Le treillis des opérateurs de réduction : applications aux bases de Gröbner non commutatives et en algèbre homologique

Cyrille Chenavier 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : In this thesis, we study unital associative algebras using rewriting methods. The theory of noncommutative Gröbner bases enables us to solve decision problems or compute homological invariants using such methods. In order to study homological properties, Berger gives a lattice characterisation of quadratic Gröbner bases. This characterisation uses reduction operators. The latter are specific projectors of a vector space equipped with a well-founded basis. When this vector space is finite dimensional, Berger proved that the set of reduction operators admits a lattice structure. He deduced a lattice formulation of confluence and used it to characterise quadratic Gröbner bases. In this thesis, we extend the approach using reduction operators to non-necessarily quadratic algebras. For that, we show that the set of reduction operators still admits a lattice structure when the underlying vector space is infinite dimensional. We deduce a general formulation of confluence as well as a lattice interpretation of completion. The algebraic formulation of confluence provides a lattice characterisation of noncommutative Gröbner bases. Moreover, we show that a completion can be obtained using a construction in the lattice of reduction operators. We deduce a method to construct noncommutative Gröbner bases. We also construct a contracting homotopy for the Koszul complex. The algebraic formulation of confluence enables us to characterise it by with equations. These equations induce representations of a family of algebras called confluence algebras. The contracting homotopy is constructed using these representations.
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Cyrille Chenavier. Le treillis des opérateurs de réduction : applications aux bases de Gröbner non commutatives et en algèbre homologique. Mathématiques [math]. Université paris Diderot, 2016. Français. ⟨tel-01415910⟩

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