On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

Tenacious Researchers Identify a Weakness in All Ebolaviruses.

The Ebolavirus genus has at least five members, four of which are known to cause deadly disease in humans. An ideal therapy or a vaccine would protect against all ebolaviruses, but identifying a common weakness in all of them has remained elusive. West et al. [B. R. West, C. L. Moyer, L. B. King, M. L. Fusco, et al., mBio 9(5):e01674-18, 2018, https://doi.org/10.1128/mBio.01674-18] make the exciting discovery of an "Achilles' heel," a cryptic and conserved pocket, on the surface antigen glycoprotein (GP) that is nearly identical in all known ebolaviruses. Key to this discovery was their study of antibody ADI-15878, the only isolated human antibody that can block infectivity of all known ebolaviruses. Following tenacious efforts in X-ray crystallography, West et al. report the high-resolution crystal structures of the Ebola virus GP and the Bundibugyo virus GP, each bound to antibody ADI-15878. These structures reveal a highly conserved but partially obscured site on the virus GP, providing a foundation for design of vaccine antigens or antiviral therapies

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

Linear connections on matrix geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9

Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23

On the first order operators in bimodules

We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.Comment: 13 pages, AMSLaTe

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

Noncommutative generalization of SU(n)-principal fiber bundles: a review

This is an extended version of a communication made at the international conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes

A common generalization of the Fr\"olicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

There is a canonical mapping from the space of sections of the bundle $\wedge T^\ast M\otimes ST\ M$ to $\Omega(T^\ast M ; T(T^\ast M))$. It is shown that this is a homomorphism on $\Omega(M;TM) for the Fr\"olicher-Nijenhuis brackets, and also on$\Gamma(ST\ M)$for the Schouten bracket of symmetric multi vector fields. But the whole image is not a subalgebra for the Fr\"olicher-Nijenhuis bracket on$\Omega(T^\ast M;T(T^\ast M))$.Comment: 14 pages, AMSTEX, LPTHE-ORSAY 94/05 and ESI 70 (1994