The coisotropic subgroup structure of SL_q(2,R)

We study the coisotropic subgroup structure of standard SL_q(2,R) and the corresponding embeddable quantum homogeneous spaces. While the subgroups S^1 and R_+ survive undeformed in the quantization as coalgebras, we show that R is deformed to a family of quantum coisotropic subgroups whose coalgebra can not be extended to an Hopf algebra. We explicitly describe the quantum homogeneous spaces and their double cosets.Comment: LaTex2e, 10pg, no figure

A global quantum duality principle for subgroups and homogeneous spaces

For a complex or real algebraic group G, with g:=Lie(G), quantizations of global type are suitable Hopf algebras F_q[G] or U_q(g) over C[q,q^{-1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G^* and a dual Lie bialgebra g^*. In this context, we introduce suitable notions of quantum subgroup and of quantum homogeneous space, in three versions: weak, proper and strict (also called "flat" in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions. The global quantum duality principle (GQDP) - cf. [F. Gavarini, "The global quantum duality principle", J. Reine Angew. Math. 612 (2007), 17-33] - associates with any global quantization of G, or of g, a global quantization of g^*, or of G^*. In this paper we present a similar GQDP for quantum subgroups or quantum homogeneous spaces. Roughly speaking, this associates with every quantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneous space, resp. a quantum subgroup, of G^*. The construction is tailored after four parallel paths - according to the different ways one has to algebraically describe a subgroup or a homogeneous space - and is "functorial", in a natural sense. Remarkably enough, the output of the constructions are always quantizations of proper type. More precisely, the output is related to the input as follows: the former is the coisotropic dual of the coisotropic interior of the latter - a fact that extends the occurrence of Poisson duality in the GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict too - so the special role of strict quantizations is respected. We end the paper with some examples and application.Comment: 43 pages, La-TeX file. Final version, published in "Documenta Mathematica

Noncommutative Instantons on the 4-Sphere from Quantum Groups

We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle S^7 --> S^4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson--Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU_q(2); it determines a new deformation of the 4-sphere Sigma^4_q as the algebra of coinvariants in S_q^7. We show that the quantum vector bundle associated to the fundamental corepresentation of SU_q(2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of Sigma^4_q, we define two 0-summable Fredholm modules and we compute the Chern-Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non trivial.Comment: 16 pages, LaTeX; revised versio

Bijectivity of the canonical map for the noncommutative instanton bundle

It is shown that the quantum instanton bundle introduced in Commun. Math. Phys. 226, 419-432 (2002) has a bijective canonical map and is, therefore, a coalgebra Galois extension.Comment: Latex, 12 pages. Published versio