Definition
A finite dimensional, unital, associative algebra A defined over a field k is said to be a Frobenius algebra if A is equipped with a nondegenerate bilinear form σ:A × A → k that satisfies the following equation: σ(a·b,c)=σ(a,b·c). This bilinear form is called the Frobenius form of the algebra.
Equivalently, one may equip A with a linear functional λ:A→k such that the kernel of λ contains no nonzero left ideal of A.
A Frobenius algebra is called symmetric if σ is symmetric, or equivalently λ satisfies λ(a·b) = λ(b·a).
There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.
Read more about this topic: Frobenius Algebra
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