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The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras. [ABSTRACT FROM AUTHOR]

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras. [ABSTRACT FROM AUTHOR]

String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs. [ABSTRACT FROM AUTHOR]

We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the twisted tensor product of bialgebras is a bialgebra, and when the twisted tensor product of Frobenius algebras is a Frobenius algebra. Our methods are purely diagrammatic, so these results hold for (braided) monoidal categories. As an application, we recover that some quantum complete intersections are Frobenius algebras, and we construct families of noncommutative symmetric Frobenius algebras. Along the way, we also characterize when twisted tensor products of separable algebras are separable, and we prove that twisted tensor products of special Frobenius algebras are special Frobenius. [ABSTRACT FROM AUTHOR]

It was shown by Ostrik (Int. Math. Res. Not. 2003(27), 1507–1520 2003) and Natale (SIGMA Symmetry Integrability Geom. Methods Appl. 13, 042 2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor Φ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties. [ABSTRACT FROM AUTHOR]

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship. Given a finite commutative Frobenius object in Span, one can obtain invariants of closed surfaces with values in the natural numbers. We explicitly compute these invariants in several examples, including examples arising from abelian groups. [ABSTRACT FROM AUTHOR]

We extend the theory of Mackey 2-functors provided by Balmer and Dell'Ambrogio [ Mackey 2-functors and Mackey 2-motives , European Mathematical Society (EMS), Zürich, 2020] by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical Green functors familiar from representation theory and topology arise by decategorification, in various ways, of some Green 2-functor occurring in Nature. [ABSTRACT FROM AUTHOR]

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutative chain ring. The former was motivated by combinatorics and the latter by coding theory over finite rings. We prove that the algebra of the multiplicative semigroup of a finite Frobenius ring is Frobenius over any field whose characteristic doesn't divide that of the ring. As a consequence we obtain an easier proof of Kovács's theorem that the algebra of the monoid of matrices over a finite field is a direct product of matrix algebras over group algebras of general linear groups (outside of the characteristic of the finite field). [ABSTRACT FROM AUTHOR]

Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by D (m , d , ξ). In this paper, we consider their quotient algebras D ′ (m , d , ξ) , which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of D ′ (m , d , ξ) when d is odd. It turns out that the Grothendieck rings are commutative rings generated by three elements subject to some relations. Then we compute the Casimir numbers of the Grothendieck rings for m = 1 and m = 3. [ABSTRACT FROM AUTHOR]

We provide a formula for computing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras. [ABSTRACT FROM AUTHOR]

This paper deals with an exterior algebra of a vector space whose base field is of positive characteristic. In this work, a minimal set of generators forming the annihilator of even neat elements of such an exterior algebra is exhibited. The annihilator of some special type of even neat elements is determined to prove the conjecture established in [3]. Moreover, a vector space basis for the annihilators under consideration is calculated. [ABSTRACT FROM AUTHOR]

We investigate conditions that are sufficient to make the Extalgebra 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 hdet(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

We investigate conditions that are sufficient to make the Extalgebra 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 hdet(μA) = 1 for any noetherian Artin-Schelter regular (hence skew Calabi-Yau) algebra A.

To each symmetric graded Frobenius superalgebra we associate a W-algebra. We then define a linear isomorphism between the trace of the Frobenius Heisenberg category and a central reduction of this W-algebra. We conjecture that this is an isomorphism of graded superalgebras. [ABSTRACT FROM AUTHOR]

Generalizing the well-known correspondence between two-sided adjunctions and Frobenius algebras, we establish a one-to-one correspondence between local adjunctions of C ∗ -correspondences, as defined and studied in prior work with Clare and Higson; and Frobenius C ∗ -algebras, a natural C ∗ -algebraic adaptation of the standard notion of Frobenius algebras that we introduce here. [ABSTRACT FROM AUTHOR]

We study the existence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra. We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama automorphisms. [ABSTRACT FROM AUTHOR]

Control theory uses 'signal-flow diagrams' to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. These diagrams can be seen as string diagrams for the symmetric monoidal category FinVectk of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVectk in terms of the generators used in signal-flow diagrams. A broader class of signal-flow diagrams also includes `caps' and `cups' to model feedback. We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRelk, where objects are still finite-dimensional vector spaces but the morphisms are linear relations. We also give a presentation for FinRelk. The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid. This sort of structure, but with tensor product replacing direct sum, is familiar from the `ZX-calculus' obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases.

We describe bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded Frobenius algebra, proving several open conjectures. Our proof uses a categorical comultiplication and generalized cyclotomic quotients of the category. We use our basis theorem to prove that the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg category recovers the lattice Heisenberg algebra associated to the Frobenius algebra. [ABSTRACT FROM AUTHOR]

Wu-Liu-Ding algebras, that is D (m , d , ξ) , are a class of non-pointed affine prime regular Hopf algebras of GK-dimension one. In this paper, we mainly study a class of quotient algebras of D (m , d , ξ) , denoted by D ′ (m , d , ξ) , which are 2 m 2 d -dimensional non-pointed semisimple Hopf algebras. For a better understanding of the structure of the Hopf algebra D ′ (m , d , ξ) , we have considered the Grothendieck rings of D ′ (m , d , ξ) and their Casimir numbers when d is odd in our previous paper. In this paper we continue dealing with the more complex case when d is even. It turns out that the Grothendieck rings of D ′ (m , d , ξ) are generated by four elements subject to some relations. Then we give the Casimir numbers of the Grothendieck rings of D ′ (1 , d , ξ) and D ′ (2 , d , ξ). [ABSTRACT FROM AUTHOR]

Strict RA semigroups are common generalizations of ample semigroups and inverse semigroups. The aim of this paper is to study algebras of strict RA semigroups. It is proved that any algebra of strict RA semigroups with finite idempotents has a generalized matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. In particular, it is proved that any algebra of finite right ample semigroups has a generalized upper triangular matrix representation whose degree is equal to the number of non-zero regular 𝓓-classes. As its application, we determine when an algebra of strict RA semigroups (right ample monoids) is semiprimitive. Moreover, we prove that an algebra of strict RA semigroups (right ample monoids) is left self-injective iff it is right self-injective, iff it is Frobenius, and iff the semigroup is a finite inverse semigroup.

The Frobenius complexity of a local ring R measures asymptotically the abundance of Frobenius operators of order e on the injective hull of the residue field of R. It is known that, for Stanley–Reisner rings, the Frobenius complexity is either −∞ or 0. This invariant is determined by the complexity sequence { c e } e of the ring of Frobenius operators on the injective hull of the residue field. We will show that { c e } e is constant for e ≥ 2 , generalizing work of Àlvarez Montaner, Boix, and Zarzuela. Our result settles an open question mentioned by Àlvarez Montaner. [ABSTRACT FROM AUTHOR]

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold. [ABSTRACT FROM AUTHOR]

We give two alternate presentations of the Frobenius Heisenberg category, ℋ e i s F , k , defined by Savage, when the Frobenius algebra F = F 1 ⊕ ⋯ ⊕ F n decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ℋ e i s F , k are given in terms of planar diagrams consisting of strands "colored" by integers i = 1 , ... , n , where a strand of color i carries tokens labelled by elements of F i. In addition, we prove that when F decomposes this way, the tensor product of Frobenius Heisenberg categories, ℋ e i s F 1 , k ⊗ ⋯ ⊗ ℋ e i s F n , k , is equivalent to a certain subcategory of the Karoubi envelope of ℋ e i s F , k that we call the partial Karoubi envelope of ℋ e i s F , k . [ABSTRACT FROM AUTHOR]

In this paper, we prove that every Khovanov homology associated to a Frobenius algebra of rank 2 can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from the category of cobordisms of oriented links to the homotopy category of complexes. [ABSTRACT FROM AUTHOR]

It has been shown in this paper that the commutative Frobenius algebra QZ5⊗Z(QS3) provides a complete invariant for two-dimensional cobordisms, i.e., that the corresponding two-dimensional quantum field theory is faithful. Zsigmondy’s Theorem is essential to the proof of this result. [ABSTRACT FROM AUTHOR]

We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this generating function explicitly as a rational function. [ABSTRACT FROM AUTHOR]

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Batalin-Vilkovisky operators on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson structure is pseudo-unimodular. The relation between modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases. [ABSTRACT FROM AUTHOR]

Let C be a finite tensor category and M an exact left C -module category. We call M unimodular if the finite multitensor category Rex C (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. [ABSTRACT FROM AUTHOR]

The monomorphism category M ( A , M , B ) induced by a bimodule M B A is the subcategory of Λ-mod consisting of [ X Y ] ϕ such that ϕ : M ⊗ B Y → X is a monic A -map, where Λ = [ A M 0 B ] , and A , B are Artin algebras. In general, M ( A , M , B ) is not the monomorphism category induced by quivers. It could describe the Gorenstein-projective Λ-modules. This monomorphism category is a resolving subcategory of Λ-mod if and only if M B is projective. In this case, it has enough injective objects and Auslander–Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting Λ-module. If M satisfies the condition (IP) (see Subsection 1.6 ), then the stable category of M ( A , M , B ) admits a recollement of additive categories, which is in fact a recollement of singularity categories if M ( A , M , B ) is a Frobenius category. Ringel–Schmidmeier–Simson equivalence between M ( A , M , B ) and its dual is introduced. If M is an exchangeable bimodule, then an RSS equivalence is given by a Λ–Λ bimodule which is a two-sided cotilting Λ-module with a special property; and the Nakayama functor N Λ gives an RSS equivalence if and only if both A and B are Frobenius algebras. [ABSTRACT FROM AUTHOR]

Let k be a commutative ring, $\mathcal {A}$ and $\mathcal {B}$ - two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ , where $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ is the homotopy category of finitely generated projective $\mathcal {A}$ -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ and a map from the set of standard G-equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)$ to $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)$ , where $\mathcal {A}/G$ denotes the orbit category. We investigate the properties of these maps and apply our results to the case where $\mathcal {A}=\mathcal {B}=R$ is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field. [ABSTRACT FROM AUTHOR]

In this work, we compute solutions of the Yang-Baxter associative equation in dimensions one and two. For these solutions, we describe the double constructions of the associated Frobenius algebras, following Bai's definitions. Besides, we determine related compatible dendriform algebras and solutions of their equations. Finally, using symmetric solutions of these equations, we build the double constructions of related Connes cocycles. [ABSTRACT FROM AUTHOR]

We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study homological properties and rigidity of group coactions on Artin–Schelter regular algebras. [ABSTRACT FROM AUTHOR]

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but for high degree polynomials, nonreal roots are typically much more numerous than the real ones. The challenge is known for a long time, and the subject has been intensively studied. Nevertheless, we propose some novel ideas and techniques and obtain dramatic acceleration of the known algorithms. In order to achieve our progress we exploit the correlation between the computations with matrices and polynomials, randomized matrix computations, and complex plane geometry, extend the techniques of the matrix sign iterations, and use the structure of the companion matrix of the input polynomial. The results of our extensive tests with benchmark polynomials and random matrices are quite encouraging. In particular in our tests the number of iterations required for convergence of our algorithms grew very slowly (if it grew at all) as we increased the degree of the input polynomials and the dimension of the input matrices from 64 to 1024. [ABSTRACT FROM AUTHOR]