N-complexes as functors, amplitude cohomology and fusion rules

We consider N-complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology only vanishes on injective functors providing a well defined functor on the stable category. For left truncated N-complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive N-complexes is proved to be isomorphic to an Ext functor of an indecomposable N-complex inside the abelian functor category. Finally we show that for the monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other words the fusion rules for N-complexes can be determined.Comment: Final versio

Examples of derivation-based differential calculi related to noncommutative gauge theories

Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry, Homology and Fundamental Interactions". To appear in a special issue of International Journal of Geometric Methods in Modern Physic

Z_3-graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo 3. We show how the notion of covariant derivation can be generalized with a 1-form A, and we give the expression in local coordinates of the curvature 3-form. Finally, the introduction of notions of a scalar product and integration of the Z_3-graded exterior forms enables us to define variational principle and to derive the differential equations satisfied by the curvature 3-form. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F_{ik} and its covariant derivatives D_i F_{km}.Comment: 13 pages, no figure

From Koszul duality to Poincar\'e duality

We discuss the notion of Poincar\'e duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincar\'e duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page

COMMENTS ABOUT HIGGS FIELDS, NONCOMMUTATIVE GEOMETRY AND THE STANDARD MODEL

We make a short review of the formalism that describes Higgs and Yang Mills fields as two particular cases of an appropriate generalization of the notion of connection. We also comment about the several variants of this formalism, their interest, the relations with noncommutative geometry, the existence (or lack of existence) of phenomenological predictions, the relation with Lie super-algebras etc.Comment: pp 20, LaTeX file, no figures, also available via anonymous ftp at ftp://cpt.univ-mrs.fr/ or via gopher gopher://cpt.univ-mrs.fr

AGN Feedback Compared: Jets versus Radiation

Feedback by Active Galactic Nuclei is often divided into quasar and radio mode, powered by radiation or radio jets, respectively. Both are fundamental in galaxy evolution, especially in late-type galaxies, as shown by cosmological simulations and observations of jet-ISM interactions in these systems. We compare AGN feedback by radiation and by collimated jets through a suite of simulations, in which a central AGN interacts with a clumpy, fractal galactic disc. We test AGN of $10^{43}$ and $10^{46}$ erg/s, considering jets perpendicular or parallel to the disc. Mechanical jets drive the more powerful outflows, exhibiting stronger mass and momentum coupling with the dense gas, while radiation heats and rarifies the gas more. Radiation and perpendicular jets evolve to be quite similar in outflow properties and effect on the cold ISM, while inclined jets interact more efficiently with all the disc gas, removing the densest $20\%$ in $20$ Myr, and thereby reducing the amount of cold gas available for star formation. All simulations show small-scale inflows of $0.01-0.1$ M$_\odot$/yr, which can easily reach down to the Bondi radius of the central supermassive black hole (especially for radiation and perpendicular jets), implying that AGN modulate their own duty cycle in a feedback/feeding cycle.Comment: 21 pages, 15 figures, 2 table