Noncommutative complex structures on quantum homogeneous spaces

Abstract

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. Throughout, the framework is applied to the quantum projective spaces endowed with the Heckenberger-Kolb calculus. * Supported by the Grant GACR P201/12/G028 Let A be a unital algebra (in what follows all algebras are assumed to be unital). A first-order differential calculus over A is a pair (Ω 1 , d), where Ω 1 is an A-A-bimodule and d : A → Ω 1 is a linear map for which the Leibniz rule holds d(ab) = a(db) + (da)b, a, b, ∈ A, and for which Ω 1 = span C {adb | a, b ∈ A}. (Where no confusion arises, we will drop explicit reference to d and denote a calculus by its bimodule Ω 1 alone.) We call an

Noncommutative Kähler structures on quantum homogeneous spaces

“…Moreover, the subfamily of irreducible quantum flag manifolds comes endowed with a differential calculus, the Heckenberger-Kolb calculus, which is uniquely characterised by a simple set of natural axioms [13,14]. These calculi have already served as the motivating examples for the theory of noncommutative complex structures [1,17,31].Metric phenomena have appeared a number of times in the literature on the noncommutative geometry of the quantum flag manifolds. A Hodge map for the Podleś sphere was defined by Majid in [25], and the induced Laplace and Dirac operators studied.…”

“…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.…”

Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections

“…Indeed, classically the flag manifolds U/K S are Kähler and hence admit these two-forms with special properties. The study of analogues of Kähler forms in the quantum setting is important from the point of view of non-commutative complex and Kähler geometry, see for instance [BeSm13,ÓBu16,ÓBu17].…”

Kähler structures on quantum irreducible flag manifolds

“…Within the realm of non-commutative geometry, the study of structures coming from complex geometry is a relatively new trend, see for instance the papers [FGR99,BeSm13,ÓBu16].…”