K-groups of the quantum homogeneous space $SU_{q}(n)/SU_{q}(n-2)$

Abstract

Quantum Steiffel manifolds were introduced by Vainerman and Podkolzin in [13]. They classified the irreducible representations of their underlying C * -algebras. Here we compute the K groups of the quantum homogeneous spaces SU q (n)/SU q (n − 2), n ≥ 3. Specializing to the case n = 3 we show that the fundamental unitary for quantum SU (3) is nontrivial and forms part of a generating set in the K 1 .

Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers

“…Such a composition series appeared already in work of Soibelman and Vaksman [30] on quantum odd-dimensional spheres. Similar results were then obtained in a number of particular cases [13,27], most recently for quantum Stiefel manifolds [6]. The main part of the proof can be thought of as an analogue of the fact that the product of symplectic leaves of dimensions n and m in G decomposes into leaves of dimensions ≤ n + m.…”

“…Following the foundational works of Woronowicz [33] and Soibelman and Vaksman [29], the algebras of functions on q-deformations of compact groups and their homogeneous spaces were extensively studied in the 90s. Later the interest moved more towards noncommutative geometry of these quantum spaces, see for example [6], [7], [21] and references therein, leaving the basic algebraic results scattered in the literature and proved mostly in particular cases with various degrees of generality limited often to SU (2) or SU (N ) and some of their homogeneous spaces, and more rarely to classical compact simple groups and the corresponding full flag manifolds. The goal of this paper is to establish the main properties of quantized algebras of functions in full generality, for arbitrary Poisson homogeneous spaces of compact semisimple Lie groups such that the stabilizer of one point is a Poisson-Lie subgroup.…”

“…A composition series similar to the one obtained in Section 3 was used already by Soibelman and Vaksman [30] to compute the K-theory of the odd-dimensional quantum spheres. Such series were later used for K-theoretic computations in [27] and [6]. The most powerful result of this sort was obtained by Nagy [18], who showed that the C * -algebra C(SU q (N )) is KK-equivalent to C(SU (N )), and remarked that similar arguments work for all classical simple compact groups.…”