# Questions tagged [frobenius-algebras]

Frobenius algebras are finite-dimensional algebras together with a compatible inner product. Commutative Frobenius algebras have attracted recent interest because they're equivalent to 2D oriented TQFTs.

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### A non-example of a graded Frobenius algebra

Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...
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### An algebra with more than one Frobenius algebra structure

Can a (finite dimenaional) $\mathbb{K}$-algebra $A$ be equipped with more than one Frobenius structure $\lambda:A \to \mathbb{K}$? Of course we identify two structures $\lambda$ and $\lambda'$ if they ...
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### Associated graded algebras and symmetric Frobenius algebras

Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this question, if $G$ is Frobenius, then $A$ is also Frobenius. If $G$ is a symmetric Frobenius algebra, ...
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### Infinite-dimensional, non-unital Frobenius algebras

A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
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### State-sum for 4d TQFT from fusion 2-categories and invariants of Morita equiavalence classes beyond Drinfeld center

If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
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### How many minimal relations are needed to obtain a Frobenius algebra?

Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$. An ...
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### A name for "anti-symmetric" Frobenius algebras?

Let $V$ be an $N$-dim vector space and denote its exterior algebra by $\Lambda^{\ast}(V)$. The algebra $\Lambda$ has an obvious Frobenius algebra structure $B(-,-)$ given by wedgeing and then ...
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### Classifying of low-dimensional Frobenius algebras

Does there exist a classification of finite-dimensional Frobenius algebras in low dimensional cases. This question is motivated by the following analogous question for Hopf algebras.
Let $A$ be a finite dimensional Frobenius algebra over a field $K$, which means that $A \cong D(A)$ as right $A$-modules. Being symmetric means that $A \cong D(A)$ as $A$-bimodules. Here $D(-)=Hom_K(-,... • 26k 6 votes 1 answer 141 views ### Commutative Frobenius algebra with non-invertible window element, but not square zero For any commutative Frobenius algebra$A$there is an associated window element$\omega \in A$. If$\mu: A \otimes A \to A$denotes the multiplication,$1 \in A$the unit,$b: A \otimes A \to k$the ... • 27.1k 3 votes 2 answers 383 views ### What is an example of a Frobenius algebra that is not Koszul? What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul? 2 votes 1 answer 129 views ### Coproduct for a Frobenius algebra The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ... 7 votes 1 answer 252 views ### Properties of a filtered algebra that can be concluded from properties of its associated graded algebra Let$F$be a filtered algebra and let$G$be its associated graded algebra. Some examples of properties of$F$that can be concluded from properties of$G$: (A) The dimension of$F$is equal to the ... • 1,144 3 votes 2 answers 194 views ### Classification of crossed$G$-algebras Added later: As Viktor Ostrik points out in a comment, what I'm looking for is a classification of so-called crossed$G$-algebras corresponding to homotopy TQFTs with homotopy target space$K(G, 1)$... • 2,901 3 votes 0 answers 98 views ### How to interpret compositional diagrams for quantum sets algebraically$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ... • 59 3 votes 0 answers 113 views ### Frobenius law in a monoidal category The Frobenius law for Frobenius algebras in a monoidal category states that $$(\mu\otimes1)\circ(1\otimes\delta)=\delta\circ\mu=(1\otimes\mu)\circ(\delta\otimes1),$$ but this only makes sense in a ... • 8,999 3 votes 0 answers 96 views ### Frobenius algebras associated to posets and coalgebra structures Let$P$be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m). Let k be a field, then the classical incidence algebra$kP$has$k$-vector ... • 26k 5 votes 0 answers 81 views ### It there an algebra of the form$B_T$with global dimension 3? Let$A$be the (symmetric Frobenius) algebra$A=K[x]/(x^3) \otimes_K K[x]/(x^3)$over a field$K$, which is isomorphic to the group algebra of$C_3 \times C_3$, with$C_3$cyclic of order 3, when$K$... • 26k 4 votes 2 answers 337 views ### Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist? I went through this list of all complex associative ... • 2,901 3 votes 1 answer 182 views ### Is the unit in the definition of a symmetric Frobenius algebra necessary? Consider a symmetric Frobenius algebra without unit, that is, a finite-dimensional complex associative algebra$\delta$with a linear functional$\epsilon$, such that$\epsilon\circ \delta$is a non-... • 2,901 1 vote 0 answers 95 views ### Are there non-semisimple complex "non-unital special Frobenius algebras"? I'm interested in "non-unital special Frobenius algebras", consisting of two linear maps (morphisms in the symmetric monoidal category of finite-dimensional complex vector spaces) $$\mu: V\... • 2,901 3 votes 1 answer 287 views ### Frobenius algebras and traces of modules \newcommand{\Hom}{\mathscr{Hom}} Let C be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable; Let A be a commutative algebra in C, ... • 13.6k 6 votes 0 answers 172 views ### Descendent Gromov-Witten invariants and Frobenius manifolds I've heard it said that genus 0 descendent Gromov-Witten invariants of a smooth projective variety X can be encoded in the structure of a Frobenius manifold on the cohomology H^*(X,\mathbb{C)}. ... 4 votes 0 answers 230 views ### Classification of special symmetric Frobenius algebras over real vector spaces Is there a general classification of special symmetric Frobenius algebras over real vector spaces? I know that n\times n matrix algebras, the quaternions, the complex numbers, the trivial algebra, ... • 2,901 11 votes 1 answer 210 views ### Are algebras with invertible linear duals always Frobenius? Let A be a finite dimensional algebra over a ground field k. The linear dual A^* = Hom_k(A,k) is naturally an A-A bimodule. I am interested in those algebras such that A^* is an invertible ... • 27.1k 11 votes 0 answers 404 views ### What is the motivation for a Frobenius manifold? A Frobenius manifold is a type of manifolds with extra structure. The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the Saito'' examples arising from singularities (... • 5,506 5 votes 1 answer 210 views ### Frobenius algebras from symmetric polynomials Let K be a field of characteristic 0 (maybe it works for more general fields) and K[x_1,...,x_n] the polynomial ring in n variables. Let e_1,e_2,...,e_n denote the elementary symmetric ... • 26k 3 votes 0 answers 117 views ### Does every special C^*-Frobenius algebra have a unit? I have a rather basic question about C^*-Frobenius algebras (also called Q-systems). Any pointers or references will be most helpful! We are given a finite-dimensional complex Hilbert space \mathbb{... 1 vote 1 answer 174 views ### A computation in a commutative Frobenius algebra This is already posted here https://math.stackexchange.com/questions/3695584/a-computation-in-a-commutative-frobenius-algebra but I didn't get any answers. Given a commutative Frobenius algebra (in ... 2 votes 0 answers 83 views ### Algebras from a basis of a Frobenius algebra Let A be a commutative Frobenius algebra over a field K (we can assume that A is local). We can assume A=K[x_1,...,x_r]/I for an ideal I with J^n \subseteq I \subseteq J^2 where J=<x_i&... • 26k 5 votes 1 answer 194 views ### Convolution algebra associated to a finite dimensional algebra Given a finite dimensional k-algebra A (we can assume it is given by a connected quiver with relations). One can form its trivial extension T(A) (see for example https://math.stackexchange.com/... • 26k 2 votes 0 answers 69 views ### Completeness results for Categorical Quantum Mechanics restricted to one \dagger-Frobenius Algebra? I've seen the various completeness results for Categorical Quantum Mechanics (CQM) axiom systems involving two interacting Frobenius algebras with various restrictions of phase-nodes. For example, the ... • 21 8 votes 0 answers 203 views ### Frobenius monads and groupoids For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ... • 1,319 4 votes 1 answer 129 views ### The existence of v\in A\otimes_{\mathbb{K}}A such that (a\otimes_{\mathbb{K}}1)v=(1\otimes_{\mathbb{K}}a)v If A is a finite dimensional commutative, associative, unital algebra over a field \mathbb{K} then does there exist a non-zero vector v\in A\otimes_{\mathbb{K}}A such that (a\otimes_{\mathbb{K}}... • 81 7 votes 0 answers 250 views ### Are the string diagrams for the Frobenius Algebra an example of a Polynomial Functor? We know that Frobenius objects in a monoidal category obey a diagrammatic string calculus. We also know that trees are polynomial functors (Kock - Polynomial functors and trees). The string calculus ... • 1,319 3 votes 0 answers 144 views ### Frobenius structure for A_n singularities I need to compute monodromy matrices M(v), associated to a Frobenius structure for A_n singularity with flat coordinates v_1,\dots,v_n, that is, f(x)=x^{n+1}. (due to Saito, Dubrovin etc.) ... • 31 5 votes 1 answer 349 views ### 2TQFT and commutative Frobenius algebras There is an equivalence between the category of commutative finite dimensional Frobenius algebras and 2 dimensional topological quantum field theories, see for example the book by Joachim Kock, which ... • 26k 2 votes 0 answers 80 views ### Characterisation of Frobenius algebras via sequences Given a commutative Frobenius algebra, finite dimensional over a field k. We assume that the algebra is connected and in fact given by quiver and relations. Let S be the unique simple modules of ... • 26k 10 votes 2 answers 445 views ### Is there a 1-dimensional analogue of the correspondence between the Levin-Wen and Turaev-Viro models? Given a spherical fusion category \mathcal C, the Levin-Wen model constructs a lattice field theory: to each oriented surface with a triangulation, it assigns a state space \mathcal H and a ... • 6,766 5 votes 0 answers 245 views ### What is the relationship between Frobenius extensions and Separable extensions Let R\to S be an extension of possibly non-commutative rings. I am interested in the relationship between R\to S being Frobenius and it being separable. If it is a Frobenius extension, then there ... • 9,539 3 votes 0 answers 187 views ### On finding simpler symmetries to differential equations I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity. It is as follows:$$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{... • 419 16 votes 3 answers 2k views ### Why is a Topological Field Theory equivalent to a Frobenius algebra? How can a physicist understand a 2-dimensional topological field theory as a Frobenius algebra? Are there some explicit examples in order to understand this relation? The definition (e.g. on ... • 651 4 votes 0 answers 182 views ### Smash Product of Frobenius Algebras We have a smash product of Hopf algebras if one acts on other (namely making it module algebra, coalgebra and Hopf algebra) with a compatibility condition (Theorem 17). Now I ask the same question ... • 191 2 votes 1 answer 191 views ### When are Morita classes represented by certain structured algebra objects? Let$\mathcal{C}$be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to$\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ... • 1,189 4 votes 1 answer 153 views ### Reference on the classification of (low rank) Gorenstein rings over$\mathbb{C}$I am interested in the question of the classification of (low rank) Gorenstein rings over$\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ... • 41 6 votes 1 answer 194 views ### Symmetric algebras of given dimension Fix an algebraically closed field$F$. Are there only finitely many symmetric algebras with unit over$F$of a given finite dimension (up to isomorphism)? By symmetric I mean a Frobenius algebra where ... • 355 11 votes 1 answer 591 views ### Are all separable algebras Frobenius algebras? Let$\mathcal C$be a [added later: semi-simple] tensor category, and let$A=(A,m:A\otimes A\to A,i:1\to A)$be an algebra object in$\mathcal C$. The algebra is... Separable if there is an$A$-$A$-... • 42.5k 8 votes 0 answers 317 views ### Structure of Lagrangian algebras in the center of a fusion category (1) Let$\mathcal F$be a spherical fusion tensor category. Then Müger showed that$R=\bigoplus_{H\in\mathrm{Irr}(\mathcal F)} H\boxtimes H^\mathrm{op}$canonically has the structure of a Frobenius ... • 2,101 2 votes 0 answers 162 views ### When the Lie algebra of matrices with zero last rows is Frobenius? Let$\mathcal{A}_{n,k}$be the Lie algebra of$n \times n$matrices over$\mathbb{C}$for which the last$k$rows are equal to zero. Suppose that$k$does not divide$n$. How to prove that$\mathcal{A}...
I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...