show that our formalism easily reproduces results existent in the mathematical literature such as the noncondensability of 5 and 10 layers of the Fibonacci TQFT.

Tensors, !-graphs, and Non-commutative Quantum Structures

!-graphs provide a means of reasoning about infinite families of string diagrams and have proven useful in manipulation of (co)algebraic structures like Hopf algebras, Frobenius algebras, and compositions thereof. However, they have previously been limited by an inability to express families of diagrams involving non-commutative structures which play a central role in algebraic quantum information and the theory of quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose’s abstract tensor notation.

Geometric representations of graded and rational Cherednik algebras

We provide geometric constructions of modules over the graded Cherednik algebra Hνgr and the rational Cherednik algebra Hνrat attached to a simple algebraic group G together with a pinned automorphism θ. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit C⁎-actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When θ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules Lν(triv) of Hνgr and of Hνrat. We give a formula for the dimension of Lν(triv) and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.

The acyclicity of the Frobenius functor for modules of finite flat dimension

Let R be a commutative Noetherian local ring of prime characteristic p and f:R⟶R the Frobenius ring homomorphism. For e≥1 let R(e) denote the ring R viewed as an R-module via fe. Results of Peskine, Szpiro, and Herzog state that for finitely generated modules M, M has finite projective dimension if and only if ToriR(R(e),M)=0 for all i>0 and all (equivalently, infinitely many) e≥1. We prove this statement holds for arbitrary modules using the theory of flat covers and minimal flat resolutions.

$\mathfrak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $\mathfrak{g}$-quasi-Frobenius Lie algebra for $\mathfrak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak{q},\beta )$ together with a left $\mathfrak{g}$-module structure which acts on $\mathfrak{q}$ via derivations and for which $\beta $ is $\mathfrak{g}$-invariant. Geometrically, $\mathfrak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\mathfrak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak{g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf{Rep}(\mathfrak{g})$. If $\mathfrak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\mathfrak{g}))$, where $D(\mathfrak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak{g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak{g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak{g})$-action which gives it the structure of a $D(\mathfrak{g})$-quasi-Frobenius Lie algebra.

New quantum obstructions to sliceness

It is well known that generic perturbations of the complex Frobenius algebra used to define Khovanov cohomology each give rise to Rasmussen's concordance invariant s. This gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot. Similar behavior has been observed in sl ( n ) Khovanov–Rozansky cohomology, where a perturbation gives rise to the concordance homomorphisms s n for each n ⩾ 2 , and where we have s 2 = s . We demonstrate that s n for n ⩾ 3 does not in fact arise generically, and that varying the chosen perturbation gives rise both to new concordance homomorphisms and to new sliceness obstructions that are not equivalent to concordance homomorphisms.

Hochschild homology of structured algebras

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication—we think of such algebras as A∞-algebras “with extra structure”. As applications, we obtain an integral version of the Costello–Kontsevich–Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler–Zeinalian and Kaufmann actions of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex.

Une généralisation de l'homologie de Khovanov

We generalize Khovanov's homology to unoriented links and with coefficients in the 2-dimensional universal Frobenius algebra. Then, we construct, for each Reidemeister move, a graduated homotopy equivalence and give the explicit formulae of these equivalences and their homotopy inverses. Further, we prove that the generalized Khovanov complex of a link diagram D mirror image is isomorphic to the dual of the Khovanov complex of D. Finally, we generalize the Rasmussen theorem on the Lee–Khovanov homology.

On Frobenius and separable algebra extensions in monoidal categories: applications to wreaths

We characterize Frobenius and separable monoidal algebra extensions i: R \to S in terms given by R and S . For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if S is a Frobenius, respectively separable, algebra in the category of bimodules over R . In the case when R is separable we show that the extension is separable if and only if S is a separable algebra. Similarly, in the case when R is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if S is a Frobenius algebra and the restriction at R of its Nakayama automorphism is equal to the Nakayama automorphism of R . As applications, we obtain several characterizations for an algebra extension associated to a wreath to be Frobenius, respectively separable.