Quantum Bundle Description of Quantum Projective Spaces

Abstract

We realise Heckenberger and Kolb's canonical calculus on quantum projective

Noncommutative Kähler structures on quantum homogeneous spaces

“…We denote these two calculi by Ω (1,0) and Ω (0,1) , and denote their direct sum by Ω 1 . For a proof of the following lemma see [30,Lemma 5.2]. (1,0) ) and V (0,1) := Φ(Ω (0,1) ) is given respectively by…”

“…Finally, a direct q-deformation of the Kähler form of CP 2 was constructed in [8].Inspired by such phenomena, this paper introduces a general framework for noncommutative Kähler geometry on quantum homogeneous spaces and applies it to quantum projective space. The manner in which this is done has three main sources of inspiration: The first is Majid's frame bundle approach to noncommutative geometry [?, 24, 25] which also underpins the author's earlier papers [30,31]. The second is Kustermanns, Murphy, and Tuset's approach to noncommutative Hodge theory [22], and the third is the presentation of classical Kähler geometry found in [39] and [15], which is both global and algebraic in style.…”

Geometry of Quantum Projective Spaces

“…As already mentioned, this first order differential calculus can be realized by commutators with a "Dirac operator" [41]. The calculus was re-obtained in [3] as the restriction of a distinguished quotient of the bicovariant calculus on A(SU q (n + 1)).…”

“…For Fredholm modules and classical characteristic classes, as well as equivariant K-theory and quantum characteristic classes, one can see [36,21,22]; differential calculi have been studied by several authors, e.g. [6,56,41,34,35,3,4]; for Dirac operators and spectral triples we refer to [25,41,18,19,21]; complex structures and positive cyclic cocycles have been studied in [37,38,39]; for monopoles and instantons in the 4-dimensional case, we refer to [22,23]. The quantum projective line has been also used as the "internal space" for a scheme of equivariant dimensional reduction leading to q-deformations of systems of non-abelian vortices in [44].…”

Holomorphic functions on the quantum polydisk and on the quantum ball

“…This line of research was further developed by M. Dieng and A. Schwarz [41] and by A. Polishchuk and A. Schwarz [112-114]. A closely related point of view was adopted by J. Rosenberg [145], M. Khalkhali, G. Landi, W. D. van Suijlekom, and A. Moatadelro [71-73], E. Beggs and S. P. Smith [13], R.Ó Buachalla [ [98][99][100]. Another approach was also initiated by Connes [33, Section VI.2], who interpreted complex structures on a compact 2-dimensional manifold M in terms of positive Hochschild cocycles on the algebra of smooth functions on M. Motivated by this, he suggested to use positivity in Hochschild cohomology as a starting point for developing noncommutative complex geometry.…”