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.

54 questions
Filter by
Sorted by
Tagged with
88 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 ...
• 26.3k
305 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?
77 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 ...
120 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 ...
• 739
161 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 $K(G, 1)$ ...
• 2,699
91 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
90 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 ...
• 7,271
89 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 ...
• 22.7k
78 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 $K$ ...
• 22.7k
232 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 ...
• 2,699
138 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-...
• 2,699
1 vote
63 views

• 359
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 ...
• 631
160 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
172 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 ...
• 1,149
145 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
176 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
546 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 $A$-$A$-...
236 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 ...
• 2,031
153 views

796 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 ...
• 2,031
563 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
467 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, ...
• 14.3k