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  "body": "\n## Summary\n\nThe Cartan matrix of a finite-dimensional algebra records how the simple modules occur as composition factors of the indecomposable [[projective modules|BFFA03GE3V]].\nIts rows are indexed by the indecomposable projectives and its columns by the [[simple modules|G00C22AEDH]].\nIn the modular representation theory of finite groups it is a basic invariant of the group algebra.\nIt is a different notion from the Cartan matrix of a semisimple Lie algebra, which shares the name.\n\n## Explanations\n\n### L1 human\n\nOver a finite-dimensional algebra, every module is built in layers from simple modules, like a number is built from primes.\nThe projective modules are the most important building blocks, and each one has a top simple module.\nThe Cartan matrix is the table that counts how often each simple module appears in the layers of each indecomposable projective module.\nFor the algebra $K[x]/(x^2)$ there is one simple module and the only indecomposable projective has two layers, so the Cartan matrix is $(2)$.\nFor a semisimple algebra the projectives are simple and the Cartan matrix is the identity matrix.\n",
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