scholarly journals Homotopy composition of cospans

2016 ◽  
Vol 19 (05) ◽  
pp. 1650047
Author(s):  
Joachim Kock ◽  
David I. Spivak
Keyword(s):  
Frobenius Algebra ◽  
Finite Sets ◽  
Frobenius Algebras

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.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

Frobenius algebra with relaxed associativity constraint

2018 ◽  
Vol 27 (07) ◽  
pp. 1841008
Author(s):  
Zbigniew Oziewicz ◽  
William Stewart Page
Keyword(s):  
Associative Algebra ◽  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frobenius Algebra ◽  
Frobenius Algebras ◽  
Frobenius Structures ◽  
Necessary And Sufficient

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra [Formula: see text] is said to be a Frobenius algebra if there exists a [Formula: see text]-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra [Formula: see text] is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided [Formula: see text]-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows [Formula: see text]-permuted opposite algebra to be extended to [Formula: see text]-permuted algebras.

scholarly journals Twin TQFTs and Frobenius Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Carmen Caprau
Keyword(s):  
Monoidal Category ◽  
Topological Quantum Field Theory ◽  
Algebra Homomorphism ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Generators And Relations ◽  
Symmetric Monoidal Category ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Algebraic Description

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

scholarly journals Braided Frobenius algebras from certain Hopf algebras

2021 ◽  
pp. 2350012
Author(s):  
Masahico Saito ◽  
Emanuele Zappala
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Frobenius Algebra ◽  
Compatibility Conditions ◽  
Ribbon Graphs ◽  
Frobenius Algebras ◽  
Ternary Operation ◽  
Compact Surfaces ◽  
Baxter Operator

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

Khovanov homology and diagonalizable Frobenius algebras

2020 ◽  
Vol 29 (01) ◽  
pp. 1950095
Author(s):  
Paul Turner
Keyword(s):  
Elementary Proof ◽  
Khovanov Homology ◽  
Frobenius Algebra ◽  
Link Homology ◽  
Frobenius Algebras

We give a short elementary proof that a Khovanov-type link homology constructed from a diagonalizable Frobenius algebra is degenerate.

Constructions of graded Frobenius algebras

2019 ◽  
Vol 19 (05) ◽  
pp. 2050081
Author(s):  
Ji-Wei He ◽  
Xue-Jun Xia
Keyword(s):  
Frobenius Algebra ◽  
Frobenius Algebras

It is proved that a connected graded Frobenius algebra is determined by a twisted superpotential, and vice versa. Some examples of connected graded Frobenius are given with few generators.

Frobenius Algebras and Their Quivers

1978 ◽  
Vol 30 (5) ◽  
pp. 1029-1044 ◽  
Author(s):  
Edward L. Green
Keyword(s):  
Representation Theory ◽  
Homomorphic Image ◽  
Frobenius Algebra ◽  
Tensor Algebras ◽  
Frobenius Algebras

This paper studies the construction of Frobenius algebras. We begin with a description of when a graded -algebra has a Frobenius algebra as a homomorphic image. We then turn to the question of actual constructions of Frobenius algebras. We give a, method for constructing Frobenius algebras as factor rings of special tensor algebras. Since the representation theory of special tensor algebras has been studied intensively ([6], see also [2; 3; 4]), our results permit the construction of Frobenius algebras which have representations with prescribed properties. Such constructions were successfully used in [9].

Twisted orthogonality relations for certain ℤ/mℤ-graded algebras

2020 ◽  
pp. 2150073
Author(s):  
Prashant Arote
Keyword(s):  
Frobenius Algebra ◽  
Graded Algebras ◽  
Irreducible Characters ◽  
Frobenius Algebras ◽  
Orthogonality Relations

In this paper, we will study the notion of a Frobenius ⋆-algebra and prove some orthogonality relations for the irreducible characters of a Frobenius ⋆-algebra. Then we will study [Formula: see text]-graded Frobenius ⋆-algebras and prove some twisted orthogonality relations for them.

scholarly journals Algebraic singularities of scattering amplitudes from tropical geometry

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
James Drummond ◽  
Jack Foster ◽  
Ömer Gürdoğan ◽  
Chrysostomos Kalousios
Keyword(s):  
Scattering Amplitudes ◽  
Natural Language ◽  
Tropical Geometry ◽  
Cluster Algebras ◽  
Finite Alphabet ◽  
Square Root ◽  
Finite Sets ◽  
Yang Mills ◽  
Square Roots ◽  
Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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