Symmetric Frobenius Algebra Research Articles

A dichotomy between twisted tensor products of bialgebras and Frobenius algebras

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.

On Yang-Baxter equation and Calabi-Yau property of Koszul algebras

–A well-known result of Beidar, Fong, and Stolin yields a description of symmetric Frobenius algebras through solutions of the quantum Yang-Baxter equation. We will prove a graded version of this theorem and produce a new characterization of Koszul Calabi-Yau algebras. We give another proof of the Calabi-Yau property for three and four-dimensional Sklyanin algebras as an application.

Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras

In this paper, we introduce the notions of quasi-triangular and factorizable antisymmetric infinitesimal bialgebras. A factorizable antisymmetric infinitesimal bialgebra leads to a factorization of the underlying associative algebra. We show that the associative double of an antisymmetric infinitesimal bialgebra naturally enjoys a factorizable antisymmetric infinitesimal bialgebra structure. Then we introduce the notion of symmetric Rota-Baxter Frobenius algebras, and show that there is a one-to-one correspondence between factorizable antisymmetric infinitesimal bialgebras and symmetric Rota-Baxter Frobenius algebras. Finally, we show that a symmetric Rota-Baxter Frobenius algebra can give rise to an isomorphism from the regular representation to the coregular representation of a Rota-Baxter associative algebra.

Reasoning about conscious experience with axiomatic and graphical mathematics

We cast aspects of consciousness in axiomatic mathematical terms, using the graphical calculus of general process theories (a.k.a symmetric monoidal categories and Frobenius algebras therein). This calculus exploits the ontological neutrality of process theories. A toy example using the axiomatic calculus is given to show the power of this approach, recovering other aspects of conscious experience, such as external and internal subjective distinction, privacy or unreadability of personal subjective experience, and phenomenal unity, one of the main issues for scientific studies of consciousness. In fact, these features naturally arise from the compositional nature of axiomatic calculus.

CFT Correlators for Cardy Bulk Fields via String-Net Models

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

Foundations of Frobenius Heisenberg categories

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.

The topological symmetric orbifold

We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

Cylindric symmetric functions and positivity

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions Λ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra slˆn and a specialised cyclotomic Hecke algebra. Using Schur–Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the k-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in Λ whose structure constants are the 3-point genus 0 Gromov–Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group G(n,1,k).

Gravity algebra structure on the negative cyclic homology of Calabi–Yau algebras

In this paper, we study the gravity algebra structure on the negative cyclic homology or the cyclic cohomology of several classes of algebras. These algebras include: Calabi–Yau algebras, symmetric Frobenius algebras, unimodular Poisson algebras, and unimodular Frobenius Poisson algebras. The relationships among these gravity algebras are also discussed under some additional conditions.

Singular Hochschild cohomology and algebraic string operations

Given a differential graded (dg) symmetric Frobenius algebra A we construct an unbounded complex \mathcal{D}^{*}(A,A) , called the Tate–Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex \mathcal{D}^*(A,A) computes the singular Hochschild cohomology of A . We construct a cyclic (or Calabi–Yau) A -infinity algebra structure, which extends the classical Hochschild cup and cap products, and an L -infinity algebra structure extending the classical Gerstenhaber bracket, on \mathcal{D}^*(A,A) . Moreover, we prove that the cohomology algebra H^*(\mathcal{D}^*(A,A)) is a Batalin–Vilkovisky (BV) algebra with BV operator extending Connes' boundary operator. Finally, we show that if two dg algebras are quasi-isomorphic then their singular Hochschild cohomologies are isomorphic and we use this invariance result to relate the Tate–Hochschild complex to string topology.

Line and surface defects in Reshetikhin–Turaev TQFT

A modular tensor category \mathcal C gives rise to a Reshetikhin–Turaev type topological quantum field theory which is defined on 3-dimensional bordisms with embedded \mathcal C -coloured ribbon graphs. We extend this construction to include bordisms with surface defects which in turn can meet along line defects. The surface defects are labelled by \Delta -separable symmetric Frobenius algebras and the line defects by “multi-modules” which are equivariant with respect to a cyclic group action. Our invariant cannot distinguish non-isotopic embeddings of 2-spheres, but we give an example where it distinguishes non-isotopic embeddings of 2-tori.

When Ext is a Batalin–Vilkovisky algebra

We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin–Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg–Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf–Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. As an application, we prove that the Hochschild cohomology of an associative algebra A is Batalin–Vilkovisky if A itself is a contramodule over its enveloping algebra A \otimes A^\mathrm {op} . This is, for example, the case for symmetric algebras and Frobenius algebras with semisimple Nakayama automorphism. We also recover the construction for Hopf algebras.

The localized skein algebra is Frobenius

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters $\chi(F)$ of the fundamental group of $F$. We localize by inverting the nonzero characters to get an algebra $S^{-1}K_N(F)$ over the function field of the character variety. We prove that if $F$ is noncompact, the algebra $S^{-1}K_N(F)$ is a symmetric Frobenius algebra. Along the way we prove $K(F)$ is finitely generated, $K_N(F)$ is a finite rank module over $\chi(F)$, and the simple closed curves that make up any simple diagram on $F$ generate a finite field extension of $S^{-1}\chi(F)$ inside $S^{-1}K_N(F)$.

An equivalence between semisimple symmetric Frobenius algebras and Calabi–Yau categories

We show that the bigroupoid of semisimple symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of Calabi–Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, semisimple Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace.

A general state-sum construction of 2-dimensional topological quantum field theories with defects

We derive a general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2D TQFTs without defects (cf. Lauda and Pfeiffer [State-sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16 (2007) 1121–1163, doi: 10.1142/S0218216507005725]). From the extended Pachner moves (Crane and Yetter [Moves on filtered PL manifolds and stratified PL spaces, arXiv:1404.3142 ]), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter [On 2-dimensional Dijkgraaf–Witten theory with defects, to appear in J. Knots Theory Ramifications], we require that triangulations be flaglike, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flaglikeness.

Consistent systems of correlators in non-semisimple conformal field theory

Based on the modular functor associated with a – not necessarily semisimple – finite non-degenerate ribbon category D, we present a definition of a consistent system of bulk field correlators for a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in D, while for surfaces of any genus they are in bijection with modular Frobenius algebras in D. This provides additional insight into structures familiar from rational conformal field theories and extends them to rigid logarithmic conformal field theories.

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 .

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.

Universal operations in Hochschild homology

AbstractWe provide a general method for finding all natural operations on the Hochschild complex ofℰ${\mathcal{E}}$-algebras, whereℰ${\mathcal{E}}$is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative orA∞${A_{\infty}}$-algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex offormal operations, for which we provide an explicit model that we can calculate in a number of cases. Whenℰ${\mathcal{E}}$encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich–Soibelman via different methods identify with all formal operations. Whenℰ${\mathcal{E}}$encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler–Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.

Self-dual and quasi self-dual algebras

Self-dual algebras are ones with an A bimodule isomorphism A → A ∨op, where A ∨ = Hom k (A, k) and A ∨op is the same underlying k-module as A ∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ∨ op). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.