Transitive Hall sets

We give the definition of Lazard and Hall sets in the context of transitive factorizations of free monoids. The equivalence of the two properties is proved. This allows to build new effective bases of free partially commutative Lie algebras. The commutation graphs for which such sets exist are completely characterized and we explicit, in this context, the classical PBW rewriting process

Poset topology and homological invariants of algebras arising in algebraic combinatorics

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interpr\'etation du nombre de Leray d'un complexe de clique en termes de la dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159 and an article in preparation. 12 pages, 3 Figure

The Tutte-Grothendieck group of a convergent alphabetic rewriting system

The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski in terms of order relations, these operations may be interpreted as a particular instance of a general theory which involves universal invariants like the Tutte polynomial, and a universal group, called the Tutte-Grothendieck group. In this contribution, Brylawski's theory is extended in two ways: first of all, the order relation is replaced by a string rewriting system, and secondly, commutativity by partial commutations (that permits a kind of interpolation between non commutativity and full commutativity). This allows us to clarify the relations between the semigroup subject to rewriting and the Tutte-Grothendieck group: the later is actually the Grothendieck group completion of the former, up to the free adjunction of a unit (this was even not mention by Brylawski), and normal forms may be seen as universal invariants. Moreover we prove that such universal constructions are also possible in case of a non convergent rewriting system, outside the scope of Brylawski's work.Comment: 17 page

Solutions of Word Equations over Partially Commutative Structures

We give NSPACE(n log n) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n log n). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups

A categorical framework for the quantum harmonic oscillator

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure