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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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Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital coas …
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Infinite-dimensional, non-unital Frobenius algebras
It is studied under the name "nearly Frobenius algebras" in this paper. In Example 3.3, the algebra $\mathbb{C}[[x,x^{-1}]]$ of Laurent series can be endowed with (countably) many unital but not co-un …