# 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.

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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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? Are there reasonable requirements for a Frobenius to be Koszul?

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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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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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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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$\newcommand{\id}{\mathrm{id}}$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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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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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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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 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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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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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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$\newcommand{\Hom}{\mathscr{Hom}}$
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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(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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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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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 ...

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Let $f_1,f_2,\ldots,f_n\in \mathbb C[z_1,\ldots, z_n]$ be such that the quotient ring
$$A:=\mathbb C[z_1,\ldots, z_n]/(f_1,f_2,\ldots,f_n)$$
is finite dimensional (in other words, it's a zero-...

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Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...

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Symmetric Frobenius algebras arise everywhere, but the non-symmetric variety seem difficult to come by. Are there any natural examples/constructions that produce non-symmetric Frobenius algebras in ...

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$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...

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Khovanov's Frobenius algebra (used in the definition of Khovanov homology) is $\mathbb{Z}[X]/X^2$ with the comultiplication. $\Delta(X)=X\otimes X, \Delta(1)=1\otimes X+X\otimes 1$ and the trace $\...

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I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me ...

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Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is ...

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Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective?
The algebra $A$ cannot be finite-dimensional, ...

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Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q-system (that is the ...