Quantum duality principle for quantum Grassmannians

Abstract

The quantum duality principle (QDP) for homogeneous spaces gives four recipes to obtain, from a quantum homogeneous space, a dual one, in the sense of Poisson duality. One of these recipes fails (for lack of the initial ingredient) when the homogeneous space we start from is not a quasi-affine variety. In this work we solve this problem for the quantum Grassmannian, a key example of quantum projective homogeneous space, providing a suitable analogue of the QDP recipe.

Quantum Principal Bundles on Projective Bases

“…The quantum homogeneous projective varietyÕ q (SL n /P) is generated in degree one, cf. Example 3.8, and one can see immediately thatÕ q (SL n /P) coincides withÕ q (P n−1 ), as defined in (17).…”

“…Similarly, in the quantum case, as in [10,17] we obtain the quantum homogeneous coordinate ringÕ q (G/P) as the O q (P)-semi-coinvariant elements of the quantum group O q (G), the quantization of the semisimple group G. Assuming Ore conditions for localizations, we then proceed to obtain fromÕ q (G/P) and O q (G) a suitable sheaf F of O q (P)-comodule algebras, which will be the quantum principal bundle over the quantum space obtained throughÕ q (G/P). More explicitly, the coinvariant subsheaf F co O q (P) will be the quantum structure sheaf associated with the (noncommutative) projective localizations ofÕ q (G/P).…”

“…A quantum homogenous projective varietỹ O q (G/P) can be similarly characterized via a quantum section d ∈ O q (G). We review this construction due to [10], see also [17], adapting, for the reader's convenience, the main definitions and results to the present setting that differs from the first reference setting (there the accent was on Poisson geometry and Quantum Duality principle).…”

“…, r . It is a quantum deformation of O(Gr) and a quantum homogeneous projective space for the quantum group O q (SL n ), (see [15,17] for more details). Again one can readily check that d = D 1...r is a quantum section and that…”

Quantum principal bundles on projective bases

“…We can in fact obtain the quantum homogeneous coordinate ring O q (G/P ) as the O q (P ) semi-coinvariant elements of the quantum group O q (G), the quantization of the semisimple group G, (see [9,16] for more details on this construction). The key is the existence of a quantization d ∈ O q (G) of t ∈ O(G), that we call a quantum section (see Def.…”