Bijectivity of the canonical map for the non-commutative instanton bundle

Abstract

It is shown that the quantum instanton bundle introduced in [Commun. Math. Phys. 226 (2002) 419] has a bijective canonical map and is, therefore, a coalgebra Galois extension.

An Explicit Formula for a Strong Connection

“…(4), (3)), and the second equality follows by (1) and by the fact that ψ −1 is the inverse of ψ. Thus we obtain the equality…”

“…Recently, several examples of strong connections have been constructed (cf. [8], [11], [13], [12]) or their form conjectured [1], but no general procedure has been established. The aim of this note is to give a direct proof of a Schneider type theorem for coalgebra extensions in which the connection is explicitly given.…”

“…[3, Theorem 3.7, Corollary 3.8] for a detailed proof in the most general case). Explicitly, the splitting s : A → B⊗A of the product is given by s(a) = a (0) ℓ(a (1) ). If, in addition, there is a group-like element e ∈ C such that ̺ A (1) = 1⊗e, then an entwined extension with a strong connection form is a principal extension.…”

Galois comodules

“…Thus in Section 5 we determine which morphisms of corings induce principal comodules from principal comodules. The importance of this induction procedure of principal comodules, and, in particular, principal extensions, in non-commutative geometry has been confirmed in recent work [1] in which it has been shown that the non-commutative 4-sphere and the corresponding instanton bundle constructed in [2] arise from a principal extension of this type. Furthermore, the functor inducing D-comodules from C-comodules features prominently in the Kontsevich-Rosenberg approach to non-commutative algebraic geometry [21], where it is understood as a pull-back of quasi-coherent sheaves over non-commutative stacks, while principal comodules are examples of covers of non-commutative spaces.…”

“…A coproduct in an A-coring C is denoted by ∆ C : C → C ⊗ A C, and the counit is denoted by ε C : C → A. To indicate the action of ∆ C we use the Sweedler sigma notation, i.e., for all c ∈ C, (1) ⊗ c (2) ,…”