Generalized differential spaces with dN=0d^N=0 and the qq-differential calculus

We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some examples. In particular we construct the $q$-analog of the simplicial differential on forms, the $q$-analog of the Hochschild differential and the $q$-analog of the universal differential envelope of an associative unital algebra.Comment: 8 pages, Latex2e, uses pb-diagram, available at http://qcd.th.u-psud.fr, to be published in the Proceedings of the 5th Colloquium ``Quantum Groups and Integrable Systems", Prague, June 199

Gravity coupled with matter and foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element $ds$. Its unitary representations correspond to Riemannian metrics and Spin structure while $ds$ is the Dirac propagator ds = \ts \!\!---\!\! \ts = D^{-1} where $D$ is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involution $J$. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.Comment: 30 pages, Plain Te

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

Spin chains with dynamical lattice supersymmetry

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension admits a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.Comment: Revised version, 52 pages, 2 figure

Cyclic homology and the Macdonald conjectures

Let A+(k) denote the ring ℂ[ t ]/ t k+1 and let G be a reductive complex Lie algebra with exponents m 1 , ..., m n . This paper concerns the Lie algebra cohomology of G ⊗ A + ( k ) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we call weight , is inherited from the obvious grading of G ⊗ A + ( k )). We conjecture that this Lie algebra cohomology is an exterior algebra with k +1 generators of homological degree 2 m s +1 for s =1,2, ..., n . Of these k +1 generators of degree 2 m s +1, one has weight 0 and the others have weights ( k +1) m s +t for t =1,2, ..., k .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46617/1/222_2005_Article_BF01391498.pd