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We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are coded in one partial permutation algebra. We propose a generalization of the partial permutation algebra to the symmetric orbifold topological quantum field theory of any seed theory and show that the theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov–Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give a complete classification of symplectic structures on six-dimensional Frobeniusian solvable Lie algebras, up to symplectomorphism. We provide a scheme to classify the isomorphism classes of six-dimensional Frobeniusian solvable Lie algebras whose exact form has a Lagrangian ideal. We complete our classification by considering Frobeniusian solvable Lie algebras without Lagrangian ideal. [ABSTRACT FROM AUTHOR]

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line S[t] as a functor defined on the 1-category of cyclotomic spectra with values in the1-category of spectra with Frobenius lifts, refining a result of Blumberg-Mandell. We define the notion of an integral topological Cartier module using Barwick's formalism of spectral Mackey functors on orbital1-categories, extending the work of Antieau-Nikolaus in the p-typical setting. As an application, we show that TR evaluated on a connective E1-ring admits a description in terms of the spectrum of curves on algebraic K-theory, generalizing the work of Hesselholt and Betley-Schlichtkrull. [ABSTRACT FROM AUTHOR]

Let G be a group. We give a categorical definition of the G-equivariant α -induction associated with a given G-equivariant Frobenius algebra in a G-braided multitensor category, which generalizes the α -induction for G-twisted representations of conformal nets. For a given G-equivariant Frobenius algebra in a spherical G-braided fusion category, we construct a G-equivariant Frobenius algebra, which we call a G-equivariant α -induction Frobenius algebra, in a suitably defined category called neutral double. This construction generalizes Rehren's construction of α -induction Q-systems. Finally, we define the notion of the G-equivariant full centre of a G-equivariant Frobenius algebra in a spherical G-braided fusion category and show that it indeed coincides with the corresponding G-equivariant α -induction Frobenius algebra, which generalizes a theorem of Bischoff, Kawahigashi and Longo. [ABSTRACT FROM AUTHOR]

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). In a previous paper, we gave an explicit description of these algebras for groups of type G 2 and F 4 in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type E 6. [ABSTRACT FROM AUTHOR]

For a positive integer r , an r -spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r -fold cover of SO 2. In particular, such a TQFT assigns a scalar invariant to every closed r -spin surface Σ. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Σ, we construct a symmetric monoidal category C and a C -valued r -spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r -spin surfaces, and we show that the Frobenius algebras associated to such TQFTs are necessarily non-semisimple. [ABSTRACT FROM AUTHOR]

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed. [ABSTRACT FROM AUTHOR]

This paper explores geometric circulant matrices whose entries are Chebyshev polynomials of the first or second kind. Motivated by our previous research on r − circulant matrices and Chebyshev polynomials, we focus on calculating the Frobenius norm and deriving estimates for the spectral norm bounds of these matrices. Our analysis reveals that this approach yields notably improved results compared to previous methods. To validate the practical significance of our research, we apply it to existing studies on geometric circulant matrices involving the generalized k − Horadam numbers. The obtained results confirm the effectiveness and utility of our proposed approach. [ABSTRACT FROM AUTHOR]

We give a complete classification of topological field theories with reflection structure and spin-statistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group G, which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time-reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite-dimensional Hermitian representations of G. In spacetime dimension 2 we give a classification in terms of strongly G-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories. [ABSTRACT FROM AUTHOR]

A Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. Using character theory, it is proved that the Frobenius kernel is a normal subgroup of its Frobenius group. In this paper, we present some group-theoretical proofs that the Frobenius kernel is a subgroup of its Frobenius group under certain conditions. [ABSTRACT FROM AUTHOR]

The tensor functor called α-induction produces a new unitary fusion category from a Frobenius algebra object, or a Q-system, in a braided unitary fusion category. In the operator algebraic language, it gives extensions of endomorphism of N to M arising from a subfactor NᑕM of finite index and finite depth, which gives a braided fusion category of endomorphisms of N. It is also understood in terms of Ocneanu's graphical calculus. We study this α-induction for bi-unitary connections, which provides a characterization of finite-dimensional nondegenerate commuting squares, and present certain 4-tensors appearing in recent studies of 2-dimensional topological order. We show that the resulting α-induced bi-unitary connections are flat if we start with a commutative Frobenius algebra, or a localQ-system. Examples related to chiral conformal field theory and the Dynkin diagrams are presented. [ABSTRACT FROM AUTHOR]

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang–Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of O-operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras. [ABSTRACT FROM AUTHOR]

We introduce the concepts of complex finite-dimensional contact and Frobenius Lie superalgebras, including some relevant examples of this type of Lie superalgebras. We establish some basic properties satisfied by both contact and Frobenius Lie superalgebras, providing characterizations for both of them in terms of its associated structure supermatrix. Finally, if g = g 0 ⊕ g 1 is a homogeneous even Frobenius Lie superalgebra satisfying dim g 0 ≠ dim g 1 , we show that such g contains a contact superideal of codimension one. [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]

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial. [ABSTRACT FROM AUTHOR]

S. Carpi et al. (Comm. Math. Phys., 402 (2023), 169-212) proved that every connected (i.e., haploid) Frobenius algebra in a tensor C*-category is unitarizable (i.e., isomorphic to a special C*-Frobenius algebra). Building on this result, we extend it to the non-connected case by showing that an algebra in a multitensor C*-category is unitarizable if and only if it is separable. [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]

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R [ t ] / (t 2) . Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q , R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F q [ t ] / (t r) . We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation. [ABSTRACT FROM AUTHOR]

We compute the structure constants of topological symmetric orbifold theories up to third order in the large N expansion. The leading order structure constants are dominated by topological metric contractions. The first order interactions are single cycles joining while at second order we can have double joining as well as splitting. At third order, single cycle joining obtains genus one contributions. We also compute illustrative small N structure constants. Our analysis applies to all second quantized Frobenius algebras, a large class of algebras that includes the cohomology ring of the Hilbert scheme of points on K3 among many others. We point out interesting open questions that our results raise. [ABSTRACT FROM AUTHOR]

As a double generalization of biFrobenius algebras and weak Hopf algebra, we introduce and study weak biFrobenius algebras (or briefly wbF algebras). A wbF algebra is a Frobenius algebra and a Frobenius coalgebra and satisfies some compatibility conditions. We discuss more properties of wbF algebras keeping as close as possible to the classical theory of weak Hopf algebras, and give conditions for finite dimensional algebras and coalgebras to be wbF algebras. Then we studies substructures in wbF algebras. Finally, we prove that our wbF algebras can give rise to Frobenius double algebras in sense of Szlachányi. [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]

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Bäcklund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Bäcklund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Bäcklund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis–Procesi (DP) equation, and construct a Lax pair of it. In addition, the Bäcklund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a Z n -DP equation. Considering the superposition principle, we obtain the solution of an associated Z n -DP equation by using Bäcklund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena. [ABSTRACT FROM AUTHOR]

A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z 2 -graded algebra E (θ) , which corresponds to a noncommutative quadric hypersurface E ! / (z) for some central regular element z ∈ E 2 !. It turns out that the bounded derived category D b (gr Z 2 E (θ)) is equivalent to the stable category of the maximal Cohen-Macaulay modules over E ! / (z) provided that E ! is noetherian. As a consequence, E ! / (z) is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E (θ) is a semisimple Z 2 -graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces. As an application, we recover Knörrer's periodicity theorem without using matrix factorizations. [ABSTRACT FROM AUTHOR]

In this paper, we classify eight-dimensional non-solvable Lie algebras that support a symplectic structure. As well as a complete classification is given, up to symplectomorphism, of eight-dimensional symplectic non-solvable Lie algebras. [ABSTRACT FROM AUTHOR]

Firstly, we prove that a quasitriangular associative D-bialgebra together with a coquasitriangular structure can induce a Nijenhuis algebra. This shows that Nijenhuis operators and associative D-bialgebras are closely related. Secondly, we investigate some properties of a representation of a Nijenhuis algebra and introduce the notion of a corepresentation of a Nijenhuis coalgebra, which also can be seen as an infinitesimal deformation of compatible bicomodule. Thirdly, a Nijenhuis associative D-bialgebra can be established by a Nijenhuis algebra and a Nijenhuis coalgebra satisfying some compatible conditions. Furthermore, we find that a Nijenhuis associative D-bialgebra is equivalent to a matched pair of Nijenhuis algebras or a double construction of a Nijenhuis Frobenius algebra. Lastly, we provide some methods to construct Nijenhuis associative D-bialgebras through admissible associative Yang-Baxter equations (aYBes for short) and O -operators on Nijenhuis algebras. [ABSTRACT FROM AUTHOR]

Let U:\mathcal {C}\rightarrow \mathcal {D} be a strong monoidal functor between abelian monoidal categories admitting a right adjoint R, such that R is exact, faithful and the adjunction U\dashv R is coHopf. Building on the work of Balan [Appl. Categ. Structures 25 (2017), pp. 747–774], we show that R is separable (resp., special) Frobenius monoidal if and only if R(\mathbb {1}_{\mathcal {D}}) is a separable (resp., special) Frobenius algebra in \mathcal {C}. If further, \mathcal {C},\mathcal {D} are pivotal (resp., ribbon) categories and U is a pivotal (resp., braided pivotal) functor, then R is a pivotal (resp., ribbon) functor if and only if R(\mathbb {1}_{\mathcal {D}}) is a symmetric Frobenius algebra in \mathcal {C}. As an application, we construct Frobenius monoidal functors going into the Drinfeld center \mathcal {Z}(\mathcal {C}), thereby producing Frobenius algebras in it. [ABSTRACT FROM AUTHOR]

Abstract–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. [ABSTRACT FROM AUTHOR]

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of Kirillov and Ostrik (Adv Math 171(2):183–227, 2002) in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by Davydov (J Algebra 323(5):1321–1348, 2010) to arbitrary characteristic. [ABSTRACT FROM AUTHOR]

We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F-regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic. [ABSTRACT FROM AUTHOR]

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category or a C*-2-category. We prove that, for C*-2-categories and , the C*-2-category F u n (,) of ∗- 2 -functors, ∗- 2 -transformations and ∗- 2 -modifications is Q-system complete, whenever is Q-system complete. We use this result to provide a characterization of Q-system complete categories in terms of ∗- 2 -functors and to prove that the 2 -category of actions of a unitary fusion category on C*-algebras is Q-system complete. [ABSTRACT FROM AUTHOR]

Gendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Suppose that a Lie algebra L admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H of order 2 such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is metabelian. Then L is solvable of derived length bounded by a constant. [ABSTRACT FROM AUTHOR]

This paper improves several previously known results. First, the results describing the R-skewsymmetric algebra and the quadratic dual of the R-symmetric algebra as Frobenius algebras are shown to be true with any restriction on the parameter q of the Hecke relation being removed. An even Hecke symmetry gives rise to a pair of graded Frobenius algebras. We describe interrelation between the Nakayama automorphisms of the two algebras. As an illustration of general technique we give full details of the verification that Artin–Schelter regular algebras of global dimension 3 and elliptic type A are not associated with any quantum GL(3). [ABSTRACT FROM AUTHOR]

Neural Architecture Search (NAS), which aims to automatically optimize the structure of a neural network for achieving excellent classification performance, has attracted considerable attention in recent years. Recently, zero-shot evaluation methods have been proposed for estimating classification performance without training to reduce the search time. However, these indices are still insufficient for finding the best-performing neural networks. In this study, we demonstrate that it is possible to evaluate convolutional neural networks (CNNs) using the robustness of the rectified linear unit (ReLU) output distribution to weights. We propose a new zero-shot CNN evaluation index based on this robustness index. [ABSTRACT FROM AUTHOR]

The aim of this work is to give necessary and sufficient conditions to extend an LSA structure on a Lie algebra h to a Lie algebra g that either contains h as a subalgebra, or is a central extension of it. We also study conditions under which a Lie algebra together with an LSA structure admit a Frobenius or a contact form. Furthermore, we describe the properties of the Reeb vector associated with the codimension one contact ideal with respect to the LSA structure on the Frobenius Lie algebra. [ABSTRACT FROM AUTHOR]

We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third-order term P 3 is Darboux–Poisson and nondegenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular,we show that each Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges and vertices labelled by numerical marks. They are also naturally related to certain pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the corresponding Nijenhuis pencils. [ABSTRACT FROM AUTHOR]

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T ∗ C n . The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them. [ABSTRACT FROM AUTHOR]

Assume that a Lie type algebra admits a Frobenius group of automorphisms with cyclic kernel of order and complement of order such that the fixed-point subalgebra with respect to is trivial and the fixed-point subalgebra with respect to is nilpotent of class . If the ground field contains a primitive th root of unity, then the algebra is nilpotent and the nilpotency class is bounded in terms of and . The result extends the well-known theorem of Khukhro, Makarenko, and Shumyatsky on Lie algebras with metacyclic Frobenius group of automorphisms. [ABSTRACT FROM AUTHOR]

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings. [ABSTRACT FROM AUTHOR]

While the discussion by Pietarinen borders on the more traditional conceptions of mathematics, part of the aim of the I Handbook i is to situate mathematics within our broader cognitive faculties and lives. Marcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including index For one acquainted with C.S. Peirce, it is hard to see Springer's recent I Handbook of Cognitive Mathematics i (editor: Marcel Danesi) through none other than a Peircean lens. Discussions of Peirce's philosophy of mathematics sometimes emphasize (as Peirce does at times) the differences between mathematics and logic. [Extracted from the article]

We construct an equivariant version of annular Khovanov homology via the Frobenius algebra associated with U (1) × U (1) -equivariant cohomology of ℂ ℙ 1 . Motivated by the relationship between the Temperley–Lieb algebra and annular Khovanov homology, we also introduce an equivariant analog of the Temperley–Lieb algebra. [ABSTRACT FROM AUTHOR]

We examine Atiyah's Hermitian axiom for two-dimensional complex topological quantum field theories. Building on the correspondence between 2D topological quantum field theories (TQFTs) and Frobenius algebras, we find the algebraic objects corresponding to Hermitian and unitary TQFTs, respectively, and prove structure theorems about them. We then clarify a few older results on unitary TQFTs using our structure theorems. [ABSTRACT FROM AUTHOR]

In this paper, assume that H is a Hopf algebra and A/B is an H-Galois extension. Firstly, by introducing the concept of an H-stable tilting complex ... over B, we show that ... is a tilting complex over A and a derived equivalence between two H-module algebras can be extended to smash product algebras under some conditions. Then we observe that ... is an H-Galois Frobenius extension if A=B is an H-Galois Frobenius extension. Finally, for any perfect recollement of derived categories of H-module algebras, we apply the above results to construct a perfect recollement of derived categories of their smash product algebras and generalize it to nrecollements. [ABSTRACT FROM AUTHOR]

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 (x , y , z) ↦ x y − 1 z , 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. [ABSTRACT FROM AUTHOR]

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category. In a recent joint work with P. Das, S. Ghosh and C. Jones, the author has categorified Bratteli diagrams and unitary connections by building a 2-category UC. We prove that every Q-system in UC splits. [ABSTRACT FROM AUTHOR]

Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U. [ABSTRACT FROM AUTHOR]

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV ∞ -algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles. [ABSTRACT FROM AUTHOR]