The vector bundle decomposition

The C*-algebra of a Vector Bundle

We prove that the Cuntz-Pimsner algebra OE of a vector bundle E of rank ≥ 2 over a compact metrizable space X is determined up to an isomorphism of C(X)algebras by the ideal (1 − [E])K(X) of the K-theory ring K(X). Moreover, if E and F are vector bundles of rank ≥ 2, then a unital embedding of C(X)-algebras OE ⊂ OF exists if and only if 1 − [E] is divisible by 1 − [F ] in the ring K(X). We intr...

Universal Vector Bundle over the Reals

Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X = XR×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR × Pic d(XR) if and only if χ(L) is odd for L ∈ Picd(XR). There is a universal quaternionic algebraic line bundle over X × Pic(X) if and only if the degree d is odd. Tak...

On Parallel Sections of a Vector Bundle

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem.

An Analysis of the Vector Decomposition Problem

The vector decomposition problem (VDP) has been proposed as a computational problem on which to base the security of public key cryptosystems. We give a generalisation and simplification of the results of Yoshida on the VDP. We then show that, for the supersingular elliptic curves which can be used in practice, the VDP is equivalent to the computational Diffie-Hellman problem (CDH) in a cyclic ...

Chain decomposition and the flag f-vector