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.
64
questions
20
votes
2
answers
1k
views
Cohomology rings and 2D TQFTs
There is a "folk theorem" (alternatively, a fun and easy exercise) which asserts that a 2D TQFT is the same as a commutative Frobenius algebra. Now, to every compact oriented manifold $X$ we can ...
- 20.6k
19
votes
1
answer
3k
views
Why did people originally like Frobenius algebras?
These days, lots of people are excited by Frobenius algebras because commutative Frobenius algebras are the same thing as 2D topological quantum field theories.
...but this seems like teaching an old ...
- 1,862
15
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 ...
- 641
13
votes
3
answers
677
views
Does every Frobenius algebra in a monoidal *-category give a Q-system?
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 ...
- 27.7k
12
votes
1
answer
497
views
How do I check that this is a Frobenius algebra?
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-...
- 42.2k
11
votes
1
answer
574
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 $A$-$A$-...
- 42.2k
11
votes
1
answer
195
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 ...
- 26.9k
11
votes
0
answers
385
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 (...
- 5,278
10
votes
5
answers
854
views
Example for non equivalent rational full CFTs with same modular invariant (partition function)
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 ...
- 2,101
10
votes
2
answers
437
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 ...
- 6,726
10
votes
1
answer
478
views
Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?
Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...
- 2,423
8
votes
1
answer
421
views
Separable and finitely generated projective but not Frobenius?
Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes_R A^{\mathrm{op}})$-algebra. ...
- 26.9k
8
votes
0
answers
202
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 ...
- 1,227
8
votes
0
answers
296
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
7
votes
1
answer
238
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 ...
- 1,104
7
votes
0
answers
246
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 ...
- 1,227
6
votes
4
answers
1k
views
Are there natural examples of non-symmetric Frobenius algebras?
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 ...
- 1,862
6
votes
1
answer
189
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
6
votes
1
answer
136
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 ...
- 26.9k
6
votes
0
answers
115
views
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-...
- 839
6
votes
0
answers
164
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 $H^*(X,\mathbb{C)}$. ...
- 81
5
votes
1
answer
206
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 ...
- 25.4k
5
votes
1
answer
486
views
Self-injective basic algebras
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, ...
- 14.9k
5
votes
1
answer
178
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/...
- 25.4k
5
votes
1
answer
340
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 ...
- 25.4k
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 $K$ ...
- 25.4k
5
votes
0
answers
241
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 ...
- 9,519
4
votes
2
answers
308
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 ...
- 2,839
4
votes
2
answers
630
views
Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19)
$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...
- 32.7k
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
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
4
votes
0
answers
203
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, ...
- 2,839
4
votes
0
answers
177
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
4
votes
0
answers
580
views
Partial order on partitions and symmetric group algebra
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 ...
- 401
3
votes
2
answers
554
views
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 ...
- 147
3
votes
2
answers
368
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?
3
votes
2
answers
184
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 $K(G, 1)$ ...
- 2,839
3
votes
1
answer
173
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-...
- 2,839
3
votes
1
answer
266
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 $C$, ...
- 12.6k
3
votes
0
answers
60
views
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(-,...
- 25.4k
3
votes
0
answers
95
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
110
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 ...
- 8,878
3
votes
0
answers
95
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 ...
- 25.4k
3
votes
0
answers
115
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{...
3
votes
0
answers
138
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
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}{...
- 389
2
votes
1
answer
397
views
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 ...
- 598
2
votes
1
answer
908
views
What is the comultiplication of a matrix frobenius algebra?
One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V → k as the co-unit (or equivalently, tr(a⋅b) as the frobenius form). This is ...
- 1,862
2
votes
1
answer
187
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
2
votes
1
answer
115
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 ...