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  "body": "\n## Summary\n\nA Frobenius algebra is a finite-dimensional [[associative algebra|3CTW1P4HBP]] over a field with a [[nondegenerate bilinear form|01NHFZF2SK]] that is compatible with the multiplication.\nEquivalently, the algebra is isomorphic to its own dual as a left module over itself.\nGroup algebras of finite groups and matrix algebras are Frobenius algebras.\nFrobenius algebras are [[self-injective|KQKJENEMSN]], and over them projective and injective modules coincide.\nCommutative Frobenius algebras also describe two-dimensional topological field theories.\n\n## Explanations\n\n### L1 human\n\nA Frobenius algebra carries a way to pair any two of its elements to a number, and the pairing respects the multiplication.\nThe matrix algebra is the model: pair two matrices $a$ and $b$ by taking the trace of $ab$.\nNo nonzero matrix pairs to zero with every other matrix, which is the nondegeneracy.\nFor the group algebra of a finite group, pair $a$ and $b$ by taking the coefficient of the identity element in $ab$.\nThis pairing gives a perfect symmetry between the algebra and its dual space.\n",
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