Narrow Results

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.

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.

Generalizing the well-known correspondence between two-sided adjunctions and Frobenius algebras, we establish a one-to-one correspondence between local adjunctions of $C^{\ast }$-correspondences, as defined and studied in prior work with Clare and Higson; and Frobenius $C^{\ast }$-algebras, a natural $C^{\ast }$-algebraic adaptation of the standard notion of Frobenius algebras that we introduce here.

Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle).

A self-dual definition of a Frobenius object without the use of a prefered dual is considered.

The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors.

Hermitian structures are modeled as a conjugation compatible with duality. It is the structure cobordism categories posses. A definition of generalized cobordism categories is proposed.

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A^op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.

We provide a combinatorial model for spin surfaces. Given a triangulation of an oriented surface, a spin structure is encoded by assigning to each triangle a preferred edge, and to each edge an orientation and a sign, subject to certain admissibility conditions. The behavior of this data under Pachner moves is then used to define a state sum topological field theory on spin surfaces. The algebraic data is a Δ-separable Frobenius algebra whose Nakayama automorphism is an involution. We find that a simple extra condition on the algebra guarantees that the amplitude is zero unless the combinatorial data satisfies the admissibility condition required for the reconstruction of the spin structure.

The Kauffman bracket skein algebra of a compact oriented surface when the variable $A$ in the Kauffman bracket is set equal to $e^{\pi i/N}$, where $N$ is an odd counting number, is a central extension of the ring of $S{L}_2ℂ$-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.

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.

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras.

These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions, in analogy with Hochschild cohomology of bialgebras, based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation, using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

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.

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 $D-$equations. Finally, using symmetric solutions of these equations, we build the double constructions of related Connes cocycles.

We give two alternate presentations of the Frobenius Heisenberg category, ${\mathscr{H}}ei{s}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus \cdots \oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ${\mathscr{H}}ei{s}_{F,k}$ are given in terms of planar diagrams consisting of strands “colored” by integers $i=1,\dots ,n$, 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, ${\mathscr{H}}ei{s}_{F_1,k}\otimes \cdots \otimes {\mathscr{H}}ei{s}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$ that we call the partial Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$.

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.

Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by $D(m,d,\xi )$. In this paper, we consider their quotient algebras $D^{\prime }(m,d,\xi ),$ which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of $D^{\prime }(m,d,\xi )$ 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 $m=3$.

We introduce a natural class of potential submanifolds in pseudo-Euclidean spaces and prove that each N-dimensional Frobenius manifold can locally be represented as an N-dimensional potential submanifold in a 2N-dimensional pseudo-Euclidean space. We show that all potential submanifolds bear natural Dubrovin–Frobenius structures on their tangent spaces; in particular, Weingarten operators of each N-dimensional potential submanifold give N-parameter integrable deformations of structural constants of Frobenius algebras. We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and define locally the class of potential submanifolds. The problem of explicit realization of an arbitrary concrete Frobenius manifold as a potential submanifold in a pseudo-Euclidean space is reduced to solving a linear system of second-order partial differential equations. For concrete Frobenius manifolds, this realization problem can be solved explicitly in elementary and special functions. Moreover, we consider a nonlinear system, which is a natural generalization of the associativity equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this system is integrable by the inverse scattering method.