---
tag: 8EVP1K2CZH
kind: concept
name: Frobenius algebra
msc: [16L60, 16D50]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Frobenius_algebra
same_as:
  wikidata: Q5505100
  wikipedia: Frobenius algebra
---

## Summary

A Frobenius algebra is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over a field with a [[nondegenerate bilinear form|01NHFZF2SK]] that is compatible with the multiplication.
Equivalently, the algebra is isomorphic to its own dual as a left module over itself.
Group algebras of finite groups and matrix algebras are Frobenius algebras.
Frobenius algebras are [[self-injective|KQKJENEMSN]], and over them projective and injective modules coincide.
Commutative Frobenius algebras also describe two-dimensional topological field theories.

## Explanations

### L1 human

A Frobenius algebra carries a way to pair any two of its elements to a number, and the pairing respects the multiplication.
The matrix algebra is the model: pair two matrices $a$ and $b$ by taking the trace of $ab$.
No nonzero matrix pairs to zero with every other matrix, which is the nondegeneracy.
For the group algebra of a finite group, pair $a$ and $b$ by taking the coefficient of the identity element in $ab$.
This pairing gives a perfect symmetry between the algebra and its dual space.