Hopf Algebras and Congruence Subgroups.

Intro -- Contents -- Abstract -- Introduction -- Chapter 1. The Modular Group -- 1.1. Generators and relations -- 1.2. Congruence subgroups -- 1.3. Orbits and congruence relations -- 1.4. Presentations of the reduced modular group -- Chapter 2. Quasitriangular Hopf Algebras -- 2.1. Quasitriangular Hopf algebras -- 2.2. The Drinfel'd double construction -- 2.3. Integrals of the Drinfel'd double -- 2.4. Twisting -- Chapter 3. Factorizable Hopf Algebras -- 3.1. Doubles of quasitriangular Hopf algebras -- 3.2. Factorizable Hopf algebras -- 3.3. The coproduct of the evaluation form -- 3.4. The double and the tensor product -- 3.5. Integrals of factorizable Hopf algebras -- Chapter 4. The Action of the Modular Group -- 4.1. The role of the integral -- 4.2. The inverse of \ -- 4.3. Ribbon elements -- 4.4. The linearity of the action -- 4.5. Integrals, ribbon elements, and the double -- 4.6. The modular group and the double -- Chapter 5. The Semisimple Case -- 5.1. The character ring -- 5.2. The Verlinde matrix -- 5.3. Matrix identities -- 5.4. A comparison -- 5.5. The exponent -- 5.6. Radford's example -- Chapter 6. The Case of the Drinfel'd Double -- 6.1. The role of the evaluation form -- 6.2. The new maps -- 6.3. The first relation -- 6.4. The second approach to the action of the modular group -- 6.5. Matrix representations of the new maps -- Chapter 7. Induced Modules -- 7.1. Induction -- 7.2. Induction and duality -- 7.3. The relation with the center construction -- 7.4. The relation of the coherence properties -- 7.5. Adjoint functors -- 7.6. More coherence properties -- Chapter 8. Equivariant Frobenius-Schur Indicators -- 8.1. Equivariant Frobenius-Schur indicators -- 8.2. Indicators and duality -- 8.3. The equivariance theorem -- 8.4. The orbit theorem -- Chapter 9. Two Congruence Subgroup Theorems -- 9.1. The action on the character ring.

9.2. Induction and multiplicities -- 9.3. The congruence subgroup theorem for the Drinfel'd double -- 9.4. The projective congruence subgroup theorem -- Chapter 10. The Action of the Galois Group -- 10.1. The Galois group and the character ring -- 10.2. The semilinear actions -- 10.3. The action on the center -- 10.4. Representations of the Drinfel'd double -- 10.5. The equivariance of the isomorphism -- Chapter 11. Galois Groups and Indicators -- 11.1. A digression on Frobenius algebras -- 11.2. The invariance of the induced trivial module -- 11.3. The action and the indicators -- 11.4. Diagonal matrices -- 11.5. The Galois group and the modular group -- Chapter 12. Galois Groups and Congruence Subgroups -- 12.1. The Hopf symbol -- 12.2. Properties of the Hopf symbol -- 12.3. The Hopf symbol and the Jacobi symbol -- 12.4. The linear congruence subgroup theorem -- Notes -- Bibliography -- Subject Index -- Symbol Index.

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