Contents Introduction 1 Chapter 1. The Calculus of Sweedler Powers 5 1.1. Monotone maps 5 1.2. The union of the symmetric groups 6 1.3. Bialgebras 7 1.4. A monoid 8 1.5. Permutations from sequences 9 1.6. Sweedler powers 10 Chapter 2. Frobenius-Schur Indicators 13 2.1. Central Sweedler powers 13 2.2. The coproduct of the Sweedler powers 14 2.3. The first formula for the Frobenius-Schur indicators 15 2.4. The Frobenius-Schur theorem 17 2.5. Frobenius-Schur indicators of the regular representation 19 Chapter 3. The Exponent 21 3.1. The exponent 21 3.2. The second formula for the Frobenius-Schur indicators 23 3.3. Sweedler powers of the integral 24 3.4. Cauchy's theorem 25 Chapter 4. The Order 29 4.1. Order and multiplicity 29 4.2. The divisibility theorem 29 4.3. An example 31 4.4. The dimension of the simple modules 33 Chapter 5. The Index 35 5.1. Indecomposable matrices 35 5.2. The normal form 37 5.3. The Perron-Frobenius theorem 38 5.4. The index formula 39 Chapter 6. The Drinfel'd Double 43 6.1. The Drinfel'd double 43 6.2. Factorizability 43 6.3. The center of the character ring 45 6.4. The third formula for the Frobenius-Schur indicators 47

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