# 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.

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### Turning a Frobenius algebra into a symmetric algebra via tensor products

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(-,...

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### 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 ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 $K(G, 1)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $K$ ...

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### 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 ...

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### 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-...

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### 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\...

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### 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 $C$, ...

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### 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 $H^*(X,\mathbb{C)}$. ...

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### 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, ...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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{...

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### 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 ...

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### 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&...

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### 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/...

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### 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 ...

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### 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 ...

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### 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}}...

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### 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 ...

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### 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.) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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}{...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $A$-$A$-...

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### 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 ...

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### 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}...

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### Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

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 ...