Petersson algebra

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A Petersson algebra is a type of composition algebra built from a Hurwitz algebra using a symmetry of order 3. Let C be a Hurwitz algebra and φ an automorphism with φ^3 = id (so applying φ three times returns the original element). Define a new multiplication on C by x * y = φ(x) φ^2(y), where φ^2(y) means applying φ twice to y. The resulting algebra (C, *) is called the Petersson algebra associated to φ.