Quantizations of Flag Manifolds and Conformal Space Time

In this letter we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2, 4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and euclidean space. 0. Introduction and summary In the quantum groups setting there has been some attention given to the quantization of the Minkowski space. In their paper [1] Zumino et al. follow an ad hoc approach to this problem: the Minkowski space gets quantized as a real 4-dimensional space with the coaction of the Poincarè group on it. In the 1980’s following the new ideas introduced by Penrose a few years earlier in [2], Manin introduced the point of view that the Minkowski space is the manifold of the real points of a big cell in the grassmannian of complex two dimensional subspaces of a complex four dimensional space (twistor space) [3]. This is the starting point for the quantizations that are described in this letter. 1 Reshetikhin and Lakshmibai have provided in their paper [4] a quantization of the flag manifold of any complex simple group; however their construction is very abstract and does not seem to provide the deformations of the Plücker relations. Taft and Towber [5] on the other hand have a paper on the same subject, though with a very different approach, in which they do have the quantized Plücker relations, but their commutation rules resolve in a tautology for the grassmannian coordinates. It is natural to define the quantized coordinate ring of grassmannians and flag manifolds of subspaces of C as the graded subring of kq[SLn] (the Manin deformation of SLn(C), where kq = C[q, q ]) generated by suitable quantum determinants in the matrix entries aij. The essential point is to determine the commutation relations between these generators and show that these relations provide a presentation of the deformed ring. We do this for the flag spaces kq[F (1, 2;n)], kq[F (1, n− 1;n)] and kq[F (2, 3; 4)] as well as the quantized grassmannian kq[G(2, n)]. After the explicit construction of the quantum ring kq[G(2, 4)] of G(2, 4), we introduce the notion of quantum big cell with coordinate ring kq[N ], which is a projective localization of kq[G(2, 4)], in complete analogy with what happens in the classical case. The quantized lower maximal parabolic kq[Pl] corresponding to the classical maximal parabolic stabilizing < e1, e2 > is then proved to have a coaction on the quantum big cell. We define the quantum complex Minkowski space to have the coordinate ring kq[N ] and the quantum complex conformal group including translations to have the coordinate ring kq[Pl]. In analogy with the commutative case we define a certain quotient of kq[Pl], as the quantum complex conformal group kq[C] without the translations. As in the classical case we have 2 that kq[N ] is isomorphic to the 2 by 2 quantum matrix ring. This fact can be used to obtain right away the holomorphic De Rham complex on kq[N ]. We now come to the theory over R. The point of view is that the real spaces and groups, both classical and quantum, are described by the complex objects together with an appropriate involution. In the spirit of Soibelman work [6], [7] it is possible to define 2 involutions ∗M,q, ∗E,q on kq[N ] that allow us to speak of real Minkowski space and real euclidean space. It is also possible to define two involutions ∗C,M , ∗C,E on kq[C] so that we can speak of quantum real conformal group. The quantum real Minkowski space and the quantum real euclidean space will still support a coaction of the real kq[C] with respect to the appropriate involution. With a simple change of coordinates, that generalizes the classical one, the real Minkowski space and the real euclidean space transform into each other. It is easy to construct a real De Rham complex on the real Minkowski space starting from the complex one and using the given involution. Our point of view is group theoretic and not at all ad hoc. One of its advantages is that it provides the quantization of the conformal group naturally. It is capable of generalizing to higher dimensions (some of which are treated here) and provides a basis for the quantization of the Penrose theory. In the special case treated here it leads to a quantization of the complex Minkowski space in tandem with a quantization of the conformal group, then of the real Minkowski and euclidean spaces in tandem with the real conformal group with a compatible coaction that quantizes the classical coaction.