Topological invariants from quantum group Usl(2|1) at roots of unity

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Modified trace from pivotal Hopf G-coalgebra

Decomposition of tensor products of the regular representations

A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra

Hopf G-coalgebra from pivotal Hopf algebra U

Table of contents :

1 Introduction
1.1 Contexte
1.2 Pr´esentation des objectifs
1.3 R´esultats principaux
2 Topological invariants from quantum group Usl(2|1) at roots of unity
2.1 Introduction
2.2 Preliminaries
2.2.1 Monoidal category
2.2.2 Pivotal category
2.2.3 Ribbon category
2.2.4 Hopf superalgebras
2.3 Quantum superalgebra Usl(2|1)
2.3.1 Hopf superalgebra Usl(2|1)
2.3.2 Pivotal Hopf superalgebra Usl(2|1)
2.4 Category of nilpotent weight modules
2.4.1 Typical module
2.4.2 Character of representations of UH sl(2|1)
2.4.3 Braided category C H
2.4.4 Ribbon category C H
2.4.5 Semi-simplicity of category C H
2.5 Modified traces on projective modules
2.5.1 Ambidextrous modules
2.5.2 Modified traces on the projective modules
2.5.3 Invariants of embedded graphs
2.6 Invariant of 3-manifolds
2.6.1 Relative G-(pre)modular categories
2.6.2 Invariants of 3-manifolds
2.6.3 Example
2.7 Relative G-modular category C H
3 Modified trace from pivotal Hopf G-coalgebra
3.1 Introduction
3.2 Pivotal Hopf G-coalgebra
3.2.1 Pivotal Hopf G-coalgebra
3.2.2 Symmetrised right and left G-integrals
3.2.3 G-unibalanced Hopf algebras
3.3 Traces on finite G-graded categories
3.3.1 Cyclic traces
3.3.2 Modified trace in pivotal category
3.3.3 Pivotal structure on H-mod
3.3.4 Applications of Theorem 3.1.1
3.4 Proof of the main theorem
3.4.1 Decomposition of tensor products of the regular representations
3.4.2 Proof of Theorem 3.1.1
3.5 Modified trace for G-graded quantum sl(2)
3.5.1 Unrestricted quantum Uqsl(2)
3.5.2 Modified trace
4 A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra
4.1 Introduction
4.2 Topological ribbon Hopf superalgebra dUH
4.2.1 Hopf superalgebra dUH
4.2.2 Topological ribbon superalgebra dUH
4.2.3 Bosonization of dUH
4.3 Universal invariant of link diagrams
4.3.1 Category of tangles
4.3.2 Universal invariant of link diagrams
4.3.3 Value of universal invariant of link diagrams
4.4 Invariant of 3-manifolds of Hennings type
4.4.1 Hopf G-coalgebra from pivotal Hopf algebra U
4.4.2 Discrete Fourier transform
4.4.3 Invariant of 3-manifolds of Hennings type
A Computations in Usl(2|1)
A.1 Proof of Lemma 2.3.3
A.2 Proof of Proposition
A.3 Proof of Lemma 4.4.19