Quantum homogeneous spaces with faithfully flat module structures

E. F. Müller and H.-J. Schneider Mathematisches Institut der Universität München Theresienstraße 39 80333 München

0 Introduction

Let A be a Hopf algebra with bijective antipode over the ground field k, and B ⊂ A a right coideal subalgebra, that is a subalgebra with Δ(B) ⊂ B ⊗ A. We can think of the inclusion B ⊂ A as defining a quotient map G → X where G is a quantum group and X is a quantum space with right G-action or a right G-space. From an algebraic point of view [T1, Sch, M1] the inclusion B ⊂ A only has good properties if A is faithfully flat as a left (or right) module over B. If A is a commutative Hopf algebra, the meaning of faithful flatness of A over B is as follows. Now G → X is a morphism of affine schemes with function algebras B ⊂ A. Since A is commutative, A/AB^{+} is a quotient Hopf algebra of A defining a closed subgroup scheme G′ of G. The quotient object G/G′ exists in the big category of sheaves for the faithfully flat Grothendieck topology (called “faisceaux durs” in [DG]). This quotient G/G′ is isomorphic to the affine scheme X if and only if A is B-faithfully flat. Hence if A is not faithfully flat over B it does not make much sense to speak of B as the function algebra of a homogeneous space of G. In general, if A is not commutative, A/AB^{+} is just a coalgebra and a left A-module but not a Hopf algebra. Thus X usually is not the quotient of G by some quantum subgroup. But if A is B-faithfully flat one still gets B back from the quotient map A → A/AB^{+} as the A/AB^{+}-coinvariant elements of A. In the last few years many interesting examples of quantum G-spaces for concrete quantum groups G have been constructed. Podleś [P] found a continuously parametrized family of SU_{q}(2)-spaces which are analogs of the classical 2-sphere SU(2)/SO(2). Dijkhuizen and Noumi [DN] defined more generally a family of U_{q}(n)-spaces called quantum projective spaces. [D] gives a survey on other quantum G-spaces which are analogs of classical compact symmetric spaces such as SU(n)/SO(n) or SU(2n)/Sp(n). In all these cases, the right coideal subalgebra B is defined by infinitesimal invariants with respect to a coideal I in U, the U-part of the quantum group G (the classical counterpart of U is the universal enveloping algebra of the Lie algebra of the algebraic group). Moreover B is a ∗-subalgebra of the Hopf ∗-algebra A which is guaranteed by the natural condition S(I)^{∗} ⊂ I. In this paper we show as a consequence of the general theorem 2.2 that in the above examples, the extension B ⊂ A has the crucial property of faithful flatness. Let U be a Hopf algebra with bijective antipode, K ⊂ U a left coideal subalgebra and C a tensor category of finite dimensional left U-modules (for example modules of type 1 for U_{q}(𝔤), where 𝔤 is a semisimple complex Lie algebra and q ∈ k not a root of 1). Let A = U_{C}^{0} be the Hopf dual of U with respect to C and

B := {a ∈ A | a · K^{+} = 0},

the algebra of infinitesimal invariants, where “·” denotes the natural (U, U)-bimodule structure on A ⊂ U^{∗}, as recalled in the beginning of section 2. In this set-up we now assume that all modules in C are semisimple over K. Then we call K C-semisimple. We show in Theorem 2.2 that

• A is faithfully flat as a left and right B-module, more precisely A as a left and right B-module is a direct sum of finitely generated and projective B-modules with one direct summand being B.

• The quotient coalgebra A/AB^{+} is cosemisimple, and if K is commutative and k is algebraically closed, then A/AB^{+} is spanned by group-like elements.

In particular, in the case of Podleś’ quantum spheres, A/AB^{+} is spanned by grouplike elements. This answers a question of Brzeziński [B]. Recall that a coalgebra C is cosemisimple [Sw] if C is the direct sum of its simple subcoalgebras. C is spanned by group-like elements if and only if C is cosemisimple and pointed, that is all its simple subcoalgebras are one-dimensional. In section 3 we show that K in fact is C-semisimple in many important cases. Let U be a Hopf ∗-algebra, and assume that U is pointed (this holds for all the quantized universal enveloping algebras U_{q}(𝔤)). If I ⊂ U is a coideal with S(I)^{∗} ⊂ I and B := {a ∈ A | a · I = 0} as in the examples described above, then IU = K^{+}U where K is the left coideal subalgebra of the right U/IU-coinvariant elements in U. Hence we may also write B = {a ∈ A | a · K^{+} = 0} as in Theorem 2.2. If we assume that all modules V ∈ C have a hermitian inner product ⟨, ⟩ such that ⟨xv, w⟩ = ⟨v, x^{∗}w⟩ for all v, w ∈ V and x ∈ U then by Corollary 3.3, K = K^{∗} is C-semisimple, and our abstract Theorem 2.2 applies. In sections 4 and 5 we consider in U := U_{q}(sl(2)) and U_{q}(𝔤), for 𝔤 a semisimple complex Lie algebra and q ∈ k not a root of unity, an arbitrary skew-primitive element x with Δ(x) = g ⊗ x + x ⊗ 1, where g is a group-like element in U. Then the subalgebra k[x] generated by x is a left coideal subalgebra in U. We take for C the class of finite dimensional representations of type 1 and define

B := {a ∈ A | a · x = 0}

in A := U_{C}^{0}, the q-deformed function algebra of the connected, simply connected algebraic group with Lie algebra 𝔤. Somewhat surprisingly we show in Theorem 5.2 that for all x (up to one case) the following conditions are equivalent:

(a) k[x] is C-semisimple, that is x acts on all finite dimensional U-modules as a diagonalizable operator.

(b) A is left or right faithfully flat over B.

(c) A/AB^{+} is spanned by group-like elements.

Moreover, condition (a) is equivalent to an explicit numerical condition on the coefficients of x. Thus we see that condition (a) which was studied by Noumi and Mimachi [NM] in connection with Podleś’ quantum spheres is equivalent to the abstract algebraic condition of faithful flatness in (b). Most calculations in section 4 and special versions of some arguments, which have been widely generalized in the proof of 2.3(1), are already contained in the first author’s diploma thesis [Mü].

1 Preliminaries and some general results

In the following we fix a field k which is the ground field for all algebras and vector space. For definitions and basic results on Hopf algebras see [Sw, M]. We use the simplified notation Δ(x) = ∑ x_{1} ⊗ x_{2} for the coproduct in a coalgebra. Let V be a vector space. In the following (except for section 3), we write V^{∗} for the dual space Hom(V, k) of V. To study sub- and quotient objects of Hopf algebras it is crucial to consider faithfully flat modules and faithfully coflat comodules. Recall that a right module M over an algebra B is called flat respectively faithfully flat if and only if the functor M ⊗_{B} −:_{B}M → _{k}M from the category of k-vector spaces preserves respectively preserves and reflects exact sequences. Dually, a right comodule V over a coalgebra C is called coflat respectively faithfully coflat if and only if the cotensor product V □_{C}−:^{C}M → _{k}M preserves respectively preserves and reflects exact sequences. Here, ^{C}M is the category of left C-comodules, and if W is a left C-comodule, then the cotensor product is defined as the kernel of

Δ_{V} ⊗ id − id ⊗ Δ_{W}: V ⊗ W → V ⊗ C ⊗ W,

where Δ_{V} and Δ_{W} are the comodule structure maps of V and W. Of course, these notions are defined in the same way for modules or comodules on the other side. Let A be a Hopf algebra. If I ⊂ A is a vector subspace with quotient map π: A → A/I (usually I will be a coideal and a left or right ideal), we let

coA/IA := {a ∈ A | ∑ π(a_{1}) ⊗ a_{2} = π(1) ⊗ a}

be the set of left A/I-coinvariant elements. If I ⊂ A is a coideal, then the quotient map π: A → A/I is a coalgebra map, and A is a right (respectively left) A/I-comodule via π with comodule structure (π⊗id)Δ (respectively (id⊗π)Δ). Dually for a subalgebra B of A we consider A as a left and a right B-module by restricting the multiplication in A. A right (respectively left) coideal subalgebra B of A is a subalgebra B ⊂ A with Δ(B) ⊂ B ⊗ A (respectively A ⊗ B).

Theorem 1.1 [T1, Theorem 2] Let A be a Hopf algebra and I ⊂ A a coideal and a left ideal with quotient map π: A → A/I. Define B := ^{coA/I}A. Assume A is faithfully coflat as a right A/I-comodule via π. Then B ⊂ A is a right coideal subalgebra, I = AB^{+}, where B^{+} = Ker(ε|_{B}) is the augmentation ideal of B, and A is faithfully flat as a right B-module.

Moreover it is shown in [T1] that M ↦ A ⊗_{B} M is an equivalence between _{B}M and the category of left (A/I, A)-Hopf modules. Later-on we will need the categories M_{B}^{A} and _{B}M^{A} of (A, B)-Hopf modules for a right coideal subalgebra B of a Hopf algebra A. Objects in M_{B}^{A} (respectively _{B}M^{A}) are right (respectively left) B-modules V which are right A-comodules such that the comodule structure map Δ_{V}: V → V ⊗ A is right (respectively left) B-linear, where V ⊗ A is a right (respectively left) B-module via (v ⊗ a)b := ∑ vb_{1} ⊗ ab_{2} (respectively b(v ⊗ a) = ∑ b_{1}v ⊗ b_{2}a) for all v ∈ V, a ∈ A, and b ∈ B. Morphisms are right A-colinear and right (respectively left) B-linear maps. Note that B ∈ M_{B}^{A} and B ∈ _{B}M^{A} where the restriction of Δ: A → A ⊗ A is the comodule structure.

Theorem 1.2 [MW, 2.1] Let A be a Hopf algebra with bijective antipode and B ⊂ A a right coideal subalgebra. Let Ā := A/AB^{+} with quotient map π: A → Ā. Then Ā is a quotient coalgebra and a quotient left A-module of A, and the following are equivalent:

(1) A is faithfully coflat as a left Ā-comodule via π, and B = ^{coĀ}A.

(2) A is faithfully flat as a left B-module.

(3) A is projective as a left B-module, and B is a left B-direct summand in A.

(4) A is flat as a left B-module, and B is a simple object in M_{B}^{A}.

(5) The functor M_{B}^{A} → MĀ, M ↦ M/MB^{+} is an equivalence.

Proof: This is [MW, 2.1] when (3) is replaced by (3′) A is a projective generator as a left B-module. We have to show the equivalence of (3) and (3′). Assume (3′). Then A is projective and faithfully flat as a left B-module by [MW] . Hence B is a left B-direct summand in A by [R, 2.11.29] □

The simplicity condition in (4) means that any non-zero right B-submodule and right A-subcomodule of B is equal to B. The categorical characterization (5) shows the significance of the equivalent conditions in 1.2. Moreover, by [M1, 1.11] the mapping

{B|B ⊂ A is a right coideal subalge- bra, A is left faithfully flat over B} → {I|I ⊂ A is a coideal and left ideal and A is left faithfully coflat over A/I}

is a bijection between sub- and quotient objects of A satisfying the corresponding conditions in 1.2.

Remark 1.3 1. If we apply 1.2 to the dual algebras B^{op} ⊂ A^{op} (A^{op} is a Hopf algebra since the antipode of A is bijective), we get the dual theorem where Ā is now A/B^{+}A, M_{B}^{A} is replaced by _{B}M^{A}, and A is considered as a right B-module.

2. Let A be a Hopf algebra and B ⊂ A a right coideal subalgebra. We note the following simplicity criterion: If B is a left B-direct summand in A then B is simple in M_{B}^{A}.

Proof: Let f: A → B be a left B-linear map such that f|_{B} = id. Let X ⊂ B be a nonzero subobject in M_{B}^{A}. Then XA is a non-zero Hopf module in M_{A}^{A}. Since M_{A}^{A} ≅ _{k}M by the fundamental theorem of Hopf modules [Sw, 4.1.1], A is simple in M_{A}^{A} and XA = A. Hence there exist finitely many elements x_{i} ∈ X, a_{i} ∈ A such that ∑_{i} x_{i}a_{i} = 1. Then 1 = f(1) = ∑_{i} x_{i}f(a_{i}) ∈ X, and X = B. □

For completeness we give the short proof of the following important observation of Koppinen.

Lemma 1.4 [Ko] Let A be a Hopf algebra with bijective antipode S and B ⊂ A a right (resp. left) coideal with 1 ∈ B. Then S(AB^{+}) = B^{+}A (resp. S(B^{+}A) = AB^{+}).

Proof: We assume that B is a right coideal (and then apply the result to the dual coalgebra of A to get the lemma for left coideals). Let {x_{j} | j ∈ J} be a basis of B^{+}. Then for all x ∈ B, Δ(x) = 1 ⊗ x + ∑_{j} x_{j} ⊗ y_{j} for some y_{j} ∈ A. Now assume x ∈ B^{+} and apply μ ∘ (S ⊗ id) and μ ∘ (id ⊗ S), where μ denotes multiplication in A:

0 = 1ε(x) = x + ∑_{j} S(x_{j})y_{j}, and 0 = S(x) + ∑_{j} x_{j}S(y_{j}).

Therefore, B^{+} ⊂ S(B^{+})A and S(B^{+}) ⊂ B^{+}A. Hence B^{+}A = S(B^{+})A and S(AB^{+}) = S(B^{+})A = B^{+}A. □

Corollary 1.5 Let A be a Hopf algebra with bijective antipode and I ⊂ A a coideal and left ideal with quotient map π: A → A/I. Assume A/I is cosemisimple, and let

A/I = ⨁_{j∈J} C_{j}

be the direct sum of the simple subcoalgebras C_{j} for j ∈ J, of A/I. Then

(1) I = AB^{+}, and A is left and right faithfully flat over B and left and right faithfully coflat over A/I via π.

(2) For all j ∈ J, let

jA := {a ∈ A | ∑ π(a_{1}) ⊗ a_{2} ∈ C_{j} ⊗ A}, A_{j} := {a ∈ A | ∑ a_{1} ⊗ π(a_{2}) ∈ A ⊗ C_{j}}.

Then A = ⨁_{j∈J} _{j}A = ⨁_{j∈J} S(A_{j}), and for all j, _{j}A respectively S(A_{j}) is a right coideal in A and a finitely generated and projective right respectively left module over B. If 1 ∈ J is the distinguished index with C_{1} = kπ(1), then B = _{1}A = S(A_{1}).

Proof:

(1) By [Sch, 1.3], A is right faithfully coflat over A/I if and only if

(a) A is right coflat over A/I and

(b) π splits as a map of right A/I-comodules.

Since A/I is cosemisimple, any exact sequence of right A/I-comodules splits. In particular, (a) and (b) hold. Thus we see that A is right and by the same argument left faithfully coflat over A/I. Since A is right faithfully coflat over A/I, we get from 1.1 that AB^{+} = I and A is right faithfully flat over B. Then B ⊂ A is a right coideal subalgebra, B = ^{coA/AB^{+}}A and A is left faithfully coflat over A/AB^{+}. Hence the equivalent conditions in 1.2 hold, and A is left faithfully flat over B, too.

(2) By (1), B is a right coideal subalgebra of A, A is left and right faithfully flat over B and Ā = A/AB^{+} is cosemisimple. Hence by 1.2,

M_{B}^{A} → MĀ, M ↦ M/MB^{+}, and MĀ → M_{B}^{A}, V ↦ V □ĀA,

are quasi-inverse category equivalences. For any right Ā-comodule V, the Hopf module structure on V □ĀA is given by multiplication and comultiplication on A. Since Ā = ⨁_{j} C_{j} is a decomposition of right Ā-comodules, Ā□ĀA ≅ ⨁_{j} C_{j}□ĀA as right A-comodules and right B-modules. Moreover, for all j the natural isomorphism

A ⟶^{≅} Ā□ĀA, A ↦ ∑ π(a_{1}) ⊗ a_{2}, maps _{j}A onto C_{j}□ĀA. Hence A = ⨁_{j} _{j}A is a

decomposition in M_{B}^{A}. By construction, _{1}A = ^{coĀ}A = B. All the _{j}A are projective right B-modules since A is projective as a right B-module by 1.3(1). They are finitely generated over B since _{j}A ≅ C_{j}□ĀA and C_{j} is finite dimensional. More generally, let V be any finite dimensional right Ā-comodule. Then M = V □ĀA is finitely generated as a right B-module. To see this write M as the ascending union of all Hopf submodules XB where X is a finite dimensional right A-subcomodule of M. Let F: M_{D}^{A} ⟶^{≅} MĀ be the category equivalence of 1.2. Then F (M) ≅ V is the ascending union of all F (XB). Since V is finite dimensional, F (M) = F (XB) for some X, hence M = XB is B-finitely generated.

To get the decomposition of left B-modules we apply the previous result to A^{op}. Then B^{op} ⊂ A^{op} is a right coideal subalgebra and A^{op} is left and right faithfully flat over B^{op}. By Koppinen’s lemma 1.4, S(AB^{+}) = B^{+}A and

σ: A/AB^{+} ⟶^{≅} A/B^{+}A, ā ↦ \overline{S(a)},

is a coalgebra antiisomorphism. Therefore A/B^{+}A = ⨁_{j} σ(C_{j}) is a direct sum of simple subcoalgebras and A^{op}/A^{op}(B^{op})^{+} = A/B^{+}A is cosemisimple. Thus we know from the previous proof that A = ⨁_{j} _{j}Ã where for all j,

jÃ := {a ∈ A | ∑ π̃(a_{1}) ⊗ a_{2} ∈ σ(C_{j}) ⊗ A}

is a finitely generated projective left B-module and a right coideal, and _{1}Ã = B. Here π̃: A → A/B^{+}A is the canonical map. Finally, for all j, _{j}Ã = S(A_{j}) since for all a ∈ A,

a ∈ _{j}Ã ⟺ ∑ π̃(a_{1}) ⊗ a_{2} ∈ σ(C_{j}) ⊗ A
⟺ ∑ π(S^{−1}(a_{1})) ⊗ S^{−1}(a_{2}) ∈ C_{j} ⊗ A
⟺ S^{−1}(a) ∈ A_{j}.
□

2 A class of homogeneous spaces defined by infinitesimal invariants

We first collect some well-known results and notations on duality (cf. [M, Chapter 9], [J, I.1.4], [T2, section 1]). Let U be an algebra. The dual coalgebra U^{0} ⊂ U^{∗} is spanned by the matrix coefficients of all finite dimensional left U-modules V. If ρ: U → End(V) is the representation of U, C^{V} denotes the image of the dual coalgebra (End(V))^{∗} under ρ^{∗}. Thus C^{V} is the k-linear span of all matrix coefficients c_{f,v} ∈ U^{∗}, f ∈ V^{∗}, v ∈ V, where c_{f,v}(u) := f(uv) for all u ∈ U. If (v_{i}), (f_{i}) are dual bases of V, V^{∗}, the coalgebra structure of C^{V} is explicitly given by Δ(c_{f,v}) = ∑_{i} c_{f,v_{i}} ⊗ c_{f_{i},v} for all f ∈ V^{∗}, v ∈ V. Then U^{0} is the sum of all the subcoalgebras C^{V}, and C^{V_{1}⊕V_{2}} = C^{V_{1}} + C^{V_{2}} for finite dimensional left Umodules V_{1}, V_{2}. The natural (U, U)-bimodule structure on U^{∗} and on all C^{V} ’s is denoted by x · a and a · x, for all x ∈ U, a ∈ U^{∗}, where (x · a)(u) := a(ux) and (a · x)(u) := a(xu) for all u ∈ U. Note that the dual algebra U^{∗} is a left and right U-module algebra with respect to these actions, since for all a, b ∈ U^{∗} and x ∈ U,

x · (ab) = ∑(x_{1} · a)(x_{2} · b), (ab) · x = ∑(a · x_{1})(b · x_{2}).

Now let U be a Hopf algebra. A tensor category C of finite dimensional left U-modules is a class C of finite dimensional left U-modules such that

k ∈ C (k as the trivial U-module via ε),

if X, Y ∈ C, then X ⊕ Y ∈ C and X ⊗ Y ∈ C (with diagonal U-action on X ⊗ Y, u(x ⊗ y) := ∑ u_{1}x ⊗ u_{2}y for all u ∈ U, x ∈ X and y ∈ Y), and X^{∗} ∈ C (where (uf)(v) := f(S(u)v) for all u ∈ U, f ∈ X^{∗}, and v ∈ X, where S denotes the antipode of U).

By definition, the dual Hopf algebra with respect to C, U_{C}^{0} ⊂ U^{0} ⊂ U^{∗}, is spanned by all matrix coefficients of all V ∈ C. Thus

U_{C}^{0} := ∑_{V ∈C} C^{V} .

A tensor category C is called semisimple if all V ∈ C are isomorphic to direct sums of simple modules in C. If C is semisimple, then U_{C}^{0} is cosemisimple, hence the antipode of U_{C}^{0} is bijective. In general, if the antipode of U is bijective and for all X ∈ C also X^{∗} with U-action given by S^{−1} (that is ∗u · f)(v) := f(S^{−1}(u)v)) is in C, then the antipode of U_{C}^{0} is bijective.

Remark 2.1 Let V be a finite dimensional left U-module with representation ρ: U → End(V).

1. V is semisimple as a U-module if and only if C^{V} is a cosemisimple coalgebra. If V is simple, then C^{V} is a simple coalgebra.

2. V ⊗ V^{∗} → C^{V}, v ⊗ f ↦ c_{f,v} is a map of (U, U)-bimodules. Here, V ⊗ V^{∗} is a (U, U)- bimodule via u(v ⊗ f) = uv ⊗ f and (v ⊗ f)u := v ⊗ fu, where (fu)(u′) := f(uu′) for all u, u′ ∈ U, v ∈ V, and f ∈ V^{∗}.

3. If V is simple and k is algebraically closed, then V ⊗ V^{∗} → C^{V}, v ⊗ f ↦ c_{f,v} is bijective.

4. Assume C is semisimple. Let E be a complete set of representatives of the simple modules in C. Then (Peter-Weyl decomposition)

U_{C}^{0} = ⨁_{V ∈E} C^{V} .

Proof:

1. If V is a semisimple U-module, then V is a faithful and semisimple ρ(U)-module, hence ρ(U) is semisimple (since the radical of ρ(U) annihilates V). If V is simple, the finite dimensional semisimple algebra ρ(U) is simple (this follows for instance from the theorem of Artin-Wedderburn). Conversely, if ρ(U) is semisimple, then V is semisimple over ρ(U) and U. This proves the claim by duality since C^{V} ≅ ρ(U)^{∗}.

2. is clear.

3. By the density theorem, ρ is onto.

4. [T2, 1.4].

□

Let C be a tensor category. A subalgebra K ⊂ U is called C-semisimple if all V ∈ C are semisimple as left K-modules (by restriction).

Theorem 2.2 Let U be a Hopf algebra, K ⊂ U a left coideal subalgebra and C a tensor category of finite dimensional left U-modules. Let A := U_{C}^{0} be the dual Hopf algebra with respect to C, B := {a ∈ A | a · K^{+} = 0} and Ā := A/AB^{+}. Assume that the antipode of A is bijective. Then

(1) B ⊂ A is a right coideal subalgebra with B = ^{coĀ}A.

(2) If K is C-semisimple, then Ā is cosemisimple and A is faithfully flat as a left and right B-module. More precisely, according to 1.5, A = ⨁_{j∈J} _{j}A respectively A = ⨁_{j∈J} S(A_{j}) is a direct sum of finitely generated and projective right respectively left B-modules and of right coideals with B = _{1}A = S(A_{1}).

(3) If K is a Hopf subalgebra of U then in (2) for all j, _{j}A is also finitely generated and projective as a left B-module.

(4) If K is commutative and k is algebraically closed, then Ā is spanned by group-like elements if and only if K is C-semisimple.

Proof:

(1) Let Ã be the image of A under the restriction map U^{0} → K^{0} which is the coalgebra map dual to the inclusion of algebras K ⊂ U. Then π: A → Ã, π(a)(u) := a(u) for all a ∈ A and u ∈ U, is a surjective coalgebra map. Moreover, the kernel of π is a left ideal in A since for all a ∈ Ker(π), c ∈ A and u ∈ K, (ca)(u) = ∑ c(u_{1})a(u_{2}) = 0. Here, the last equality holds because K is a left coideal in U. Thus π: A → Ã is a surjective map of coalgebras and left A-modules. We first note that B = ^{coÃ}A. For if a ∈ A, then a is left Ã-coinvariant if and only if ∑ a_{1}(x)a_{2}(y) = ε(x)a(y) for all x ∈ K, y ∈ U. By definition of the right U-module structure on A (as a submodule of U^{∗}), the last equation is equivalent to a · K^{+} = 0 or a ∈ B. Since π is a map of left A-modules, B = ^{coÃ}A is a right coideal subalgebra of A, and therefore B ⊂ ^{coĀ}A. To see that B = ^{coĀ}A, note that AB^{+} ⊂ Ker(π) since π is left A-linear and for all b ∈ B^{+}, ∑ π(b_{1}) ⊗ b_{2} = π(1) ⊗ b, hence π(b) = π(1)ε(b) = 0. Thus π can be factorized as A → A/AB^{+} = Ā → Ã, and ^{coĀ}A ⊂ ^{coÃ}A = B.

(2) We now assume that K is C-semisimple. Then Ã is cosemisimple. For by definition, Ã is the sum of all C^{V} restricted to K, for V ∈ C. Since K is C-semisimple, any V ∈ C is K-isomorphic to some direct sum X_{1} ⊕ ⋯ ⊕ X_{n} of simple K-modules X_{i}. Hence π(C^{V}) is cosemisimple as the sum of the simple subcoalgebras C^{X_{i}} of K^{0}. Thus Ã ⊂ K^{0} is a cosemisimple subcoalgebra. Since Ã is cosemisimple and π: A → Ã is a surjective map of coalgebras and left A-modules, we conclude from 1.5 that Ã = A/AB^{+} = Ā and A is left and right faithfully flat over B and we have the decompositions of 1.5.

(3) Assume K is a Hopf subalgebra of U. Then U^{0} → K^{0} is a Hopf algebra map and Ā = A/AB^{+} is a quotient Hopf algebra of A. The antipode defines a bijection in the set of all simple subcoalgebras C_{j} of Ā. By (2) it suffices to show that S(C_{j}) = C_{l} for j, l ∈ J, implies S(_{j}A) = A_{l}. Indeed, for any a ∈ A,

a ∈ _{j}A ⟺ ∑ ā_{1} ⊗ a_{2} ∈ C_{j} ⊗ A
⟺ ∑ S(ā_{1}) ⊗ S(a_{2}) ∈ S(C_{j}) ⊗ A
⟺ ∑ S(a)_{1} ⊗ \overline{S(a)}_{2} ∈ A ⊗ C_{l}
⟺ S(a) ∈ A_{l}.

(4) Assume K is commutative. Then K^{0} and Ã are cocommutative. If K is Csemisimple, by (2), Ā = Ã is cosemisimple hence spanned by group-like elements because k is algebraically closed. Conversely assume that A/AB^{+} is spanned by group-like elements. Since Ã is a coalgebra quotient of A/AB^{+}, also Ã is spanned by group-like elements and hence cosemisimple. By definition, Ã is the sum of all the dual coalgebras ρ_{V} (K), where ρ_{V}: U → End(V) is the representation corresponding to V ∈ C. Hence for all V ∈ C, ρ_{V} (K)^{∗} is cosemisimple or equivalently ρ_{V} (K) is a semisimple algebra or V is a semisimple module over K. Thus K is C-semisimple.

□

In the situation of 2.2(4) we now assume that also C is semisimple. Then the decompositions in 2.2 can be described more concretely. In this case we will give an alternative proof not using 2.2. We will explicitly determine dual bases of the finitely generated and projective B-module summands of A and the set G(Ā) of group-like elements of the quotient coalgebra Ā. For any character χ ∈ Alg(K, k) of K, a left K-module M, and a right K-module N we will denote the eigenspaces of χ by

χM := {m ∈ M | ∀x ∈ K: xm = χ(x)m}

and

N_{χ} : te = {n ∈ N | ∀x ∈ K: nx = nχ(x)}.

Corollary 2.3 Let U be a Hopf algebra with bijective antipode over an algebraically closed field k, K ⊂ U a left coideal subalgebra and C a tensor category of finite dimensional left Umodules. Define A, B and Ā as in 2.2. Assume that K is commutative and C-semisimple and C is semisimple. Let X(K, C) be the set of all χ ∈ Alg(K, k) with _{χ}V ≠ {0} for some simple U-module V ∈ C. Then

(1) A = ⨁_{χ∈X(K,C)} A_{χ} and A = ⨁_{χ∈X(K,C)} S(_{χ}A) are direct sums of right coideals and finitely generated and projective right (left, respectively) B-modules with B = A_{ε} = S(_{ε}A).

(2) The natural coalgebra map Ā → K^{0} given by restriction defines a bijection

G(Ā) ⟶^{≅} X(K, C), g ↦ χ_{g},

and for any group-like element g ∈ Ā, the right eigenspace A_{χ_{g}} of χ_{g} is the space of left g-invariant elements _{g}A := {a ∈ A | ∑ ā_{1} ⊗ a_{2} = g ⊗ a}, and _{χ_{g}}A = A_{g} := {a ∈ A | ∑ a_{1} ⊗ ā_{2} = a ⊗ g}.

Proof:

(1) We first consider the decomposition A = ⨁_{χ} A_{χ}. Let E be a set of representatives of the isomorphism classes of the simple modules in C. By 2.1, A = ⨁_{V ∈E} C^{V} is a decomposition into finite dimensional right (and left) U-modules. Since K is commutative and k is algebraically closed, all finite dimensional simple K-modules are 1-dimensional and given by characters of K. By assumption any V ∈ E is Ksemisimple and has a basis (v_{i}) of eigenvectors v_{i} ∈ V_{χ_{i}} for some characters χ_{i} of K. Hence xv_{i} = χ_{i}(x)v_{i} for all i and all x ∈ K. Let (f_{i}) be the dual basis of (v_{i}) in V^{∗}. Then f_{i} · x = χ_{i}(x)f_{i} for all i and all x ∈ K. Thus it follows from 2.1, (2) and (3) that for all V ∈ E, C^{V} ≅ V ⊗ V^{∗} is a semisimple right K-module, and we can write

A = ⨁_{χ} A_{χ} = ⨁_{V ∈E} ⨁_{χ}(C^{V})_{χ} .

By definition, A_{ε} = B. For any χ, A_{χ} is a right B-submodule of A since for all a ∈ A_{χ}, b ∈ B and x ∈ K,

(ab) · x = ∑(a · x_{1})(b · x_{2})
= ∑(a · x_{1})bε(x_{2}), since x_{2} ∈ K
= (a · x)b
= abχ(x).

It is easy to check that all eigenspaces A_{χ} are right coideals in A.

Let χ be a character of K. It remains to show that A_{χ} is finitely generated and projective as a right B-module. We can assume that A_{χ} ≠ {0} or equivalently χ ∈ X(K, C). Hence there exists a simple module V ∈ E with (C^{V})_{χ} ≠ {0}. Let (v_{i}), (f_{i}) be dual bases of eigenvectors of V and V^{∗} as before. Thus f_{j} · x = χ(x)f_{j} and xv_{j} = χ(x)v_{j} for some j and all x ∈ K. Define v := v_{j}, f := f_{j} and consider the matrix coefficient c_{f,v}. Since (v_{i}), (f_{i}) are dual bases and

Δ(c_{f,v}) = ∑_{i} c_{f,v_{i}} ⊗ c_{f_{i},v},

we get 1 = f(v) = ε(c_{f,v}) = ∑_{i} c_{f,v_{i}}S(c_{f_{i},v}), where S is the antipode of A. Define a_{i} := c_{f,v_{i}}, b_{i} := S(c_{f_{i},v}) and ϕ_{i}(a) := b_{i}a for all a ∈ A_{χ}. By the dual basis lemma it suffices to prove that a_{i} ∈ A_{χ} and ϕ_{i}(a) = b_{i}a ∈ B for all i and a ∈ A_{χ}.

For all u ∈ U and x ∈ K,

(a_{i} · x)u = f(xuv_{i}) = (f · x)(uv_{i}) = χ(x)f(uv_{i}) = χ(x)a_{i}(u),

hence a_{i} ∈ A_{χ}.

Finally, for all a ∈ A_{χ} and x ∈ K,

(b_{i}a) · x = ∑(b_{i} · x_{1})(a · x_{2})
= ∑(b_{i} · x_{1})aχ(x_{2}) since x_{2} ∈ K
= (b_{i} · ∑ x_{1}χ(x_{2}))a
= (b_{i}a)ε(x),

since (b_{i} · ∑ x_{1}χ(x_{2})) = b_{i}ε(x). Thus b_{i} · a ∈ B. The last equality holds since for all u ∈ U,

(b_{i} · ∑ x_{1}χ(x_{2}))(u) = S(c_{f_{i},v})(∑ x_{1}χ(x_{2})u) = c_{f_{i},v}(S(u) ∑ S(x_{1})χ(x_{2})) =
= f_{i}(S(u) ∑ S(x_{1})χ(x_{2})v)
= f_{i}(S(u) ∑ S(x_{1})x_{2}v) since x_{2} ∈ K
= f_{i}(S(u)v)ε(x) = b_{i}(u)ε(x).

In the same way we have a decomposition into left eigenspaces A = ⨁_{χ} _{χ}A. Hence A = ⨁_{χ} S(_{χ}A) since S is bijective. One easily sees that all the S(_{χ}A) are right coideals. From Koppinen’s lemma for left coideals (apply 1.4 to the dual coalgebra) we get

K^{+}U = S^{−1}(UK^{+}) = S^{−1}(K^{+})U and B = {a ∈ A | a · S^{−1}(K^{+}) = {0}}.

Hence S(_{ε}A) = B since for all a ∈ A, x ∈ K and u ∈ U:

(S(a) · S^{−1}(x)) (u) = (x · a)(S(u)).

Then we see that S(_{χ}A) is a left B-submodule of A for any χ since for all a ∈ _{χ}A, b ∈ B and x ∈ K,

x · S^{−1}(bS(a)) = x · (aS^{−1}(b)) = ∑(x_{1} · a)(x_{2} · S^{−1}(b)) =
= ∑(x_{1} · a)ε(x_{2})S^{−1}(b) since S^{−1}(B) = _{ε}A
= (x · a)S^{−1}(b) =
= ε(x)S^{−1}(bS(a)).

Let χ ∈ X(K, C). We have to show that S(_{χ}A) is finitely generated and projective as a left B-module. Using the notations above it suffices to show that b_{i} ∈ S(_{χ}A) and S(a)a_{i} ∈ B for all i and a ∈ _{χ}A. For all u ∈ U and x ∈ K,

(x · c_{f_{i},v})(u) = f_{i}(uxv) = χ(x)f_{i}(uv) = χ(x)c_{f_{i},v}(u),

hence b_{i} = S(c_{f_{i},v}) ∈ S(_{χ}A). For all a ∈ _{χ}A, S(a)a_{i} ∈ B = S(_{ε}A) since a_{i} = c_{f,v_{i}} and for all x ∈ K,

x · S^{−1}(S(a)a_{i}) = x · (S^{−1}(c_{f,v_{i}})a) =
= ∑(x_{1} · S^{−1}(c_{f,v_{i}}))(x_{2} · a) =
= ∑(x_{1} · S^{−1}(c_{f,v_{i}}))χ(x_{2})a since a ∈ _{ε}A
= ((∑ χ(x_{2})x_{1}) · S^{−1}(c_{f,v_{i}}))a =
= ε(x)S^{−1}(c_{f,v_{i}})a =
= ε(x)S^{−1}(S(a)a_{i}).

In the proof we used the equality

∑ χ(x_{2})x_{1} · S^{−1}(c_{f,v_{i}}) = ε(x)S^{−1}(c_{f,v_{i}})

which holds since for all u ∈ U

(∑ χ(x_{2})x_{1} · S^{−1}(c_{f,v_{i}}))(u) = ∑ c_{f,v_{i}}(χ(x_{2})S^{−1}(x_{1})S^{−1}(u)) =
= ∑ f(χ(x_{2})S^{−1}(x_{1})S^{−1}(u)v_{i}) =
= ∑ f(x_{2}S^{−1}(x_{1})S^{−1}(u)v_{i}) since f ∈ (V^{∗})_{χ}
= ε(x)f(S^{−1}(u)v_{i}) = ε(x)S^{−1}(c_{f,v_{i}})(u).

(2) As in the proof of 2.2, let Ã be the image of A under the restriction map U^{0} → K^{0} and π: A → Ã the induced surjective coalgebra map. By 2.2, the kernel of π is AB^{+}. The group-like elements in A^{0} are the characters of K. Hence the group-like elements G(Ã) of Ã are the characters of K which can be extended to a linear map a: U → k with a ∈ A.

Let χ be a character of K with A_{χ} ≠ 0. In the notation of the proof of (1), the matrix coefficient c_{f,v} is an element in A such that for all x ∈ K, c_{f,v}(x) = χ(x). Therefore χ is a group-like element of Ã.

By definition, the space of left χ-invariants is the eigenspace A_{χ}, since for all a ∈ A, ∑ π(a_{1})⊗a_{2} = χ⊗a if and only if for all x ∈ K and u ∈ U, a(xu) = ∑ a_{1}(x)a_{2}(u) = χ(x)a(u). Similarly, the space of right χ-invariants is _{χ}A. In particular for all a ∈ A_{χ}, π(a) = χε(a), and we see that π(A_{χ}) = kχ.

Hence by (1), Ã = π(A) = ⨁_{χ∈X(K,C)} kχ, and X(K, C) is the set of all group-like elements of Ã. This finishes the proof of (2) since Ā ≅ Ã.

□

In the next theorem we consider the special case of 2.1 when K = k[x] for some (g, 1)- primitive element x ∈ U, that is Δ(x) = g ⊗ x + x ⊗ 1, g group-like in U. Then K is a commutative left coideal subalgebra of U. If C is a tensor category of finite dimensional left U-modules, we call an element x ∈ U C-semisimple if for all V ∈ C, the linear map V → V, v ↦ xv, is diagonalizable. When x is (1, g)-primitive, that is Δ(x) = 1⊗x+x⊗g, then k[x] is a right coideal subalgebra of U.

Theorem 2.4 Let U be a Hopf algebra over an algebraically closed field k, g a group-like element and x a (g, 1)-primitive (resp. (1, g)-primitive) element of U and C a tensor category of finite dimensional left U-modules. Define A := U_{C}^{0}, B := {a ∈ A | a · x = 0} and Ā := A/AB^{+} (resp. A/B^{+}A). Assume that the antipode of A is bijective.

(1) The following are equivalent:

(a) x is C-semisimple.

(b) Ā is spanned by group-like elements.

If these conditions hold, then A is faithfully flat as a right and as a left B-module.

(2) If C is semisimple and B is a simple object in M_{B}^{A} (resp. _{B}M^{A}), then B := {a ∈ A | ∃n ≥ 1: a · x^{n} = 0}.

Proof: We assume that x is (g, 1)-primitive (in the case of (1, g)-primitive elements we then apply the result to the dual coalgebra of U). Then K := k[x] is a left coideal subalgebra of U and (1) is a special case of 2.2(4) and (2) since x is C-semisimple if and only if K is C-semisimple. For (2) it is enough to show that B = A^{(2)} := {a ∈ A | a · x^{2} = 0}. Assume B ⊊ A^{(2)}. Let ϕ: A → A, ϕ(a) := a · x, be right multiplication with x. The map ϕ is right A-colinear since for all a ∈ A and u, u′ ∈ U,

∑(a_{1} · x)(u)a_{2}(u′) = ∑ a_{1}(xu)a_{2}(u′) = a(xuu′) = (a · x)(uu′),

hence Δ(a·x) = ∑ a_{1} ·x⊗a_{2}. Since A is a right U-module algebra, ϕ is also right B-linear, because for all a ∈ A and b ∈ B,

(ab) · x = ∑(a · x_{1})(b · x_{2}) = (a · g)(b · x) + (a · x)b = (a · x)b

since b·x = 0 by the definition of B. Thus ϕ is a morphism in M_{B}^{A}. Hence also A^{(2)} → B, a ↦ a · x, is a morphism in M_{B}^{A}. This map is non-zero since B ⊊ A^{(2)}, hence it is onto because B is a simple object in M_{B}^{A}. By assumption, C is semisimple, hence A is cosemisimple. Therefore the surjective map A^{(2)} → B splits as a map of right A-comodules and there exists a right A-colinear map γ: B → A^{(2)} such that γ(b) · x = b for all b ∈ B. Since Δ(γ(1)) = γ(1) ⊗ 1, γ(1) is a scalar multiple of the identity in B and we get the contradiction 1 = γ(1) · x = γ(1)ε(x) = 0. □

3 Unitarizable tensor categories

In this short section we want to show that for Hopf ∗-algebras there is a natural condition which guarantees that a left coideal subalgebra K is C-semisimple. Hence in many examples for quantum homogeneous spaces with ∗-structures our main result 2.2 can be applied. A Hopf ∗-algebra H [KS, 1.2.7] is a Hopf algebra over the field of complex numbers with an involution ∗: H → H such that for x, y ∈ H and a complex number α (with complex conjugate ᾱ)

(x + y)^{∗} = x^{∗} + y^{∗}, (αx)^{∗} = ᾱx^{∗}, (xy)^{∗} = y^{∗}x^{∗}, Δ(x^{∗}) = ∑ x_{1}^{∗} ⊗ x_{2}^{∗}.

Then 1^{∗} = 1, ε(x^{∗}) = \overline{ε(x)} and S(x^{∗})^{∗} = S^{−1}(x) for all x ∈ H. In this section, for all subsets T of a Hopf ∗-algebra H let T^{∗} := {t^{∗} | t ∈ T }. A coalgebra is called pointed if each of its simple subcoalgebras is one-dimensional. A Hopf algebra is pointed if it is generated as an algebra by group-like and skew-primitive elements [M, 5.5.1]. In particular, the q-deformed enveloping algebras of semisimple Lie algebras are all pointed.

Proposition 3.1 Let U be a Hopf ∗-algebra and I ⊂ U a coideal. Define K := U^{co U/IU}. Then

(1) K is a left coideal subalgebra of U, and if U is pointed (or more generally U is faithfully left coflat over U/IU), then IU = K^{+}U.

(2) If S(I)^{∗} ⊂ I, then K^{∗} = K.

Proof:

(1) Since IU is a right ideal and coideal, the set of right U/IU-coinvariant elements K is a left coideal subalgebra. If U is pointed, then by [M2, 1.3] U is left (and right) faithfully coflat over U/IU. Hence we know from Theorem 1.1 (applied to A^{op cop}) that IU = K^{+}U.

(2) First note that K^{+} ⊂ IU since for any x ∈ K, ∑ x_{1} ⊗ x̄_{2} = x ⊗ ¯1, hence x̄ = ε(x)¯1 in U/IU. By Koppinen’s lemma 1.4, S(K^{+}U) = UK^{+}. Then

(K^{+})^{∗} ⊂ (UK^{+})^{∗} = S(K^{+}U)^{∗}, since UK^{+} = S(K^{+}U)
⊂ S(IU)^{∗}, since K^{+} ⊂ IU
= S(I)^{∗}U
⊂ IU, since S(I)^{∗} ⊂ I.

Clearly K^{∗} is again a left coideal subalgebra with (K^{∗})^{+} = (K^{+})^{∗}, and we have shown that (K^{∗})^{+}U ⊂ IU. Therefore

K^{∗} ⊂ U^{co U/(K^{∗})^{+}U} ⊂ U^{co U/IU} = K.

Since ∗ is an involution, we get K^{∗} = K.

□

As an illustration of 3.1(1), let g, h be group-like elements of U and x ∈ U with Δ(x) = g ⊗ x + x ⊗ h. Then kx is a one-dimensional coideal, k[xh^{−1}] is a left coideal subalgebra, and xU = k[xh^{−1}]^{+}U.

Remark 3.2 Let U be a Hopf ∗-algebra and C a tensor category of finite dimensional left U-modules. For any left U-module V let V̄ be V as an additive group with the following left U-module structure, denoted by ⋆: For any v ∈ V and u ∈ U,

u ⋆ v := S(u)^{∗}v.

In particular, V̄ is a complex vector space with α ⋆ v := ᾱv for complex numbers α. V̄ is the restriction of V to U via the ring isomorphism

U → U, u ↦ S(u)^{∗}.

A tensor category C will be called a tensor ∗-category if for all V ∈ C also V̄ ∈ C. Let C be a tensor ∗-category. Then (1) A := U_{C}^{0} is a Hopf ∗-algebra with ∗-structure defined by a^{∗}(u) := \overline{a(S(u)^{∗})} for all

a ∈ A and u ∈ U.

(2) If I ⊂ U is a coideal with S(I)^{∗} ⊂ I, then B := {a ∈ A | a · I = 0}, the algebra of infinitesimal invariants defined by I, is a ∗-subalgebra of A.

Proof:

(1) The full Hopf dual U^{0} is a Hopf ∗-algebra with ∗-structure as described above. A is closed under the ∗-structure since for all V ∈ C, v ∈ V, linear functionals f on V and u ∈ U,

c_{f,v}^{∗}(u) = \overline{f(S(u)^{∗}v)} = c_{f̄,v}(u),

where c_{f̄,v} is the matrix coefficient of V̄ and the linear functional f̄ on V̄ is defined by f̄(w) := \overline{f(w)} for all w ∈ V̄ = V.

(2) is easy to check [KD, 1.9].

□

Let U be a Hopf ∗-algebra and C a tensor category of finite dimensional left U-modules. We call C unitarizable if for all V ∈ C there is a hermitian inner product ⟨, ⟩: V × V → ℂ, conjugate linear in the first and linear in the second variable such that for all x ∈ U and v, w ∈ V, ⟨xv, w⟩ = ⟨v, x^{∗}w⟩.

Corollary 3.3 Let U be a pointed Hopf ∗-algebra, C a unitarizable tensor category of finite dimensional left U-modules, and I ⊂ U a coideal with S(I)^{∗} ⊂ I. Define A := U_{C}^{0} and B := {a ∈ A | a · I = 0}. Then

(1) K := U^{co U/IU} is a left coideal subalgebra of U, B := {a ∈ A | ∀x ∈ K: a · x = aε(x)} is a right coideal subalgebra of A and K is C-semisimple.

(2) A/AB^{+} is cosemisimple, A is faithfully flat as a left and a right B-module and a direct sum of finitely generated and projective left B-modules and right B-modules as in 2.2.

Proof:

(1) Since U is pointed, we get from 3.1(1) that IU = K^{+}U. Hence B is also the set of infinitesimal invariants with respect to K^{+}. Moreover K^{∗} = K by 3.1(2). Let V ∈ C and ⟨, ⟩ the hermitian inner product on V × V. We have to show that V is semisimple as a module over K. Let W ∈ V be a K-submodule. Then W^{⊥} = {v ∈ V | ∀w ∈ W: ⟨v, w⟩ = 0} is a K-submodule because for all x ∈ K, v ∈ W^{⊥}, and w ∈ W,

⟨xv, w⟩ = ⟨v, x^{∗}w⟩ = 0

since x^{∗} ∈ K. Hence V = W ⊕ W^{⊥} is a direct sum of K-modules, and V is K-semisimple.

(2) follows from (1) and 2.2

□

Remark 3.4 Corollary 3.3 applies to many recent examples of quantum homogeneous spaces.

1. In general, if 𝔤 is a semisimple Lie algebra and U = U_{q}(𝔤) is the q-deformed universal enveloping algebra with positive real q ≠ 1, then U is a pointed Hopf ∗-algebra with standard ∗-structure, and the tensor category of finite dimensional left U-modules of type 1 is unitarizable [CP, 10.1.21]

2. Let A = A(SU_{q}(2)) be the function algebra of the q-deformed special unitary group SU(2) and assume 0 < q ≤ 1. For any parameter ρ ∈ [0, ∞], Dijkhuizen and Koornwinder [KD], [KS, 4.5] defined a right coideal subalgebra B = B_{ρ} ⊂ A by infinitesimal invariants with respect to one skew-primitive element x_{ρ}. They show that the algebras B_{ρ} can be identified with the function algebras of Podleś’ quantum spheres S_{qc}^{2} for 0 ≤ c ≤ ∞, [P]. In this situation all the assumptions in 3.3 are satisfied for the extension B ⊂ A. In particular we know from 3.3 and 2.2(4) that A/AB^{+} is spanned by group-like elements. This answers a question of Brzeziński [B].

3. A more general family of examples has been introduced by Dijkhuizen and Noumi [DN], [KS, 11.6.6]. Let n ≥ 2 and 0 < q ≤ 1. For any non-negative real numbers c, d which do not vanish simultaneously, they define a coideal I = I^{(c,d)} in U_{q}(𝔤𝔩(n)). Then they define the quantum projective space ℂℙ_{q}^{n−1}(c, d) with a natural transitive action of the quantum unitary group U_{q}(n). Here ℂℙ_{q}^{n−1}(c, d) is given by a right coideal subalgebra B = B^{(c,d)} in A := A(U_{q}(n)), the function algebra of U_{q}(n). B is defined by infinitesimal invariants with respect to I^{(c,d)}. These homogeneous U_{q}(n)- spaces have first been studied by Vaksman and Korogodsky [KV] as a q-analog of the Hopf fibration S^{2n−1} → ℂℙ^{n−1}. Again the extension B ⊂ A satisfies the assumptions in 3.3.

4. [D] contains a survey of similar constructions of quantum homogeneous spaces which are analogs of compact symmetric spaces.

4 Semisimple skew-primitive elements in U_{q}(sl(2))

Let k be an algebraically closed field and let q be a non-zero element of k, which is not a root of unity. For all positive integers n let E_{n} be the unit matrix with n rows and columns. First we recall some definitions and results of [K]. For any integer n, set [K, p. 121]

[n] = \frac{q^{n} − q^{−n}}{q − q^{−1}} = q^{n−1} + q^{n−3} + ⋯ + q^{−n+3} + q^{−n+1}.

The Hopf algebra U_{q}(sl(2)) is generated as an algebra by E, F, K, K^{−1} with relations

KK^{−1} = K^{−1}K = 1, KEK^{−1} = q^{2}E, KF K^{−1} = q^{−2}F, EF − F E = \frac{K − K^{−1}}{q − q^{−1}}

and comultiplication defined by

Δ(K) = K ⊗K, Δ(K^{−1}) = K^{−1}⊗K^{−1}, Δ(E) = 1⊗E +E ⊗K, Δ(F) = K^{−1}⊗F +F ⊗1.

All simple U_{q}(sl(2)) left modules are isomorphic to some V_{e,n}, where e = ±1 and n is a nonnegative integer. The (n + 1)-dimensional module V_{e,n} has a basis {v_{0}, v_{1}, …, v_{n}}, such that the left action of the generators E, F, and K of U_{q}(sl(2)) can be represented on this basis by the matrices [K, p. 129]

ρ_{e,n}(E) = e ( \begin{matrix}0 & [n] & 0 & ⋯ & 0 \\ 0 & 0 & [n − 1] & ⋯ & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & 0 & ... & ... & 1 \\ 0 & 0 & ⋯ & 0 & 0 \end{matrix} ) , ρ_{e,n}(F) = ( \begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & [2] & ... & 0 & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & 0 & ⋯ & [n] & 0 \end{matrix} ) ,

and

ρ_{e,n}(K) = e ( \begin{matrix}q^{n} & 0 & ⋯ & 0 & 0 \\ 0 & q^{n−2} & ⋯ & 0 & 0 \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & q^{−n+2} & 0 \\ 0 & 0 & ⋯ & 0 & q^{−n} \end{matrix} ) .

Here we need the right action and use the transposed matrices. We first consider the case e = 1. Let

x = α(K^{−1} − 1) + βEK^{−1} + γF

be a (K^{−1}, 1)-primitive element of U_{q}(sl(2)), where α, β, and γ are fixed elements of the ground field k, which do not vanish simultaneously. Then the (right) action of x on the chosen basis of V_{1,n} can be represented by the matrix

M_{n} := ( \begin{matrix}(q^{−n} − 1)α & γ & 0 & ⋯ & 0 \\ q^{−n}[n]β & (q^{2−n} − 1)α & [2]γ & ..... & ⋮ \\ 0 & q^{2−n}[n − 1]β & (q^{4−n} − 1)α & ..... & 0 \\ ⋮ & ..... & ..... & ..... & [n]γ \\ 0 & ⋯ & 0 & q^{n−2}[1]β & (q^{n} − 1)α \end{matrix} ) .

The quantum plane k_{q}[a, b] is the k-algebra generated by the elements a and b with the relation ba = qab [K, Chapter IV]. It is a U_{q}(sl(2)) left module algebra, where the action of U_{q}(sl(2)) is given in [K, Chapter VII.3]. Here we need the corresponding right action given by

b·E = 0, a·E = b, b·F = a, a·F = 0, b·K = qb, a·K = q^{−1}a, b·K^{−1} = q^{−1}b, a·K^{−1} = qa.

The quantum plane has a natural gradation, given by the degrees of the monomials, therefore there are no zero divisors. Let k_{q}[a, b]_{n} denote the vector subspace of homogeneous polynomials of degree n in k_{q}[a, b]. Then k_{q}[a, b]_{n} is a simple U_{q}(sl(2)) right module isomorphic to V_{1,n} [K, Theorem VII.3.3].

Lemma 4.1 (1) There is a non zero element ξ ∈ k_{q}[a, b]_{2} such that ξ · x = 0.

(2) For all z, ξ ∈ k_{q}[a, b], where ξ · x = 0, the equation (zξ) · x = (z · x)ξ holds, that is, the action of x is B_{X}-right linear, where B_{X} = {ξ ∈ k_{q}[a, b]: ξ · x = 0}.

(3) The eigenspaces of the action of x on k_{q}[a, b]_{n} are one-dimensional.

Proof:

(1) The determinant of M_{2} vanishes.

(2) The quantum plane is a module algebra of U_{q}(sl(2)):

(zξ) · x = (z · K^{−1})(ξ · x) + (z · x)(ξ · 1) = (z · x)ξ.

(3) Let λ ∈ k. and let E_{n+1} be the unit matrix with n + 1 rows and columns. In the cases β ≠ 0 or γ ≠ 0 the first or last n columns respectively of M_{n} − λE_{n+1} are linearly independent. In the case β = γ = 0 there must be u ≠ 0, and in the diagonal matrix M_{n} − λE_{n+1} at most one entry in the diagonal vanishes, for since q is not a root of unity, all entries in the diagonal are pairwise distinct.

□

For convenience, we fix a solution \sqrt{q} of the equation x^{2} = q and define q^{l} := \sqrt{q}^{2l} for l ∈ \frac{1}{2}ℤ. For each nonnegative integer n set I_{n} := \{−\frac{n}{2}, 1 − \frac{n}{2}, …, \frac{n}{2} − 1, \frac{n}{2}\}.

Proposition 4.2 (1) Let

P_{n}(Y) = det(M_{n} − Y E_{n+1}) = (−1)^{n+1}(Y^{n+1} + z_{n}Y^{n} + d_{n−1}Y^{n−1} + …)

be the characteristical polynomial of M_{n}. Then for n ≥ 2 the polynomial P_{n−2} divides P_{n}.

(2) Fix R ∈ k such that

R^{2} = α^{2} + \frac{4βγq^{−1}}{(q − q^{−1})^{2}} .

Then M_{n} has the n + 1 (not necessarily distinct) eigenvalues

λ_{r} := \frac{α}{2}(q^{r} − q^{−r})^{2} + \frac{1}{2}(q^{2r} − q^{−2r})R,

for r ∈ I_{n}.

Proof:

(1) Let μ be an f-fold zero of P_{n−2}, i. e. the dimension of the generalized eigenspace for the eigenvalue μ in k_{q}[a, b]_{n−2} is f, and let v_{1}, …, v_{f} be a basis of V_{μ}^{n−2}. Therefore there exists a positive integer t such that v_{ν} · (x − μ)^{t} = 0 for ν = 1, …, f. Let ξ be as in Lemma 4.1. Then v_{1}ξ, …, v_{f}ξ are linearly independent elements of k_{q}[a, b]_{n}, since k_{q}[a, b] is an integral domain. Moreover by part (2) of the lemma,

(v_{ν}ξ) · (x − μ)^{t} = (v_{ν} · (x − μ)^{t}) ξ = 0,

whence v_{ν}ξ for ν = 1, …, f belongs to the generalized eigenspace for the eigenvalue μ in k_{q}[a, b]_{n}. Therefore μ is at least an f-fold zero of P_{n}.

(2) For n = 0 and n = 1 one easily computes

P_{0}(Y) = 0−Y = −Y and P_{1}(Y) = Y^{2}−α(\sqrt{q}−\sqrt{q}^{−1})^{2}Y −q^{−1}βγ−α^{2}(\sqrt{q}−\sqrt{q}^{−1})^{2}.

Now assume n ≥ 2. Due to deg P_{n}(Y) = 2 + deg P_{n−2}(Y) and part (a) only the two extra zeros a_{n} and b_{n} of P_{n} have to be determined. Vieta’s Theorem says

z_{n−2} − a_{n} − b_{n} = z_{n}, a_{n}b_{n} − (a_{n} + b_{n})z_{n−2} + d_{n−2} = d_{n} ⇒
a_{n} + b_{n} = z_{n−2} − z_{n}, a_{n}b_{n} = d_{n} − d_{n−2} + z_{n−2}(z_{n−2} − z_{n}).

Therefore a_{n} and b_{n} must satisfy the quadratic equation in T

T^{2} − (z_{n−2} − z_{n})T + d_{n} − d_{n−2} + z_{n−2}(z_{n−2} − z_{n}) = 0.

Let l = n/2. We compute the coefficients of this equation:

(a) The coefficient z_{n} is the negative trace of M_{n}. Therefore

z_{n} − z_{n−2} = −u(q^{n} − 1 + q^{−n} − 1) = −u(q^{l} − q^{−l})^{2}.

(b) The coefficient d_{n} is the sum of D_{n} and N_{n}, where D_{n} is the sum of all products of two distinct entries of the main diagonal and N_{n} is the sum of products of entries, which do not belong to the main diagonal. All indices run over I_{n} if there are no limits of summation.

D_{n} := α^{2} ∑_{r<s}(q^{−2r} − 1)(q^{−2s} − 1) =
= \frac{1}{2}(∑_{r} α(q^{−2r} − 1))^{2} − \frac{1}{2}(α^{2} ∑_{r}(q^{−2r} − 1)^{2}) =

= \frac{1}{2}z_{n}^{2} − \frac{1}{2}α^{2} ∑_{r}(q^{−2r} − 1)^{2}.
N_{n} := −q^{−2}βγ∑_{t=1}^{n} q^{2t−n}[t][n + 1 − t] ⇒
\frac{−N_{n}}{q^{−1}βγ/(q − q^{−1})^{2}} = ∑_{t=1}^{n} q^{2t−n−1}(q^{t} − q^{−t})(q^{n+1−t} − q^{t−n−1}) =
= (q^{n+1} + q^{−n−1}) ∑_{t=1}^{n} q^{2t−n−1} − ∑_{t=1}^{n}(q^{4t−2n−2} + 1) =
= (q^{n+1} + q^{−n−1})\frac{q^{n} − q^{−n}}{q − q^{−1}} − ∑_{t=1}^{n}(q^{4t−2−2n} + 1) =
= \frac{q^{2n+1} − q^{−2n−1}}{q − q^{−1}} − 1 + ∑_{t=1}^{n}(q^{4t−2−2n} + 1)
⇒ \frac{N_{l−1} − N_{l}}{q^{−1}βγ/(q − q^{−1})^{2}} = \frac{q^{2n+1} − q^{2n−3} + q^{3−2n} − q^{−2n−1}}{q − q^{−1}} − q^{2n−2} − q^{2−2n} − 2 =
= (1 + q^{2})(q^{2n−2} + q^{−2n}) − q^{2n−2} − q^{2−2n} − 2 = (q^{n} − q^{−n})^{2}.

Now the constant coefficient of the quadratic equation can be computed: d_{l} − d_{l−1} + z_{l−1}(z_{l−1} − z_{l}) =

= N_{l} − N_{l−1} + D_{l} − D_{l−1} + z_{l−1}(z_{l−1} − z_{l}) =
= N_{l} − N_{l−1} − \frac{1}{2}α^{2} ((q^{n} − 1)^{2} + (q^{−n} − 1)^{2}) + \frac{1}{2}z_{l}^{2} − \frac{1}{2}z_{l−1}^{2} + z_{l−1}^{2} − z_{l}z_{l−1} =
= N_{l} − N_{l−1} − \frac{1}{2}α^{2}(q^{n}(q^{l} − q^{−l})^{2} + q^{−n}(q^{l} − q^{−l})^{2}) + \frac{1}{2}(z_{l−1} − z_{l})^{2} =
= N_{l} − N_{l−1} − \frac{1}{2}α^{2}(q^{n} + q^{−n})(q^{l} − q^{−l})^{2} + \frac{1}{2}α^{2}(q^{l} − q^{−l})^{4} =
= \frac{−βγ}{q(q − q^{−1})^{2}} (q^{2l} − q^{−2l})^{2} − α^{2}(q^{l} − q^{−l})^{2}.

Therefore the quadratic equation is

x^{2} − α(q^{l} − q^{−l})^{2}x − \frac{q^{−1}βγ}{(q − q^{−1})^{2}} (q^{2l} − q^{−2l})^{2} − α^{2}(q^{l} − q^{−l})^{2} = 0.

Note that for n = 1 this equation equals P_{1}(Y) = 0. It has the discriminant

α^{2}(q^{l}−q^{−l})^{4}+4(\frac{q^{−1}βγ}{(q − q^{−1})^{2}} (q^{2l}−q^{−2l})^{2}+α^{2}(q^{l}−q^{−l})^{2}) = (q^{2l}−q^{−2l})^{2}(α^{2}+\frac{4q^{−1}βγ}{(q − q^{−1})^{2}} ).

If R solves the equation

x^{2} = α^{2} + \frac{4q^{−1}βγ}{(q − q^{−1})^{2}} ,

we get the zeros λ_{n/2} and λ_{−n/2}, where λ_{r} = \frac{α}{2}(q^{r} − q^{−r})^{2} + \frac{1}{2}(q^{2r} − q^{−2r})R for 2r ∈ ℤ. For n = 0 this is the zero 0 of P_{0}.

□

Remark 4.3 In [NM, Theorem 1] another method is sketched to get the zeros of P_{n}(Y) if x acts diagonally. Starting from a matrix g = ( α̃ ; γ̃ β̃ ; δ̃) in GL(2; ℂ) (but ℂ can be replaced by k), the element

D = −α̃β̃E + (α̃δ̃ + β̃γ̃)\frac{K − 1}{q − q^{−1}} + γ̃δ̃q^{−1}F K

is considered (the authors write q^{H} for K, \sqrt{q}X_{+}q^{H/2} for E and \sqrt{q}X_{−}q^{−H/2} for F). By construction of the eigenvectors of the left action of D on the quantum plane, it is shown that the distinct eigenvalues are

λ_{m}(g) = \frac{q^{m} − q^{−m}}{q − q^{−1}} · (α̃δ̃q^{m} − β̃γ̃q^{−m})

where 2m ∈ ℤ, if α̃δ̃ − q^{2t}β̃γ̃ ≠ 0 for all t ∈ ℤ (then D is called diagonalizable). This left action is represented by a matrix obtained from M_{n} by swapping the j-th column with the (n + 1 − j)-th column and the j-th row with the (n + 1 − j)-th row for 1 ≤ j ≤ (n+1)/2 and therefore has the same characteristical polynomial as M_{n}, where α = \frac{α̃δ̃+β̃γ̃}{q−q^{−1}}, β = q^{−1}γ̃δ̃, γ = −α̃β̃. Consequently, in the case α̃δ̃ − q^{2t}β̃γ̃ ≠ 0 for all t ∈ ℤ there is the factorization

P_{n}(Y) = (λ_{n/2}(g) − Y)(λ_{n/2−1}(g) − Y) ⋯ (λ_{−n/2}(g) − Y).

Up to the cases D ∈ kE or D ∈ kF K, the products α̃δ̃ and β̃γ̃ do not vanish simultaneously, and if exactly one product vanishes, then D is always diagonalizable. Now assume that α̃β̃˜−γδ̃ ≠ 0. In the factorization of P_{n}(Y) both sides are polynomials in α̃, β̃, γ̃, δ̃. The equation is valid for an infinite number of values for α̃ (when β̃, γ̃, δ̃ are fixed), therefore it is an identity of polynomials in α̃. A similar reasoning holds for β̃, γ̃, δ̃. Thus λ_{n/2}(g), …, λ_{−n/2}(g) are the not necessarily distinct zeros of P_{n}(Y) in any case, even if g is an arbitrary 2 × 2 matrix. Translating this into the notation of the Proposition, one gets the zeros λ_{r} considered in the Proposition. The condition R ≠ 0 is equivalent to α̃δ̃ − β̃γ̃ ≠ 0 (this is excluded by the choice of g in [NM], but also in this case D is skew primitive).

Theorem 4.4 Fix R ∈ k such that R^{2} = α^{2} + 4q^{−1}βγ/(q − q^{−1})^{2}. Then the following statements are equivalent:

(1) For all nonnegative integers n, the matrix M_{n} can be diagonalized.

(2) R ≠ 0, and for all nonnegative integers m, the kernel of the action of x on V_{1,2m} equals the generalized eigenspace for the eigenvalue 0 (i. e. if v · x^{t} = 0 for some t ≥ 1 and v ∈ V_{1,2m}, then v · x = 0).

(3) There is no nonnegative integer n satisfying

q^{−1}βγ (\frac{q^{n} + q^{−n}}{q − q^{−1}} )^{2} = −α^{2}.

Proof: The conclusion (1) ⇒ (2) is trivial (R = 0 implies that λ_{r} = λ_{−r} for all r ∈ \frac{1}{2}ℤ\\{0\}, and by the Lemma, the action of x on V_{1,n} for n > 1 cannot be diagonalized). In order to prove (2) ⇒ (1), assume there is a polynomial P_{n} with a double zero. Then by 4.2 there are distinct half integers l and m such that l + m is an integer (because l, m ∈ I_{n}) and λ_{l} = λ_{m}. Hence

\frac{α}{2}(q^{l} − q^{−l})^{2} + \frac{1}{2}(q^{2l} − q^{−2l})R = \frac{α}{2}(q^{m} − q^{−m})^{2} + \frac{1}{2}(q^{2m} − q^{−2m})R ⇒

⇒ \frac{α}{2}(q^{2l} + q^{−2l} − q^{2m} − q^{−2m}) + \frac{R}{2}(q^{2l} − q^{−2l} − q^{2m} + q^{−2m}) = 0

⇒ (q^{l−m} − q^{m−l})(\frac{α}{2}(q^{l+m} − q^{−l−m}) + \frac{R}{2}(q^{l+m} + q^{−l−m})) = 0 ⇒ (q^{l−m} − q^{m−l})λ_{l+m} = 0.

Since q is not a root of unity, this yields λ_{l+m} = 0. The assumption R ≠ 0 implies l+m ≠ 0, whence 0 is a double zero of P_{2|l+m|}, and by the Lemma, the generalized eigenspace of 0 does not equal the eigenspace in V_{1,2|l+m|}. Finally, condition (3) is equivalent to R ≠ 0 and λ_{n}λ_{−n} ≠ 0 for all positive integers n, i. e. to condition (2). □

There is a similar statement about (1, K)-primitive elements of U_{q}(sl(2)).

Remark 4.5 1. We consider y := xK. Then y is (1, K)-primitive and

M′_{n} := ρ_{e,n}(y) = ρ_{e,n}(K)M_{n} =

= ( \begin{matrix}(1 − q^{−n})α & q^{n−2}γ & 0 & ⋯ & 0 \\ [n]β & (1 − q^{2−n})α & q^{n−4}[2]γ & ..... & ⋮ \\ 0 & [n − 1]β & (1 − q^{4−n})α & ..... & 0 \\ ⋮ & ..... & ..... & ..... & q^{−n}[n]γ \\ 0 & ⋯ & 0 & [1]β & (1 − q^{n})α \end{matrix} ) .

Let M′′_{n} the matrix obtained from M′_{n} after replacing α by −α and interchanging β and γ. Comparison with M_{n} yields that for all i, j ≤ n+1, the i, j-entry of M_{n} is the (n+2−i, n+2−j)-entry of M′′_{n}, whence the characteristic polynomials of M_{n} and M′′_{n} are identical. If P_{n}(Y) is the characteristic polynomial of M′′_{n} then the characteristic polynomial of M′_{n} is P_{n}(−Y) because all eigenvalues remain unchanged when β, γ are swapped, and replacing α by −α can be performed by multiplying the first, third, fifth, … row and the second, fourth, sixth, … column by −1. Thus 4.2(2) explicitly gives the zeros −λ_{r} of the characteristic polynomial of the action of y on V_{1,m} and two of them coincide if and only if this is true for x, too. An analog of 4.1 holds for y, too (the equation in part (2) becomes (ξz)·y = ξ(z ·y)), in particular all eigenspaces are one-dimensional. Therefore 4.4 can be proved analogously for y.

2. If there exists a nonnegative integer n such that

q^{−1}βγ (\frac{q^{n} + q^{−n}}{q − q^{−1}} )^{2} = −α^{2},

then direct computation as in 4.4 gives λ_{r} = λ_{n−r} for all r. If n is even then the eigenspace for λ_{r/2} equals the generalized eigenspace and the eigenspaces for other eigenvalues are not the generalized eigenspaces. This explains why the case R = 0, where the eigenspace for the eigenvalue 0 is the generalized eigenspace but x is not diagonalizable, is not too special from this point of view. If n is odd then all eigenspaces are different from the generalized eigenspaces.

3. If and only if there is an element n ∈ \frac{1}{2}ℕ \ ℕ satisfying

q^{−1}βγ (\frac{q^{n} + q^{−n}}{q − q^{−1}} )^{2} = −α^{2},

then there is a non zero element ξ′ ∈ k_{q}[a, b]_{2n} such that ξ′ · x = 0. This means that the algebras generated by 3 generators and 4 relations in analogy to Podleś’ quantum spheres are proper right coideal subalgebras of the algebra B = {a ∈ A | x · a = 0} from the construction in 2.3, cf. [Mü]. These cases have been excluded in [KD] by the assumption that x should be ∗-invariant. The precise correspondence between the parameters α, β, γ used in this section and the parameter c (including the case c = ∞) in Podleś’ original quantum spheres S_{μc} (for μ = q) is as follows:

c = \{ \begin{matrix}\frac{βγq^{−1}}{α^{2}(q−q^{−1})^{2}} & if α ≠ 0 \\ ∞ & if α = 0 (and βγ ≠ 0) \end{matrix}

(in the ∗-invariant case we automatically have α ∈ ℝ and qγ̄ = β, whence c = (\frac{|γ|}{α(q−q^{−1})})^{2} is nonnegative. The special cases for negative c in [P] do not occur because then the finite dimensional quantum spheres cannot be canonically embedded into the function algebra as ∗-invariant subalgebras.

If x (or y respectively) is diagonalizable on all modules V_{1,n}, then the following general argument shows that it is diagonalizable on all modules V_{e,n}. Let kw be a onedimensional U_{q}(sl(2)) module with basis {w}, such that E · w = 0, F · w = 0, K · w = ew. Then V_{e,n} ˜= kw⊗V_{1,n} ˜= V_{1,n} ⊗kw, where U_{q}(sl(2)) acts diagonally on the latter two modules and the second isomorphism is due to the Hopf algebra automorphism of U_{q}(sl(2)), which maps E and F to −E and −F respectively and leaves K unchanged.

Proposition 4.6 Let U be a Hopf algebra, g ∈ U a group-like element, x ∈ U a (g, 1)- primitive (respectively (1, g)-primitive) element, kw a one-dimensional left U-module with basis {w} and V a finite dimensional left U-module. If the action of x on V is diagonalizable then so is the diagonal action on kw ⊗ V (respectively V ⊗ kw).

Proof: Assume x is (g, 1)-primitive (The proof for (1, g)-primitive elements is similar). Since kw is a one-dimensional U-module, there is a map χ: U → k such that u·w = χ(u)w for all u ∈ U. Let v_{1},…, v_{n} be a basis of V consisting of eigenvectors of the action of x, i. e. x · v_{j} = λ_{j}v_{j} for j = 1,…, n and λ_{j} ∈ k. Then

x·(w⊗v_{j}) = (g·w)⊗(x·v_{j})+(x·w)⊗v_{j} = χ(g)w⊗λ_{j}v_{j}+χ(x)w⊗v_{j} = (χ(g)λ_{j}+χ(x))w⊗v_{j},

whence the set {w ⊗ v_{1},…, w ⊗ v_{n}} is a basis of kw ⊗ V consisting of eigenvectors of x. □

5 Application to U_{q}(𝔤)

As in the last section, we assume that k is an algebraically closed field of characteristic 0. Let (a_{ij}) be an n × n matrix with integer coefficients such that a_{ii} = 2 for all i, a_{ij} ≠ 0 for all i ≠ j, and there are relatively prime integers d_{1}, …, d_{n} ∈ {1, 2, 3} such that (d_{i}a_{ij}) is a symmetric positive definite matrix. Thus (a_{ij}) is the Cartan matrix of a finite dimensional semisimple Lie algebra 𝔤. Let q ∈ k \ {0} be not a root of 1 and define q_{i} := q^{d_{i}}. The standard q-deformation U_{q}(𝔤) is the algebra generated by E_{i}, F_{i}, K_{i}, K_{i}^{−1} subject to the following relations

K_{i}K_{i}^{−1} = 1 = K_{i}^{−1}K_{i}, K_{i}K_{j} = K_{j}K_{i},
K_{i}E_{j}K_{i}^{−1} = q_{i}^{a_{ij}}E_{j}, K_{i}F_{j}K_{i}^{−1} = q_{i}^{−a_{ij}}F_{j},

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K_{i} − K_{i}^{−1}}{q_{i} − q_{i}^{−1}}

for all i, j, and the q-deformed Serre relation (see [J, 5.1.1(vi)] between the E_{i}’s resp. the F_{i}’s which we do not need explicitly. U_{q}(𝔤) is a Hopf algebra where all the elements K_{i} are group-like and

Δ(E_{i}) = 1 ⊗ E_{i} + E_{i} ⊗ K_{i}, Δ(F_{i}) = K_{i}^{−1} ⊗ F_{i} + F_{i} ⊗ 1

for all i. For 1 ≤ i ≤ n let U_{i} be the subalgebra of U_{q}(𝔤) generated by K_{i}, K_{i}^{−1}, E_{i}, F_{i}. Then U_{i} ≅ U_{q_{i}}(sl(2)) as Hopf algebras. Let C be the semisimple tensor category of all finite dimensional left U_{q}(𝔤)-modules of type 1, that is all eigenvalues of the left multiplication with K_{i} for all i are of the form q^{m}, m ∈ ℤ (see for example [J, 4.3]). The dual Hopf algebra U_{q}(𝔤)_{C}^{0} is the q-deformed algebra of regular functions on the simply connected, connected semisimple algebraic group with Lie algebra 𝔤. Let x ∈ U_{q}(𝔤) be a (g, 1)-primitive element which is not a scalar multiple of g − 1. Then there is some i such that g = K_{i}^{−1} and x ∈ U_{i} is a k-linear combination of K_{i}^{−1} − 1, E_{i}K_{i}^{−1} and F_{i} [CM, Theorem A]. We call x semisimple if multiplication with x is a diagonalizable operator on all finite dimensional left U_{q}(𝔤)-modules.

Lemma 5.1 For any 1 ≤ i ≤ n let V be a finite dimensional left U_{i}-module of type 1 and x ∈ U_{i} ⊂ U_{q}(𝔤) a (K_{i}^{−1}, 1)-primitive element.

(1) There exists a finite dimensional left U_{q}(𝔤)-module W of type 1 and a one-dimensional left U_{q}(𝔤)-module ka with E_{j}a = 0, F_{j}a = 0 and K_{j}a = α_{j}a where α_{j} ∈ {1, −1} for all 1 ≤ j ≤ n such that V is isomorphic to a U_{i}-submodule of ka ⊗ W and α_{i} = 1.

(2) x is semisimple in U_{i} if and only if x is semisimple in U_{q}(𝔤).

Proof:

(1) By [J, 10.1.14] or the classification of highest weight modules of U_{q}(𝔤), V is contained in a finite dimensional left U_{q}(𝔤)-module Ṽ. From the description of finite dimensional left U_{q}(𝔤)-modules it is known that Ṽ ≅ ka ⊗ W, where W is a left U_{q}(𝔤)-module of type 1 and E_{j}a = 0, F_{j}a = 0, K_{j}a = α_{j}a, α_{j} ∈ {1, −1} for all j (see [J, 4.3]). In remains to show that α_{i} = 1. Since V is of type 1, there is a non-zero v ∈ V such that K_{i}v = q^{m}v for some m ∈ ℤ. Let a ⊗ w be the image of v in ka ⊗ W. Then

q^{m}(a ⊗ w) = K_{i}(a ⊗ w) = K_{i}a ⊗ K_{i}w = α_{i}a ⊗ K_{i}w,

hence K_{i}w = α_{i}^{−1}q^{m}w. Since W is of type 1, we conclude α_{i} = 1.

(2) If x is semisimple in U_{i}, then trivially x is semisimple as an element in U_{q}(𝔤). Conversely assume x is semisimple in U_{q}(𝔤). Let V be any finite dimensional left U_{i}-module. By [J, 10.1.14] or the classification of highest weight modules of U_{q}(𝔤), V is contained in a finite dimensional left U_{q}(𝔤)-module Ṽ. Hence multiplication with x is diagonalizable on Ṽ and then on V, too.

□

Theorem 5.2 Let 1 ≤ i ≤ n, α, β, γ ∈ k, and x = α(K_{i}^{−1} − 1) + βK_{i}^{−1}E_{i} + γF_{i} ∈ U_{q}(𝔤) \ {0} a (K_{i}^{−1}, 1)-primitive element. Assume α^{2} + 4q^{−1}βγ/(q − q^{−1})^{2} ≠ 0. Let C be the tensor category of finite dimensional left U_{q}(𝔤)-modules of type 1. Define A := U_{q}(𝔤)_{C}^{0} and B := {a ∈ A | a · x = 0}. Then the following are equivalent:

(1) x is semisimple.

(2) There is no nonnegative integer n satisfying

α^{2} + βγq^{−1} (\frac{q^{n} + q^{−n}}{q − q^{−1}} )^{2} = 0.

(3) A/AB^{+} is spanned by group-like elements

(3)′ A/B^{+}A is spanned by group-like elements

(4) A is faithfully flat as a left B-module (4)′ A is faithfully flat as a right B-module

(5) B is a B-direct summand in A as a left B-module

(5)′ B is a B-direct summand in A as a right B-module

(6) B is simple in M_{B}^{A}

(6)′ B is simple in _{B}M^{A}.

Proof: (1) ⟺ (2): By 5.1(2) and section 3.

(1) ⇒ (3) and (3) ⇒ (4) follow from 2.4.

(4) ⇒ (5) follows from 1.2.

(5) ⇒ (6) is 1.3(2).

(6) ⇒ (1): By 2.4(2), assumption (6) implies B = {a ∈ A | ∃n ≥ 1: a · x^{n} = 0}. Let W be any finite dimensional simple left U_{q}(𝔤)-module of type 1. By 2.1, part (2) and (3), W ⊗ W^{∗} ≅ C^{W} as right (and left) U_{q}(𝔤)-modules. Hence if f · x^{n} = 0 for some n ≥ 1 and f ∈ W^{∗}, then f · x = 0. Or equivalently, if ϕ: W^{∗} → W^{∗} is right multiplication with x, then Ker(ϕ) = Ker(ϕ^{n}) for all n ≥ 1. If ψ: W → W is left multiplication with x, then ψ^{∗} = ϕ, and we get im(ψ) = im(ψ^{n}), or equivalently, Ker(ψ) = Ker(ψ^{n}) for all n ≥ 1. Thus for all simple modules in C, hence for all modules W in C (because C is semisimple), we have shown

{w ∈ W | x · w = 0} = {w ∈ W | ∃n ≥ 1: x^{n} · w = 0}.

We want to show the same statement over U_{i}. Let V be a finite dimensional left U_{i}-module of type 1. By 5.1(1), V is isomorphic to a U_{i}-submodule of ka ⊗ W, where W is a left U_{q}(𝔤)-module of type 1 and E_{i}a = F_{i}a = 0, K_{i}a = a. Hence

xa = α(K_{i}^{−1} − 1)a + βK_{i}^{−1}E_{i}a + γF_{i}a = 0,

and the action of x on any element a ⊗ w, w ∈ W, is given by

x(a ⊗ w) = K_{i}^{−1}a ⊗ xw + xa ⊗ w since Δ(x) = K_{i}^{−1} ⊗ x + x ⊗ 1,
= a ⊗ xw.

In particular,

{v ∈ V | x · v = 0} = {v ∈ V | ∃n ≥ 1: x^{n} · v = 0},

since this equality holds for W. Thus we see that condition (2) of theorem 4.4 is satisfied (here we use the assumption α^{2} +4q^{−1}βγ/(q −q^{−1})^{2} ≠ 0). Hence by 4.4(1), multiplication with x is diagonalizable on all U_{q_{i}}(sl(2))-modules of type 1, and x is diagonalizable as an element in U_{q_{i}}(sl(2)) by 4.6. Define y := xK_{i}. Then y is (1, K_{i})-primitive, and B = {a | a · y = 0}. We now repeat the previous arguments with x replaced by y. Consider the statement (1)′ y is semisimple. We have shown in 4.5(1) and 4.6 that x is semisimple in U_{i} if and only if y is semisimple in U_{i}. Hence (1) ⟺ (1)′ by 5.1(2). (1)′ ⇒ (3)′ and (3)′ ⇒ (4)′ follow from 2.4. (4)′ ⇒ (5)′ follows from 1.2 (for A^{op}). (5)′ ⇒ (6)′ is 1.3(2) (for A^{op}). (6)′ ⇒ (2): By 2.4(2), B = {a ∈ A | ∃n ≥ 1: a · y^{n} = 0}. Hence for all W ∈ C,

{w ∈ W | y · w = 0} = {w ∈ W | ∃n ≥ 1: y^{n} · w = 0}.

Let V be a finite dimensional U_{i}-module of type 1. Then by 5.1(1), V is a U_{i}-submodule of ka ⊗ W for some left U_{q}(𝔤)-module W and E_{i} · a = 0, F_{i} · a = 0, K_{i} · a = a. Since ya = xa = 0, Δ(y) = 1 ⊗ y + y ⊗ K_{i}, and Δ(x) = K_{i}^{−1} ⊗ x + x ⊗ 1, we have

y(a ⊗ w) = a ⊗ yw + ya ⊗ K_{i}w = a ⊗ yw.

Hence {v ∈ V | y ·v = 0} = {v ∈ V | ∃n ≥ 1: y^{n} ·v = 0}, since this holds for W. Therefore we get from 4.4 for y, that (2) is satisfied.

□

Remark 5.3 1. Now assume α^{2} +4q^{−1}βγ/(q −q^{−1})^{2} = 0. This actually gives just one exception:

• If α ≠ 0 consider x̃ = 2(K_{i}^{−1} − 1) + (q − q^{−1})K_{i}^{−1}E_{i} + (q^{2} − 1)F_{i}. Then

B = \{a ∈ A | a · x = 0\} = \{a ∈ A | a · \frac{2}{α}x = 0\}

and the Hopf algebra automorphism of U_{q}(𝔤)

(E_{j}, F_{j}, K_{j}) ↦ \{ \begin{matrix}(\frac{β2}{(q−q^{−1})α}E_{i}, \frac{γ2}{(q^{2}−1)α}F_{i}, K_{i}) & if j = i, \\ (E_{j}, F_{j}, K_{j}) & if j ≠ i \end{matrix}

maps \frac{2}{α}x to x̃ and induces an isomorphism of B to the algebra \{a ∈ A | a·x̃ = 0\} of infinitesimal invariants with respect to x̃.

• If α = βγ = 0 then x is a scalar multiple of K_{i}^{−1}E_{i} or F_{i}. Then x (and K_{i}x) acts nilpotently on all finite dimensional U_{i}-modules of type 1 and both x and K_{i}x cannot be semisimple. Moreover the numerical condition (2) does not hold. In this case 5.2 remains true.

2. In the case 𝔤 = sl_{2} the module structure is flat for any choice of α, β, γ. (It can be shown that A is the ascending union of free modules over B).

References

[B] T. Brzeziński, Quantum homogeneous spaces as quantum quotient spaces, J. Math. Phys. 37 (1996), 2388–2399

[CM] W. Chin, I. M. Musson, The Coradical Filtration for Quantized Enveloping Algebras, J. London Math. Soc. (2), 53 (1996), 50–62

[CP] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press 1994

[DG] M. Demazure, P. Gabriel, Groupes Algébriques, Tome I, Masson, Paris, 1970

[D] M. Dijkhuizen, Some remarks on the construction of quantum symmetric spaces, Acta Appl. Math. 44 (1996), 59–80

[DN] M. Dijkhuizen, M. Noumi, A family of quantum projective spaces and related qhypergeometric orthogonal polynomials, to appear in Trans. Amer. Math. Soc.

[J] A. Joseph, Quantum groups and theis primitive ideals, Springer, Berlin–New York, 1995

[K] C. Kassel, Introduction to Quantum Groups, GTM 155, Springer New York 1995

[Ko] M. Koppinen, Coideals subalgebras in Hopf algebras: freeness, integrals, smash products, Comm. Algebra 21 (1993), 427–444

[KD] T. K. Koornwinder, M. Dijkhuizen, Quantum Homogeneous Spaces, Quantum Duality and Quantum 2-Spheres, Geom. Dedicata 52 (1994), 291–315

[KS] A. U. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer, Berlin 1997 [KV] L. I. Korogodsky, L. L. Vaksman, Quantum G-spaces and Heisenberg algebra, in:

Quantum Groups, Proceedings of Workshops held in the Euler International Mathematical Institute 1990, P. P. Kulish (ed.), Lecture notes in Math. 1510, pp. 56–66, Springer, Berlin, 1992

[M1] A. Masuoka, Quotient theory of Hopf algebras, in: Advances in Hopf algebras, J. Bergen, S. Montgomery (eds.), Dekker, 1994

[M2] A. Masuoka, On Hopf algebras with cocommutative coradicals, J. Algebra 144 (1991), 451–466

[MW] A. Masuoka, D. Wigner, Faithful flatness of Hopf algebras, J. Algebra 170 (1994), 156–164

[M] S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Math. 82, Amer. Math. Soc. 1993

[Mü] E. Müller, Konstruktion von Rechtscoidealunteralgebren, Diploma Thesis, Munich 1995

[NM] M. Noumi, K. Mimachi, Askey-Wilson Polynomials as spherical functions on SU_{q}(2), in: Quantum Groups, Proceedings of Workshops held in the Euler International Mathematical Institute 1990, P. P. Kulish (ed.), Lecture notes in Math. 1510, pp. 98–103, Springer, Berlin, 1992

[P] P. Podleś, Quantum Spheres, Lett. Math. Phys. 14 (1987), 193–202

[R] L. Rowen, Ring Theory, Vol. I, Academic Press, Boston, 1988

[Sch] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), 167–195

[Sw] Sweedler, Hopf algebras, Benjamin New York 1969

[T1] M. Takeuchi, Relative Hopf modules—equivalences and freeness criteria, J. Algebra 60 (1979), 452–471

[T2] M. Takeuchi, Hopf algebra techniques applied to the quantum group U_{q}(sl(2)), Contemp. Math. 134 (1992), 309–323