Quantum Homogeneous Spaces as Quantum Quotient Spaces

We show that certain embeddable homogeneous spaces of a quantum group that do not correspond to a quantum subgroup still have the structure of quantum quotient spaces. We propose a construction of quantum fibre bundles on such spaces. The quantum plane and the general quantum two-spheres are discussed in detail. 0. Introduction A homogeneous space X of a Lie group G may be always identified with the quotient space G/G0, where G0 is a Lie subgroup of G. When the notion of a homogeneous space is generalised to the case of quantum groups or non-commutative Hopf algebras the situation becomes much more complicated. A general quantum homogeneous space of a quantum group H need not be a quotient space of H by its quantum subgroup. By a quantum subgroup of H we mean a Hopf algebra H0 such that there is a Hopf algebra epimorphism π : H → H0. The quotient space is then understood as a subalgebra of H of all points that are fixed under the coaction of H0 on H induced by π. A quantum homogeneous space B of H might be such a quotient space but it is not in general. There is, however, a certain class of quantum homogeneous spaces, of which the quantum two sphere of Podleś [P1] is the most prominent example, that not being quotient spaces by a quantum subgroup of H, may be embedded in H. One terms such homogeneous spaces embeddable [P2]. The general quantum two sphere S q (μ, ν) is such an embeddable Most of this paper was written during my stay at Universite Libre de Bruxelles supported by the European Union Human Capital and Mobility grant. This work is also supported by the grant KBN 2 P302 21706 p01 Typeset by AMS-TEX 2 TOMASZ BRZEZIŃSKI homogeneous space of the quantum group SUq(2), and it is a quantum quotient space in the above sense when ν = 0. In the latter case the corresponding subgroup of SUq(2) may be identified with the algebra of functions on U(1). In this paper we show that certain embeddable quantum homogeneous spaces, and the general quantum two sphere S q (μ, ν) among them, can still be understood as quotient spaces or fixed point subalgebras. Precisely we show that there is a coalgebra C and a coalgebra epimorphism π : H → C such that the fixed point subspace of H under the coaction of C on H induced from the coproduct in H by a pushout by π is a subalgebra of H isomorphic to B. The interpretation of embeddable quantum homogeneous spaces as quantum quotient spaces allows one to develop the quantum group gauge theory of such spaces following the lines of [BM]. The study of such a gauge theory becomes even more important once the appearance of the quantum homogeneous spaces in the A. Connes geometric description of the standard model was annonunced [C]. For this purpose, however, one needs to generalise the notion of a quantum principal bundle of [BM] so that a Hopf algebra playing the role of a quantum structure group there may be replaced by a coalgebra. We propose such a generalisation. Since the theory of quantum principal bundles is strictly related to the theory of Hopf-Galois extensions (cf. [S]), we thus propose a generalisation of such extensions. The paper is organised as follows. In Section 1 we describe the notation we use in the sequel. In Section 2 we show a fixed point subalgebra structure of embeddable quantum homogeneous spaces. Next we propose a suitable generalisation of the notion of a quantum principal bundle in Section 3. Sections 4 and 5 are devoted to careful study of two examples of quantum embeddable spaces, namely the quantum plane Cq [M] and the quantum sphere S 2 q (μ, ν) [P1]. 1. Preliminaries In the sequel all the vector spaces are over the field k of characteristic not 2. C denotes a coalgebra with the coproduct ∆ : C → C ⊗ C and the counit ǫ : C → k which satisfy the standard axioms, cf. [Sw]. For the coproduct we use the Sweedler sigma notation ∆c = c ⊗ c , (∆⊗ id) ◦∆c = c ⊗ c ⊗ c , etc., QUANTUM HOMOGENEOUS SPACES AS QUANTUM QUOTIENT SPACES 3 where c ∈ C, and the summation sign and the indices are suppressed. A vector space A is a left C-comodule if there exists a map ∆L : A → C ⊗ A, such that (∆⊗ id) ◦∆L = (id⊗∆L) ◦∆L, and (ǫ⊗ id) ◦∆L = id. For ∆L we use the explicit notation ∆La = a(1) ⊗ a(∞), where a ∈ A and all a(1) ∈ C and all a(∞) ∈ A. Similarly we say that a vector space A is a right C-comodule if there exists a map ∆R : A → A⊗C, such that (∆R⊗ id)◦∆R = (id⊗∆)◦∆R, and (id⊗ ǫ)◦∆R = id. For ∆R we use the explicit notation ∆Ra = a(0) ⊗ a(1), where a ∈ A and all a(1) ∈ C and all a(0) ∈ A. H denotes a Hopf algebra with product m : H ⊗ H → H, unit 1, coproduct ∆ : H → H ⊗H, counit ǫ : H → k and antipode S : H → H. We use Sweedler’s sigma notation as before. Similarly as for a coalgebra we can define right and left H-comodules. For a right H-comodule A we denote by A a vector subspace of A of all elements a ∈ A such that ∆Ra = a ⊗ 1. We say that a right (resp. left) H-comodule A is a right (resp. left) H-comodule algebra if A is an algebra and ∆R (resp. ∆L) is an algebra map. A vector subspace J of H such that ǫ(J) = 0 and ∆J ⊂ J ⊗H ⊕H ⊗ J is called a coideal in H. If J is a coideal in H then C = H/J is a coalgebra with a coproduct ∆ given by ∆ = (π⊗π)◦∆, where π : H → C is a canonical surjection. The counit ǫ in C is defined by the commutative diagram H ǫ −−−−→ k π 