Quantization of projective homogeneous spaces and duality principle

Abstract

We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.

Graded quantum cluster algebras and an application to quantum Grassmannians

“…The definition of the quantum Grassmannian is made by analogy with the classical case, where an alternative algebraic realization of the homogeneous coordinate ring of the Grassmannian Gr(k, n) is given by taking the subalgebra of O(Mat(k, n)) generated by all k × k minors. Alternative approaches are possible (for example, [5]) but these are known to yield isomorphic algebras in this specific case.…”

The quantum chiral Minkowski and conformal superspaces

“…The relations (31,32) are the quantum super Plücker relations. If one specifies q = 1, the superalgebra becomes commutative and the quantum super Plücker relations become the standard ones (16).…”

“…where I Gr is the two-sided ideal generated by the commutations relations (28,29,30) and the quantum super Plücker relations (31,32) where D ij is substituted by the indeterminates X ij . Moreover Gr q /(q − 1) ∼ = O(Gr) (see Section 4.2).…”

The Segre embedding of the quantum conformal superspace

“…By the method of parabolic induction one can see that equivariant sections of O(G) with respect to the n th power of such character give the degree n subspace of the graded coordinate ring O(G/P ) associated to the projective embedding of G/P . Luckily, it is possible to translate this approach to the quantum realm by means of a quantum section [36,37,38]. We then achieve a characterization of the coordinate superring of F associated to the super Segre embedding.…”