1,053 research outputs found
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
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
Quantum Orthogonal Planes: ISO_{q,r}(N) and SO_{q,r}(N) -- Bicovariant Calculi and Differential Geometry on Quantum Minkowski Space
We construct differential calculi on multiparametric quantum orthogonal
planes in any dimension N. These calculi are bicovariant under the action of
the full inhomogeneous (multiparametric) quantum group ISO_{q,r}(N), and do
contain dilatations.
If we require bicovariance only under the quantum orthogonal group
SO_{q,r}(N), the calculus on the q-plane can be expressed in terms of its
coordinates x^a, differentials dx^a and partial derivatives \partial_a without
the need of dilatations, thus generalizing known results to the multiparametric
case. Using real forms that lead to the signature (n+1,m) with m = n-1, n, n+1
, we find ISO_{q,r}(n+1, m) and SO_{q,r}(n+1,m) bicovariant calculi on the
multiparametric quantum spaces. The particular case of the quantum Minkowski
space ISO_{q,r}(3,1)/SO_{q,r}(3,1) is treated in detail.
The conjugated partial derivatives \partial_a* can be expressed as linear
combinations of the \partial_a. This allows a deformation of the phase-space
where no additional operators (besides x^a and p_a) are needed.Comment: LaTeX, 36 pages. Considered more real forms, added some explicit
formulas, used simpler definition of hermitean momenta. To be published in
European Phys. Jou.
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