I. Heckenberger, S. Kolb: Differential Calculus on some Quantum Spaces

In the framework of quantum groups and quantum spaces there appear many examples of q-deformed coordinate algebras which allow well behaved covariant first order differential calculi (FODC) in the sense of Woronowicz [Wor89]. On the other hand there exists no general construction of a deformation of classical Kähler differentials in this setting. Therefore the task of classification of all covariant FODC over quantum spaces naturally arises and has been settled for many examples.

There are several techniques to perform classification. For any Hopf algebra A there exists a one to one correspondence between differential calculi and certain right (or left) ideals of ker(ε) ⊂ A [Wor89]. In [Her00] U. Hermisson generalized this method to certain right (or left) coideal subalgebras B ⊂ A.

A reformulation of the method of right ideals involves the notion of quantum tangent space. We consider right coideal subalgebras B of a quantum group A = O_{q}(G) obtained as right K invariants for certain left coideal subalgebras K of U_{q}(𝔤). In this situation a quantum tangent space is essentially a finite dimensional right K invariant left subcomodule of the dual coalgebra B^{∘} containing ε. Thus to classify FODC over B one either has to gain a good understanding of B^{∘} or to show that the quantum tangent space lies in a simpler subcoalgebra.

In [HK01] this program is performed for quantum Grassmann manifolds O_{q}(Gr(r, N)) in the so called quantum subgroup case. All covariant FODC over O_{q}(Gr(r, N)) of dimension up to 2r(N−r) are classified. It turns out that up to special cases in low dimensions there exists exactly one covariant FODC which has the same dimension as its classical counterpart. This calculus has been constructed and investigated in [Kol01].

The above method is also used to classify all covariant first order differential calculi over Podleś’ quantum sphere.

References

[Wor89] S.L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1989), no. 1, 125–170.

[Her00] U. Hermisson, Derivations with quantum group action, to appear in

       Comm. Alg., math.QA/0005106.

[Kol01] S. Kolb, Differential calculus on quantum complex Grassmann manifolds I: Construction, math.QA/0112154.

[HK01] I. Heckenberger, S. Kolb, Differential calculus on quantum complex Grassmann manifolds II: Classification, math.QA/0112155.