Levi–Civita Connections on Quantum Spheres

Abstract

We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi–Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.

1 Introduction

In recent years, a lot of progress has been made in understanding Riemannian aspects of noncommutative geometry. These are not only mathematically interesting, but also important in physics where noncommutative geometry is expected to play a key role, notably in a theory of quantum gravity. In Riemannian geometry the Levi–Civita connection and its curvature have a central role, and it turns out that there are several different ways of approaching these objects in the noncommutative setting (see e.g. [2, 4, 5, 7,8,9,10, 12, 14, 18, 21]).

From an algebraic perspective, the set of vector fields and the set of differential forms are (finitely generated projective) modules over the algebra of functions, a viewpoint which is also adopted in noncommutative geometry. However, considering vector fields as derivations does not immediately carry over to noncommutative geometry, since the set of derivations of a (noncommutative) algebra is in general not a module over the algebra but only a module over the center of the algebra. Therefore, one is led naturally to focus on differential forms and define a connection on a general module as taking values in the tensor product of the module with the module of differential forms. More precisely, let M be a (right) $\mathcal {A}$-module and let $\Omega ^1(\mathcal {A})$ denote a module of differential forms together with a differential $d:\mathcal {A}\rightarrow \Omega ^1(\mathcal {A})$. A connection on M is a linear map $\nabla :M\rightarrow M\otimes \Omega ^1(\mathcal {A})$ satisfying a version of Leibniz rule

$$\begin{aligned} \nabla (mf) = (\nabla m)f + m\otimes df \end{aligned}$$

(1.1)

for $f\in \mathcal {A}$ and $m\in M$. In differential geometry, for a vector field X one obtains a covariant derivative $\nabla _X:M\rightarrow M$, by pairing differential forms with X (as differential forms are dual to vector fields). In a noncommutative version of the above, there is in general no canonical way of obtaining a “covariant derivative” $\nabla _X:M\rightarrow M$. In a derivation based approach to noncommutative geometry (see e.g. [13, 14]), one puts emphasis on the choice of a Lie algebra $\mathfrak {g}$ of derivations of the algebra $\mathcal {A}$. Given a (right) $\mathcal {A}$-module M one defines a connection as a map $\nabla :\mathfrak {g}\times M\rightarrow M$, usually writing $\nabla (\partial ,m) = \nabla _\partial m$ for $\partial \in \mathfrak {g}$ and $m\in M$, satisfying

$$\begin{aligned} \nabla _{\partial }(mf) = (\nabla _{\partial }m)f + m\,\partial (f) \end{aligned}$$

for $f\in \mathcal {A}$ and $m\in M$, in parallel with (1.1). We stress that in general $\mathfrak {g}$ is not a module over $\mathcal {A}$ when $\mathcal {A}$ is not commutative. Thus we do not require $\mathcal {A}$-linearity in the argument $\partial$ of $\nabla _{\partial }$. This is in contrast with the braided geometry framework [4, 6] where for a braided commutative algebra the braided Lie algebra of its braided derivations is a module over the algebra and such a $\mathcal {A}$-linearity on the connection can be stated. Braided commutativity of a Hopf algebra is a feature of its being cotriangular (and not just coquasitriangular).

For quantum groups, it turns out that natural analogues of vector fields are not quite derivations, but rather maps satisfying a twisted Leibniz rule. For instance, as we shall see, for the quantum 3-sphere $S^{3}_{q}$ one defines maps $X_a:S^{3}_{q}\rightarrow S^{3}_{q}$ satisfying

$$\begin{aligned} X_a(fg) = X_a(f) \sigma _a(g) + f X_a(g) \end{aligned}$$

(1.2)

for $f,g\in S^{3}_{q}$, and $\sigma _a:S^{3}_{q}\rightarrow S^{3}_{q}$, for $a=1,2,3$, are algebra morphisms. In this note we explore the possibility of introducing a corresponding q-affine connection on a (right) $S^{3}_{q}$-module M. Motivated by (1.2) we introduce a covariant derivative $\nabla _{X_a}:M\rightarrow M$ such that

$$\begin{aligned} \nabla _{X_a}(mf) = (\nabla _{X_a}m)\sigma _a(f) + m X_a(f) \end{aligned}$$

for $f\in S^{3}_{q}$ and $m\in M$. In the following, we make these ideas precise and prove that there exist q-affine connections on projective modules. Again, we will not ask for $S^{3}_{q}$-linearity in the argument X of $\nabla _X$. Furthermore, we introduce a condition for metric compatibility, and in the particular case of a left covariant calculus over $S^{3}_{q}$, we investigate a derivation based definition of torsion. Then we explicitly construct a Levi–Civita connection, that is a torsion free and metric compatible connection. Moreover, we construct metric connections on a class of projective modules over the quantum 2-sphere. We mention that the Riemannian geometry of quantum spheres was studied [7] from the point of view of a bimodule connection on differential forms satisfying (1.1) as well as a right Leibniz rule twisted by a braiding map. In a final section we sketch a way to generalise (some of) the constructions of the present paper to any Hopf algebra with a (left) covariant differential calculus and corresponding quantum tangent space of twisted derivations.

The present paper is an alternative and extended version of the paper [1] where the left module structure of differential forms was used to construct q-affine connections, rather than the right module structure considered in the following.

2 The Quantum 3-Sphere

In this section we recall a few basic properties of the quantum 3-sphere [22]. The algebra $S^{3}_{q}$ is a unital $*$-algebra generated by $a,a^*,c,c^*$ fulfilling

$$\begin{aligned}&ac = qca \quad c^{*}a^{*} =qa^{*}c^{*} \quad ac^{*} =qc^{*}a, \\&ca^{*} =qa^{*}c \quad cc^{*} = c^{*}c \quad a^{*}a+c^{*}c = aa^{*}+q^{2}cc^{*} =\mathbb {1}, \end{aligned}$$

for a real parameter q. The identification of $S^{3}_{q}$ with the quantum group $SU_q(2)$ is via the Hopf algebra structure given by

$$\begin{aligned}&\Delta (a) = a\otimes a-qc^*\otimes c \quad \Delta (c) = c\otimes a+a^*\otimes c,\\&\Delta (a^*) = -qc\otimes c^*+a^*\otimes a^*\quad \Delta (c^*) = a\otimes c^*+c^*\otimes a^*, \end{aligned}$$

with antipode and counit

$$\begin{aligned}&S(a) = a^*\quad S(c) = -qc \quad \epsilon (a)=1 \quad \epsilon (c)=0, \\&S(a^*) = a \quad S(c^*) = -q^{-1}c^*\quad \epsilon (a^*)=1 \quad \epsilon (c^*)=0. \end{aligned}$$

We also need the dual quantum enveloping algebra $\mathcal {U}_q(\text {su}(2))$, which is the $*$-algebra with generators $E,F,K,K^{-1}$ satisfying

$$\begin{aligned}&K^{\pm 1}E = q^{\pm 1}EK^{\pm 1}\quad K^{\pm 1}F = q^{\mp 1}FK^{\pm 1}\quad [E,F] = \frac{K^2-K^{-2}}{q-q^{-1}} . \end{aligned}$$

The corresponding Hopf algebra structure is given by the coproduct,

$$\begin{aligned} \Delta (E) = E\otimes K + K^{-1}\otimes E \quad \Delta (F) = F\otimes K + K^{-1}\otimes F \quad \Delta (K^{\pm 1}) = K^{\pm 1}\otimes K^{\pm 1} \end{aligned}$$

together with antipode and counit

$$\begin{aligned}&S(K) = K^{-1} \quad S(E) = -qE \quad S(F) = -q^{-1}F , \\&\epsilon (K) = 1 \quad \epsilon (E) = 0 \quad \epsilon (F) = 0. \end{aligned}$$

We recall that there is a unique bilinear pairing between $\mathcal {U}_q(\text {su}(2))$ and $S^{3}_{q}$ given by

$$\begin{aligned} \left\langle K^{\pm 1}, a \right\rangle = q^{\mp \, 1/2} , \quad \left\langle K^{\pm 1}, a^* \right\rangle = q^{\mp \, 1/2} , \quad \left\langle E,c \right\rangle = 1 , \quad \left\langle F,c^* \right\rangle = -q^{-1}, \end{aligned}$$

with the remaining pairings being zero.

The algebra $S^{3}_{q}$ is a noncommutative algebra which is not quasi-commutative. This stems from the Hopf algebra $SU_q(2)$ being coquasitriangular and not simply cotriangular. Dually, the Hopf algebra $\mathcal {U}_q(\text {su}(2))$ is quasitriangular and not triangular [16,  §8, §10].

The pairing above induces a $\mathcal {U}_q(\text {su}(2))$-bimodule structure (left and right actions) on $S^{3}_{q}$:

$$\begin{aligned} h\triangleleft f= f_{(1)}\left\langle h,f_{(2)} \right\rangle \quad \text {and}\quad f\triangleleft h = \left\langle h,f_{(1)} \right\rangle f_{(2)} \end{aligned}$$

(2.1)

for $h\in \mathcal {U}_q(\text {su}(2))$ and $f\in S^{3}_{q}$, with Sweedler’s notation $\Delta (f)=f_{(1)}\otimes f_{(2)}$ (and implicit sum). The $*$-structure on $\mathcal {U}_q(\text {su}(2))$, denoted here by $\dag$ (to distinguish it from the $*$-structure of the algebra), is given by $(K^{\pm 1})^\dag = K^{\pm 1}$ and $E^\dag = F$. The action of $\mathcal {U}_q(\text {su}(2))$ is compatible with the $*$-algebra structures in the following sense

$$\begin{aligned} h\triangleright f^*= \big (S(h)^\dag \triangleright f\big )^*\quad f^*\triangleleft h = \big (f\triangleleft S(h)^\dag \big )^*. \end{aligned}$$

(2.2)

Let us for convenience list the left and right actions of the generators:

$$\begin{aligned}&K^{\pm 1}\triangleright a^{n} = q^{\mp \frac{n}{2}}\,a^{n} \quad K^{\pm 1}\triangleright c^{n} = q^{\mp \frac{n}{2}}\,c^{n},\\&K^{\pm 1}\triangleright a^{*}\,^{n} = q^{\pm \frac{n}{2}}(a^{*})^{n}\quad K^{\pm 1}\triangleright c^{*}\,^{n} = q^{\pm \frac{n}{2}}(c^{*})^{n},\\&E\triangleright a^{n} = -q^{(3-n)/2} [n]a^{n-1}c^{*} \quad E\triangleright c^{n} = q^{(1-n)/2}[n]c^{n-1}a^*,\\&E\triangleright (a^{*})^{n} = 0 \quad E\triangleright (c^{*})^{n} = 0,\\&F\triangleright a^{n} = 0 \quad F\triangleright c^{n} = 0,\\&F\triangleright (a^{*})^{n} = q^{(1-n)/2}[n]c(a^{*})^{ n-1} \quad F\triangleright (c^{*})^{n} = -q^{-(1+n)/2}[n]a(c^{*})^{n-1}, \end{aligned}$$

and

$$\begin{aligned}&a^n\triangleleft K^{\pm 1} = q^{\mp \frac{n}{2}}a^n \quad (a^*)^n\triangleleft K^{\pm 1} = q^{\pm \frac{n}{2}}(a^*)^n,\\&c^n\triangleleft K^{\pm 1} = q^{\pm \frac{n}{2}}c^n \quad (c^*)^n\triangleleft K^{\pm 1} = q^{\mp \frac{n}{2}}(c^*)^n,\\&a^n\triangleleft F = q^{\frac{n-1}{2}}[n]ca^{n-1} \quad (a^*)^n\triangleleft F = 0,\\&c^n\triangleleft F = 0 \quad (c^*)^n\triangleleft F = -q^{\frac{n-3}{2}}[n]a^*(c^*)^{n-1},\\&a^n\triangleleft E = 0 \quad (a^*)^n\triangleleft E = -q^{\frac{n-3}{2}}[n]c^*(a^*)^{n-1},\\&c^n\triangleleft E = q^{\frac{n-1}{2}}[n]c^{n-1}a \quad (c^*)^n\triangleleft E = 0, \end{aligned}$$

where $[n] = (q^n-q^{-n})/(q-q^{-1})$.

2.1 The Covariant Calculus and the Quantum Tangent Space

It is well known [22] that there is a left covariant (first order) differential calculus on $S^{3}_{q}$, denoted by $\Omega ^1(S^{3}_{q})$, generated as a left $S^{3}_{q}$-module by

$$\begin{aligned} \omega _1=\omega _{+}= a\,dc-qc\,da\quad \omega _2=\omega _{-}= c^*da^*-qa^*dc^*\quad \omega _3=\omega _{z}= a^*da + c^*dc . \end{aligned}$$

In fact, $\Omega ^1(S^{3}_{q})$ is a free left module with a basis given by $\{\omega _{+},\omega _{-},\omega _{z}\}$. Moreover, $\Omega ^1(S^{3}_{q})$ is a bimodule with respect to the relations

$$\begin{aligned}&\omega _{z}a = q^{-2}a\omega _{z} \quad \omega _{z}a^* = q^{2}a^*\omega _{z} \quad \omega _{z}c = q^{-2}c\omega _{z} \quad \omega _{z}c^* = q^{2}c^*\omega _{z}, \\&\omega _{\pm } a = q^{-1}a\omega _{\pm } \quad \omega _{\pm }a^* = q a^*\omega _{\pm }\quad \omega _{\pm }c = q^{-1}c\omega _{\pm } \quad \omega _{\pm }c^* = qc^*\omega _{\pm }, \end{aligned}$$

and, furthermore, $\Omega ^1(S^{3}_{q})$ is a $*$-bimodule with

$$\begin{aligned}&\omega _{+}^\dag = -\omega _{-}\quad \omega _{z}^\dag =-\omega _{z}\end{aligned}$$

satisfying $(f\omega g)^\dagger =g^*\omega ^\dag f^*$ for $f,g\in S^{3}_{q}$ and $\omega \in \Omega ^1(S^{3}_{q})$.

The differential $d:S^{3}_{q}\rightarrow \Omega ^1(S^{3}_{q})$ is computed using a dual basis $\{X_{+},X_{-},X_{z}\}$ of twisted derivations (the corresponding quantum tangent space [16,  §14.1.2]),

$$\begin{aligned} df = (X_{+}\triangleright f)\omega _{+}+ (X_{-}\triangleright f)\omega _{-}+ (X_{z}\triangleright f)\omega _{z}, \quad f\in S^{3}_{q}, \end{aligned}$$

(2.3)

with explicitly,

$$\begin{aligned} X_{+}= \sqrt{q}EK\quad X_{-}= \frac{1}{\sqrt{q}}FK\quad X_{z}= \frac{1-K^4}{1-q^{-2}} . \end{aligned}$$

Their twisted derivation properties are easily found. For $f,g\in S^{3}_{q}$, and $a=\pm , z$ one has,

$$\begin{aligned} X_a \triangleright fg = f(X_a \triangleright g) + (X_a\triangleright f)(\sigma _a \triangleright g) \end{aligned}$$

(and similarly for the right action), with

$$\begin{aligned} \sigma _{+}=\sigma _{-}=K^2\quad \text {and}\quad \sigma _{z}=K^4. \end{aligned}$$

Furthermore, these maps satisfy the following q-deformed commutation relations

$$\begin{aligned}&X_{-}X_{+}- q^2X_{+}X_{-}= X_{z}, \end{aligned}$$

(2.4)

$$\begin{aligned}&q^2X_{z}X_{-}-q^{-2}X_{-}X_{z}=(1+q^2) X_{-}, \end{aligned}$$

(2.5)

$$\begin{aligned}&q^2X_{+}X_{z}-q^{-2}X_{z}X_{+}=(1+q^2)X_{+}. \end{aligned}$$

(2.6)

As for the $*$-structures, one checks that $X_{\pm }^\dagger = X_{\mp }$ and $K^\dagger =K$. From this, using (2.2) one computes, for $f \in S^{3}_{q}$, that

$$\begin{aligned} \begin{aligned}&X_{\pm }\triangleleft f^* = -(K^{-2} X_{\mp }\triangleleft f)^* = - K^{2} \triangleleft (X_{\mp }\triangleleft f)^* \\&X_{z}\triangleleft f^* = -(K^{-4} X_{z}\triangleleft f)^* = - K^{4} \triangleleft (X_{z}\triangleleft f)^* . \end{aligned} \end{aligned}$$

(2.7)

In the classical limit of $q=1$, the above reduces to the Lie algebra of $\text {su}(2)$ and the calculus is the usual calculus on the sphere $S^3$ given in terms of left invariant one-forms.

3 q-Affine Connections

In differential geometry, a connection extends the action of derivatives to vector fields, and for $S^{3}_{q}$ a natural set of (q-deformed) derivations is given by $\{X_{+},X_{-},X_{z}\}$. In this section, we will introduce a framework extending the action of $X_a$ to a connection on $S^{3}_{q}$-modules. Let us first define the set of q-deformed derivations we shall be interested in.

Definition 3.1

The quantum tangent space of $S^{3}_{q}$ is defined as

$$\begin{aligned} TS^{3}_{q}= \mathbb {C}\left\langle X_{+},X_{-},X_{z} \right\rangle , \end{aligned}$$

that is the complex vector space generated by $X_a$ for $a=\pm , z$.

We point out that $TS^{3}_{q}$ is not a module over $S^{3}_{q}$.

Considering $TS^{3}_{q}$ to be the analogue of a (complexified) tangent space of $S^{3}_{q}$, we would like to introduce a covariant derivative $\nabla _X$ on a (right) $S^{3}_{q}$-module M, for $X\in TS^{3}_{q}$. Since the basis elements of $TS^{3}_{q}$ act as q-deformed derivations, the connection should obey an analogous q-deformed Leibniz rule. The motivating example is when $M=S^{3}_{q}$ and the action of $TS^{3}_{q}$ is simply $\nabla _{X}f = X< (f) = X(f)$ for $X\in TS^{3}_{q}$ and $f\in S^{3}_{q}$. (To lighten notation, in the following we shall drop the symbol < for the left action when there is no risk of ambiguities.)

In fact, let us be slightly more general and consider the action on a free module of rank n. Thus, we let M be a free right $S^{3}_{q}$-module with basis $\{e_i\}_{i=1}^n$, and write an arbitrary element $m\in M$ as $m=e_im^i$ for $m^i\in S^{3}_{q}$, implicitly assuming a summation over i from 1 to n.

Let us define $\nabla ^0:TS^{3}_{q}\times M\rightarrow M$ by setting

$$\begin{aligned} \nabla ^0_{X_a}(m) = e_iX_a(m^i) \end{aligned}$$

(3.1)

for $m=e_im^i\in M$ (and extending it linearly to all of $TS^{3}_{q}$). Now, it is easy to check that

$$\begin{aligned}&\nabla ^0_{X_a}(mf) = (\nabla ^0_{X_a}m)\sigma _a(f) + mX_a(f) \end{aligned}$$

for $f\in S^{3}_{q}$ and $m\in M$. Let us generalize these concepts to arbitrary right $S^{3}_{q}$-modules.

Definition 3.2

Let M be a right $S^{3}_{q}$-module. A right q-affine connection on M is a map $\nabla :TS^{3}_{q}\times M\rightarrow M$ such that

• (1) $\nabla _{X}(\lambda _1m_1+\lambda _2m_2) = \lambda _1 \nabla _X m_1 + \lambda _2\nabla _Xm_2$,

• (2) $\nabla _{\lambda _1 X+\lambda _2Y}m = \lambda _1\nabla _Xm + \lambda _2\nabla _Ym$,

• (3) $\nabla _{X_a}(mf) = (\nabla _{X_a}m)\sigma _a(f)+mX_a(f)$,    $a=\pm , z$,

for $m,m_1,m_2\in M$, $f\in S^{3}_{q}$, $X\in TS^{3}_{q}$ and $\lambda _1,\lambda _2\in \mathbb {C}$.

Remark 3.3

As mentioned previously, the space $TS^{3}_{q}$ is not a module over $S^{3}_{q}$. Thus we are not requiring $S^{3}_{q}$-linearity ‘in the first argument’, that is we are not requiring the connection to satisfy the relation $\nabla _{X f} \, m = (\nabla _X \, m) f)$ for $f\in S^{3}_{q}$. This is in contrast with what happens in braided geometry [4, 6] where for a braided commutative algebra the braided Lie algebra of its braided derivations is a module over the algebra and such a relation on the connection can be stated. Braided commutativity of a Hopf algebra is a feature of its being cotriangular.

Definition 3.4

A Hermitian form on a right $S^{3}_{q}$-module M is a map $h:M\times M\rightarrow S^{3}_{q}$ such that

$$\begin{aligned}&h(m_1,m_2f)=h(m_1,m_2)f \quad h(m_1,m_2)^*= h(m_2,m_1) ,\\&h(m_1+m_2,m_3) = h(m_1,m_3)+h(m_2,m_3), \end{aligned}$$

for $f\in S^{3}_{q}$ and $m_1,m_2,m_3\in M$. Moreover, h is said to be invertible if the induced map $\hat{h}:M\rightarrow M^*$, defined by $\hat{h}(m_1)(m_2)=h(m_1,m_2)$, is bijective.

On a free module with basis $\{e_i\}_{i=1}^n$, a Hermitian form is given by $h_{ij}=h_{ji}^*\in S^{3}_{q}$ by setting

$$\begin{aligned} h(m_1,m_2) = (m_1^i)^*h_{ij}m_2^j \end{aligned}$$

for $m_1=e_im_1^i\in (S^{3}_{q})^n$ and $m_2=e_im_2^i\in (S^{3}_{q})^n$. Moreover, if h is invertible, then there exist $h^{ij}\in S^{3}_{q}$ such that $h^{ij}h_{jk}=\delta ^i_k\mathbb {1}$. In case the module is projective (but not necessarily free) and generated by $\{e_i\}_{i=1}^n$, one can find $h^{ij}\in S^{3}_{q}$ such that $e_ih^{ij}h_{jk}=e_k$ if the Hermitian form is invertible (see e.g. [3]).

Next, we will introduce a notion of compatibility between a q-affine connection and a Hermitian form. To motivate Definition 3.5, let us study the case of free modules. For the q-affine connection $\nabla ^0$ in (3.1), one finds that

$$\begin{aligned} X_{+}\big ( h(m_1,m_2)\big )&= X_{+}\big ((m_1^i)^*h_{ij}m_2^j\big ) \\&= (m_1^i)^*X_{+}(h_{ij}m_2^j) + X_{+}\big ((m_1^i)^*\big )K^2(h_{ij}m_2^j)\\&= (m_1^i)^*h_{ij}X_{+}(m_2^j) + (m_1^i)^*X_{+}(h_{ij})K^2(m_2^j)\\&\quad +X_{+}\big ((m_1^i)^*\big )K^2(h_{ij}m_2^j). \end{aligned}$$

For the connection $\nabla ^0$, a natural requirement for the compatibility with h is to demand that $X_{+}(h_{ij})=0$. Then, from (2.7) $X_{+}(f^*)=-(K^{-2}X_{-}(f))^*= -K^{2}(X_{-}(f))^*$, and one has,

$$\begin{aligned} X_{+}\big (h(m_1,m_2)\big )&= (m_1^i)^*h_{ij}X_{+}(m_2^j) +X_{+}\big ((m_1^i)^*\big )K^2(h_{ij}m_2^j)\\&=(m_1^i)^*h_{ij}X_{+}(m_2^j) - \big (K^{-2}X_{-}(m_1)\big )^*K^2(h_{ij}m_2^j)\\&= (m_1^i)^*h_{ij}X_{+}(m_2^j) - K^2\big (X_{-}(m_1)^*\big ) K^2(h_{ij}m_2^j)\\&= (m_1^i)^*h_{ij}X_{+}(m_2^j) - K^2\big (X_{-}(m_1)^*h_{ij}m_2^j\big )\\&= h\big (m_1,\nabla ^0_{X_{+}}m_2\big )-K^2\big (h(\nabla ^0_{X_{-}}m_1,m_2)\big ) . \end{aligned}$$

Corresponding formulas are easily worked out for $\nabla ^0_{X_{-}},\nabla ^0_{X_{z}}$, and we shall take this as a motivation for the following definition.

Definition 3.5

A q-affine connection $\nabla$ on a right $S^{3}_{q}$-module M is compatible with the Hermitian form $h:M\times M\rightarrow S^{3}_{q}$ if

$$\begin{aligned}&X_{+}\big (h(m_1,m_2)\big ) = h\big (m_1,\nabla _{X_{+}}m_2\big )-K^2\big (h(\nabla _{X_{-}}m_1,m_2)\big ), \end{aligned}$$

(3.2)

$$\begin{aligned}&X_{-}\big (h(m_1,m_2)\big ) = h\big (m_1,\nabla _{X_{-}}m_2\big )-K^2\big (h(\nabla _{X_{+}}m_1,m_2)\big ), \end{aligned}$$

(3.3)

$$\begin{aligned}&X_{z}\big (h(m_1,m_2)\big ) =h\big (m_1,\nabla _{X_{z}}m_2\big )-K^4\big (h(\nabla _{X_{z}}m_1,m_2)\big ), \end{aligned}$$

(3.4)

for $m_1,m_2\in M$.

Note that (3.2) and (3.3) are equivalent since

$$\begin{aligned} \Big (&X_{+}\big (h(m_2,m_1)\big ) -h(m_2,\nabla _{X_{+}}m_1)+K^2\big (h(\nabla _{X_{-}}m_2,m_1)\big ) \Big )^*\\&\quad =-K^{-2}\Big ( X_{-}\big (h(m_1,m_2)\big )+K^2\big (h(\nabla _{X_{+}}m_1,m_2)\big )-h(m_1,\nabla _{X_{-}}m_2) \Big ). \end{aligned}$$

In the case of a q-affine connection on a free module, one can derive a convenient parametrization of all connections that are compatible with a given Hermitian form. To this end, let us introduce some notation. Let $(S^{3}_{q})^n$ be a free right $S^{3}_{q}$-module with basis $\{e_i\}_{i=1}^n$. A q-affine connection $\nabla$ on $(S^{3}_{q})^n$ can be determined by specifying the Christoffel symbols

$$\begin{aligned} \nabla _{X_a}e_i = e_j\Gamma _{ai}^j, \end{aligned}$$

with $\Gamma _{ai}^j\in S^{3}_{q}$ for $a=\pm , z$ and $i,j=1,\ldots ,n$, and setting

$$\begin{aligned}&\nabla _{X_a}(e_im^i) = (\nabla _{X_a}e_i)\sigma _a(m^i) + e_iX_a(m^i) = e_j\big (\Gamma ^j_{ai}\sigma _a(m^i) + X_a(m^j)\big ). \end{aligned}$$

The next result gives the form of the Christoffel symbols for a q-affine connection compatible with an invertible Hermitian form on a free module.

Proposition 3.6

Let $(S^{3}_{q})^n$ be a free right $S^{3}_{q}$-module with a basis $\{e_i\}_{i=1}^n$ and let $\nabla$ be a q-affine connection on $(S^{3}_{q})^n$ given by the Christoffel symbols $\nabla _ae_i = e_j\Gamma _{ai}^j$. Furthermore, assume that h is an invertible Hermitian form on $(S^{3}_{q})^n$ and set $h_{ij}=h(e_i,e_j)$. Then $\nabla$ is compatible with h if and only if there exist $\gamma _{ij},\rho _{ij}\in S^{3}_{q}$ such that $\rho _{ij}^*=\rho _{ji}$ and

$$\begin{aligned}&\Gamma ^i_{+j} = h^{ik}\big (\tfrac{1}{2}X_{+}(h_{kj})+K(\gamma _{kj})\big ), \end{aligned}$$

(3.5)

$$\begin{aligned}&\Gamma ^i_{-j} = h^{ik}\big (\tfrac{1}{2}X_{-}(h_{kj})+K(\gamma _{jk}^*)\big ), \end{aligned}$$

(3.6)

$$\begin{aligned}&\Gamma ^i_{zj} = h^{ik}\big (\tfrac{1}{2}X_{z}(h_{kj})+K^2(\rho _{kj})\big ). \end{aligned}$$

(3.7)

Proof

Let us start by showing that if (3.2)–(3.4) in Definition 3.5 hold for a set of generators of the module, then the equations hold for all elements of the module. Thus, for $m_1=e_im_1^i$ and $m_2=e_jm_2^j$, one computes

$$\begin{aligned}&h\big (e_im_1^i,\nabla _{X_{+}}(e_jm_2^j)\big )-K^2\big (h(\nabla _{X_{-}}(e_im_1^i),e_jm_2^j)\big )\\&\quad =(m_1^i)^*h\big (e_i,(\nabla _{X_{+}}e_j)\sigma _{+}(m_2^j)+e_jX_{+}(m_2^j)\big )\\&\qquad -K^2\big (h((\nabla _{X_{-}}e_i)\sigma _{-}(m_1^i)+e_iX_{-}(m_1^i),e_j)m_2^j\big )\\&\quad = (m_1^i)^*h\big (e_i,\nabla _{X_{+}}e_j\big )\sigma _{+}(m_2^j)+ (m_1^i)^*h\big (e_i,e_j\big )X_{+}(m_2^j)\\&\qquad -K^2\big (\sigma _{-}(m_1^i)^*h(\nabla _{X_{-}}e_i,e_j)m_2^j\big ) -K^2\big (X_{-}(m_1^i)^*h(e_i,e_j)m_2^j\big ), \end{aligned}$$

and using that $\sigma _{+}=\sigma _{-}=K^2$, $K(f)^*=K^{-1}(f^*)$ and $X_{-}(f)^*=-K^{-2}X_{+}(f^*)$ one may rewrite the above expression as

$$\begin{aligned}&h\big (e_im_1^i,\nabla _{X_{+}}(e_jm_2^j)\big )-K^2\big (h(\nabla _{X_{-}}(e_im_1^i),e_jm_2^j)\big )\\&\quad =(m_1^i)^*\Big (h(e_i,\nabla _{X_{+}}e_j)-K^2\big (h(\nabla _{X_{-}}e_i,e_j)\big )\Big )K^2(m_2^j)\\&\qquad +(m_1^i)^*h\big (e_i,e_j\big )X_{+}(m_2^j) +X_{+}(m_1^*) K^2\big (h(e_i,e_j)m_2^j\big ). \end{aligned}$$

Now, assuming that (3.2) holds for $m_1=e_i$ and $m_2=e_j$, i.e.

$$\begin{aligned} X_{+}(h(e_i,e_j)) = h(e_i,\nabla _{X_{+}}e_j)-K^2\big (h(\nabla _{X_{-}}e_i,e_j)\big ), \end{aligned}$$

one obtains

$$\begin{aligned}&h\big (e_im_1^i,\nabla _{X_{+}}(e_jm_2^j)\big )-K^2\big (h(\nabla _{X_{-}}(e_im_1^i),e_jm_2^j)\big )\\&\quad =(m_1^i)^*X_{+}\big (h(e_i,e_j)\big )K^2(m_2^j) +(m_1^i)^*h\big (e_i,e_j\big )X_{+}(m_2^j) +X_{+}(m_1^*) K^2\big (h(e_i,e_j)m_2^j\big )\\&\quad = (m_1^i)^*X_{+}\big (h(e_i,e_j)m_2^j\big )+X_{+}(m_1^*) K^2\big (h(e_i,e_j)m_2^j\big )\\&\quad =X_{+}\big ((m_1^i)^*h(e_i,e_j)m_2^j\big ) = X_{+}\big (h(e_im_1^i,e_jm_2^j)\big ), \end{aligned}$$

by using that $fX_{+}(g)+X_{+}(f)\sigma _{+}(g)=X_{+}(fg)$. An analogous computation corresponding to (3.4) shows that one indeed only needs to check (3.2)–(3.4) for a set of generators.

It is then straight forward to check that the q-affine connection $\nabla$, defined by

$$\begin{aligned} \nabla _{X_a}e_i = e_j\Gamma ^j_{ai} \end{aligned}$$

with $\Gamma ^j_{+i},\Gamma ^j_{-i},\Gamma ^j_{zi}$ given by (3.5)–(3.7) defines a connection compatible with h. For instance,

$$\begin{aligned}&h\big (e_i,\nabla _{X_{+}}e_j\big )-K^2\big (h(\nabla _{X_{-}}e_i,e_j)\big ) =h_{ik}\Gamma ^k_{+j}-K^2\big ((\Gamma ^k_{-i})^*h_{kj}\big )\\&\quad =h_{ik}\Gamma ^k_{+j}-K^2\big ((h_{jk}\Gamma ^k_{-i})^*\big )\\&\quad =h_{ik}h^{kl}\big (\tfrac{1}{2}X_{+}(h_{lj})+K(\gamma _{lj})\big ) -K^2\Big (\big (h_{jk}h^{kl}\left( \tfrac{1}{2}X_{-}(h_{li})+K(\gamma _{il}^*)\right) \big )^*\Big )\\&\quad =\tfrac{1}{2}X_{+}(h_{ij})+K(\gamma _{ij}) -K^2\big (\tfrac{1}{2}X_{-}(h_{ji})^*+K(\gamma _{ij}^*)^*\big )\\&\quad =\tfrac{1}{2}X_{+}(h_{ij})+K(\gamma _{ij}) -K^2\big (-\tfrac{1}{2}K^{-2}X_{+}(h_{ij})+K^{-1}(\gamma _{ij})\big )\\&\quad =\tfrac{1}{2}X_{+}(h_{ij})+K(\gamma _{ij}) +\tfrac{1}{2}X_{+}(h_{ij})-K(\gamma _{ij}) = X_{+}(h_{ij}). \end{aligned}$$

Conversely, assume that the connection $\nabla$ is compatible with h, and write $\nabla _{X_a}e_i=e_j\Gamma ^j_{ai}$. From the compatibility condition (3.2) one finds that the Christoffel symbols satisfy

$$\begin{aligned}&X_{+}(h_{ij})= h_{ik}\Gamma ^k_{+j} - K^2\big ((\Gamma ^k_{-i})^*h_{kj}\big ) \\&=h_{ik}\Gamma ^k_{+j} - K^2\big ((h_{jk}\Gamma ^k_{-i})^*\big ) \\&= \Gamma _{+,ij} - K^{2}\big (\Gamma _{-,ji}^*\big ), \end{aligned}$$

with $\Gamma _{a,ij}=h_{ik}\Gamma ^k_{aj}$, which can be written as

$$\begin{aligned} \Gamma _{+,ij} = X_{+}(h_{ij})+K^2\big (\Gamma _{-,ji}^*\big ). \end{aligned}$$

(3.8)

Defining

$$\begin{aligned} \gamma _{ij} = K^{-1}(\Gamma _{-,ji})^*+\tfrac{1}{2}K^{-1}X_{+}(h_{ij}) \end{aligned}$$

it follows immediately that $\Gamma _{-,ij} = \tfrac{1}{2}X_{-}(h_{ij}) + K(\gamma _{ji}^*)$, and (3.8) implies that

$$\begin{aligned} \Gamma _{+,ij} = \tfrac{1}{2}X_{+}(h_{ij}) + K(\gamma _{ij}), \end{aligned}$$

giving (3.5) and (3.6) via $\Gamma ^i_{aj}=h^{ik}\Gamma _{a,kj}$. Similarly, (3.4) implies that

$$\begin{aligned} X_{z}(h_{ij}) = \Gamma _{z,ij}-K^4\big ((\Gamma _{z,ji})^*\big ), \end{aligned}$$

(3.9)

and defining

$$\begin{aligned} \rho _{ij} = K^{-2}(\Gamma _{z,ij})-\tfrac{1}{2}K^{-2}X_{z}(h_{ij}) \end{aligned}$$

it follows immediately that $\Gamma _{z,ij} = \tfrac{1}{2}X_{z}(h_{ij})+K^2(\rho _{ij})$, and (3.9) implies that $\rho _{ij}=\rho _{ji}^*$. $\square$

Thus the previous proposition gives the general class of q-affine connections which are compatible with an invertible Hermitian form on a free right $S^{3}_{q}$-module. Later on in Sect. 4, on the right free $S^{3}_{q}$-module $\Omega ^1(S^{3}_{q})$ we shall select a subclass of these that are also torsion free.

3.1 q-Affine Connections on Projective Modules

As expected, one can construct q-affine connections on projective modules. More precisely, one proves the following result.

Proposition 3.7

Let M be a right $S^{3}_{q}$-module and let $\nabla$ be a q-affine connection on M. Given a projection on M,  i.e. an endomorphism $p:M\rightarrow M$ such that $p^2=p,$ then $p\circ \nabla$ is a q-affine connection on the right $S^{3}_{q}$-module p(M).

Proof

Since $\nabla$ is a q-affine connection and p is an endomorphism, it is immediate that $\widetilde{\nabla }=p\circ \nabla$ satisfies properties (3.2) and (3.2) in Definition 3.2. Moreover, for $m\in p(M)$

$$\begin{aligned} \widetilde{\nabla }_{X_a}(m f)&=p\big (\nabla _{X_a}(m f)\big ) = p\big ((\nabla _{X_a}m) \, \sigma _a(f) \big ) + p \big ( m X_a(f) \big )\\&=(\widetilde{\nabla }_{X_a}m) \, \sigma _a(f) + m X_a(f), \end{aligned}$$

since $p(m)=m$ when $m\in p(M)$. We conclude that $\widetilde{\nabla }$ is a q-affine connection on p(M). $\square$

Since we have shown in the previous section that q-affine connections exist on free modules, Proposition 3.7 implies that every projective $S^{3}_{q}$-module can be equipped with a q-affine connection. Moreover, let $\nabla$ and $\widetilde{\nabla }$ be q-affine connections on a $S^{3}_{q}$-module M and define

$$\begin{aligned} \alpha (X,m) = \nabla _Xm-\widetilde{\nabla }_Xm. \end{aligned}$$

Then $\alpha :TS^{3}_{q}\times M\rightarrow M$ satisfies

$$\begin{aligned}&\alpha (\lambda X+\mu Y,m_1) = \lambda \alpha (X,m_1)+\mu \alpha (Y,m_1), \end{aligned}$$

(3.10)

$$\begin{aligned}&\alpha (X, m_1 f + m_2 g) = \alpha (X,m_1) f + \alpha (X,m_2) g, \end{aligned}$$

(3.11)

for $m_1,m_2\in M$, $X,Y\in TS^{3}_{q}$, $f,g\in S^{3}_{q}$ and $\lambda ,\mu \in \mathbb {C}$. Conversely, every q-affine connection on a projective module M can be written as

$$\begin{aligned} \nabla _Xm = p(\nabla ^0_X m) + \alpha (X,m), \end{aligned}$$

where $\nabla ^0$ is the connection defined in (3.1) and $\alpha :TS^{3}_{q}\times M\rightarrow M$ is an arbitrary map satisfying (3.10) and (3.11). Next, let us show that a connection on a projective module is compatible with the restricted metric if the projection is orthogonal.

Proposition 3.8

Let $\nabla$ be a q-affine connection on the $S^{3}_{q}$-module M and assume furthermore that $\nabla$ is compatible with a Hermitian form h on M. If $p:M\rightarrow M$ is an orthogonal projection, i.e. p is a projection such that, for all $m_1,m_2\in M,$

$$\begin{aligned} h\big (p(m_1),m_2\big ) = h\big (m_1,p(m_2)\big ) \end{aligned}$$

then $\widetilde{\nabla }=p\circ \nabla$ is a q-affine connection on p(M) that is compatible with h restricted to p(M).

Proof

First of all, it follows from Proposition 3.7 that $\widetilde{\nabla }=p\circ \nabla$ is a q-affine connection on p(M). Since p is an orthogonal projection, one finds that for $m_1,m_2\in p(M)$

$$\begin{aligned}&h\big (m_1,\widetilde{\nabla }_{X_{+}}m_2\big )-K^2\big (h(\widetilde{\nabla }_{X_{-}}m_1,m_2)\big )\\&\quad = h\big (m_1,p(\nabla _{X_{+}}m_2)\big )-K^2\big (h(p(\nabla _{X_{-}}m_1),m_2)\big )\\&\quad =h\big (p(m_1),\nabla _{X_{+}}m_2\big )-K^2\big (h(\nabla _{X_{-}}m_1,p(m_2))\big )\\&\quad =h\big (m_1,\nabla _{X_{+}}m_2\big )-K^2\big (h(\nabla _{X_{-}}m_1,m_2)\big ) = X_{+}\big (h(m_1,m_2)\big ) \end{aligned}$$

by using that $\nabla$ is compatible with h. A similar computation shows that

$$\begin{aligned} X_{z}\big (h(m_1,m_2)\big ) = h\big (m_1,\widetilde{\nabla }_{X_{z}}m_2\big )-K^4\big (h(\widetilde{\nabla }_{X_{z}}m_1,m_2)\big ), \end{aligned}$$

from which we conclude that $\widetilde{\nabla }$ is compatible with h restricted to p(M). $\square$

4 A q-Affine Levi–Civita Connection on $\Omega ^1(S^{3}_{q})$

In this section we shall construct a q-affine connection on $\Omega ^1(S^{3}_{q})$, compatible with an invertible Hermitian form h and satisfying a certain torsion freeness condition. The module $\Omega ^1(S^{3}_{q})$ is a free $S^{3}_{q}$-module of rank 3 with basis $\omega _{+},\omega _{-},\omega _{z}$ which implies that the results of Proposition 3.6 may be used. Although $\Omega ^1(S^{3}_{q})$ has a bimodule structure, we shall only consider the right module structure of $\Omega ^1(S^{3}_{q})$ in what follows. In the case of a q-affine connection on $\Omega ^1(S^{3}_{q})$, there is a natural definition of torsion freeness, suggested by the relations (2.4)–(2.6).

We have already mentioned that those relations reduce to the Lie algebra of $\text {su}(2)$ in the classical limit of $q=1$. These relations are reflected in the notion of the torsion T. For instance one would have $T(X_-,X_+):=\nabla _{-}X_{+}- \nabla _{+}X_{-}- [X_{-}, X_{+}]$ and its vanishing is just the condition $\nabla _{-}X_{+}- \nabla _{+}X_{-}= [X_{-}, X_{+}] = X_{z}$; for dual forms this translates into $\nabla _{-}\omega _{+}- \nabla _{+}\omega _{-}= \omega _{z}$. There are similar expressions the other two cases. Given the duality between the derivations $X_a$ and the basis forms $\omega _a$, for $a=\pm , z$, we propose the following definition for a torsion freeness condition on the connection.

Definition 4.1

A q-affine connection $\nabla$ on $\Omega ^1(S^{3}_{q})$ is torsion free if

$$\begin{aligned}&\nabla _{-}\omega _{+}-q^2\nabla _{+}\omega _{-}=\omega _{z}, \end{aligned}$$

(4.1)

$$\begin{aligned}&q^2\nabla _{z}\omega _{-}-q^{-2}\nabla _{-}\omega _{z}=(1+q^2)\omega _{-}, \end{aligned}$$

(4.2)

$$\begin{aligned}&q^2\nabla _{+}\omega _{z}-q^{-2}\nabla _{z}\omega _{+}=(1+q^2)\omega _{+}. \end{aligned}$$

(4.3)

In the following, we will construct a torsion free q-affine connection on $\Omega ^1(S^{3}_{q})$ that is compatible with a Hermitian form. We call a connection satisfying these conditions a q-affine Levi–Civita connection. As it turns out, for such connections to exist, the Hermitian form needs to satisfy a compatibility condition.

Deriving a family of metric and torsion free connections under some conditions undermines in general the classical uniqueness result for such a (Levi–Civita) connection. This seems to be a common feature of the study of linear connections in the framework of truly non commutative algebras (and not just braided-commutative). It is however interesting to see when and why metric and torsion free connections are unique or not. A natural question would then be under which additional ‘natural’ conditions is it possible to single out a canonical connection. This problem will be addressed elsewhere.

Proposition 4.2

Let h be an invertible Hermitian form on the (right) $S^{3}_{q}$-module $\Omega ^1(S^{3}_{q})$ and write $h_{ab}=h(\omega _a,\omega _b)$. A q-affine Levi–Civita connection on $\Omega ^1(S^{3}_{q})$ exists if and only if

$$\begin{aligned}&X_{z}(h_{++}-q^2h_{--}) =K^2X_{-}(h_{z+})-q^2X_{-}(h_{-z}) -q^2K^2X_{+}(h_{z-})+X_{+}(h_{+z}). \end{aligned}$$

(4.4)

Proof

Assume that h is an invertible Hermitian form on $\Omega ^1(S^{3}_{q})$ and write $h_{ab}=h(\omega _a,\omega _b)$ with inverse $h^{ab}$. Furthermore, we write $\nabla _a=\nabla _{X_a}$ and

$$\begin{aligned} \nabla _{a}\omega _b = \omega _c\Gamma _{ab}^c \end{aligned}$$

for $a,b=\pm , z$. In terms of $\Gamma _{a,bc}=h_{bp}\Gamma ^p_{ac}$ the torsion free equations (4.1)–(4.3) become

$$\begin{aligned}&\Gamma _{-,a+}-q^2\Gamma _{+,a-}=h_{az}, \end{aligned}$$

(4.5)

$$\begin{aligned}&q^{2}\Gamma _{z,a-}-q^{-2}\Gamma _{-,az}=(1+q^2)h_{a-}, \end{aligned}$$

(4.6)

$$\begin{aligned}&q^{2}\Gamma _{+,az}-q^{-2}\Gamma _{z,a+}=(1+q^2)h_{a+}. \end{aligned}$$

(4.7)

Since $\Omega ^1(S^{3}_{q})$ is a free (right) module, one can apply the results of Proposition 3.6 to obtain

$$\begin{aligned}&\Gamma _{+,ab} = \tfrac{1}{2}X_{+}(h_{ab})+K(\gamma _{ab}),\\&\Gamma _{-,ab} = \tfrac{1}{2}X_{-}(h_{ab})+K(\gamma _{ba}^*),\\&\Gamma _{z,ab} = \tfrac{1}{2}X_{z}(h_{ab})+K(\rho _{ab}) , \end{aligned}$$

for “parameters” $(\gamma _{ab}, \rho _{ab}=\rho _{ba}^*)$ in $S^{3}_{q}$, giving all q-affine connections compatible with h.

Inserting the above expressions into (4.5)–(4.7) gives

$$\begin{aligned}&\gamma _{+a}^*-q^2\gamma _{a-} = K^{-1}(h_{az}) -\tfrac{1}{2}K^{-1}X_{-}(h_{a+})+\tfrac{1}{2}q^2K^{-1}X_{+}(h_{a-})\equiv A_a,\\&q^2K(\rho _{a-})-q^{-2}\gamma _{za}^*=K^{-1}\Big [(1+q^2)h_{a-} -\tfrac{1}{2}q^2X_{z}(h_{a-})+\tfrac{1}{2}q^{-2}X_{-}(h_{az})\Big ]\equiv B_a,\\&q^2\gamma _{az}-q^{-2}K(\rho _{a+}) =K^{-1}\Big [(1+q^2)h_{a+}-\tfrac{1}{2}q^2X_{+}(h_{az})+\tfrac{1}{2}q^{-2}X_{z}(h_{a+})\Big ] \equiv C_a. \end{aligned}$$

Note that the right hand sides $A_a$, $B_a$ and $C_a$ only depend on the metric components $h_{ab}$.

The above nine equations can be grouped into three independent sets:

Group 1

Group 2

Group 3

where for notational convenience we denoted $\tilde{\rho }_{ab}=K(\rho _{ab})$.

The equations in Group 1 can be solved as

$$\begin{aligned}&\gamma _{++} = A_+^*+q^{2}\gamma _{+-}^*, \end{aligned}$$

(4.8)

$$\begin{aligned}&\gamma _{--} = q^{-2}\gamma _{+-}^*-q^{-2}A_-, \end{aligned}$$

(4.9)

and the equations in Group 2 can be solved as

$$\begin{aligned}&\gamma _{z+} = q^4\tilde{\rho }_{+-}^*-q^2B_+^*, \end{aligned}$$

(4.10)

$$\begin{aligned}&\rho _{z-} = q^{-2}K^{-1}(B_z)+q^{-4}K^{-1}(\gamma _{zz}^*), \end{aligned}$$

(4.11)

$$\begin{aligned}&\gamma _{-z} = q^{-2}C_-+q^{-4}\tilde{\rho }_{-+}, \end{aligned}$$

(4.12)

$$\begin{aligned}&\rho _{z+} = q^4K^{-1}(\gamma _{zz})-q^2K^{-1}(C_z). \end{aligned}$$

(4.13)

Note that the condition $\rho _{ab}^*=\rho _{ba}$ will not pose a problem here, since neither $\rho _{-z}$ nor $\rho _{+z}$ appear in any other equation, and may simply be defined as $\rho _{-z}=\rho _{z-}^*$ and $\rho _{+z}=\rho _{z+}^*$.

For the equations in Group 3, the fact that we require $\rho _{++}^*=\rho _{++}$ and $\rho _{--}=\rho _{--}^*$ gives a non-trivial condition for solutions to exist. From (G3.2) and (G3.3) one obtains

$$\begin{aligned}&\gamma _{z-} = q^4K^{-2}(\tilde{\rho }_{--})-q^2B_-^*, \end{aligned}$$

(4.14)

$$\begin{aligned}&\gamma _{+z} = q^{-2}C_++q^{-4}\tilde{\rho }_{++}, \end{aligned}$$

(4.15)

and inserted into (G3.1) this gives

$$\begin{aligned}&q^{-4}\rho _{++}-q^6\rho _{--} = K(A_z)-q^4K(B_-^*)-q^{-2}K(C_+^*)\quad \Leftrightarrow \quad \nonumber \\&\rho _{++} = q^{10}\rho _{--}+q^4K(A_z)-q^8K(B_-^*)-q^2K(C_+^*). \end{aligned}$$

(4.16)

A necessary (and sufficient) condition for solutions to exist, is that the right hand side of the above equation is Hermitian. From

$$\begin{aligned}&A_z = K^{-1}\Big [h_{zz} -\tfrac{1}{2}X_{-}(h_{z+})+\tfrac{1}{2}q^2X_{+}(h_{z-})\Big ],\\&B_- = K^{-1}\Big [(1+q^2)h_{--} -\tfrac{1}{2}q^2X_{z}(h_{--})+\tfrac{1}{2}q^{-2}X_{-}(h_{-z})\Big ],\\&C_+ = K^{-1}\Big [(1+q^2)h_{++}-\tfrac{1}{2}q^2X_{+}(h_{+z})+\tfrac{1}{2}q^{-2}X_{z}(h_{++})\Big ], \end{aligned}$$

one obtains

$$\begin{aligned} K(A_z)&= h_{zz} -\tfrac{1}{2}X_{-}(h_{z+})+\tfrac{1}{2}q^2X_{+}(h_{z-}),\\ K(B_-^*)&= (1+q^2)K^2(h_{--}) +\tfrac{1}{2}q^2K^{-2}X_{z}(h_{--})-\tfrac{1}{2}q^{-2}X_{+}(h_{z-})\\&= \tfrac{q^2}{2(1-q^{-2})}\big (K^2(h_{--})+K^{-2}(h_{--})\big ) -\tfrac{q^{-2}}{1-q^{-2}}K^2(h_{--})-\tfrac{1}{2}q^{-2}X_{+}(h_{z-}),\\ K(C_+^*)&= (1+q^2)K^2(h_{++})+\tfrac{1}{2}q^2X_{-}(h_{z+})-\tfrac{1}{2}q^{-2}K^{-2}X_{z}(h_{++})\\&= -\tfrac{q^{-2}}{2(1-q^{-2})}\big (K^2(h_{++})+K^{-2}(h_{++})\big )+\tfrac{q^2}{1-q^{-2}}K^2(h_{++}), +\tfrac{1}{2}q^2X_{-}(h_{z+}), \end{aligned}$$

by using that $X_{z}=(1-K^4)/(1-q^{-2})$.

Since $\rho _{--}$ and $h_{zz}$, as well as $K^2(h_{--})+K^{-2}(h_{--})$ and $K^2(h_{++})+K^{-2}(h_{++})$, are Hermitian, the non-Hermitian terms of (4.16), which we denote by S, become

$$\begin{aligned} S&= q^6X_{+}(h_{z-})-q^4X_{-}(h_{z+})+\tfrac{q^{6}}{1-q^{-2}}K^2(h_{--}) -\tfrac{q^4}{1-q^{-2}}K^2(h_{++}). \end{aligned}$$

Thus, a necessary and sufficient condition for $\rho _{++}$ to be Hermitian is that

$$\begin{aligned}&0= S-S^*= q^6X_{+}(h_{z-})+q^6K^{-2}X_{-}(h_{-z}) -q^4X_{-}(h_{z+})-q^4K^{-2}X_{+}(h_{+z})\\&\quad +\tfrac{q^{6}}{1-q^{-2}}K^2(h_{--})-\tfrac{q^{6}}{1-q^{-2}}K^{-2}(h_{--}) -\tfrac{q^4}{1-q^{-2}}K^2(h_{++})+\tfrac{q^4}{1-q^{-2}}K^{-2}(h_{++}). \end{aligned}$$

By using that $X_z=(1-K^4) (1-q^{-2})^{-1}$, the above condition can be written as

$$\begin{aligned}&q^6X_{+}(h_{z-})+q^6K^{-2}X_{-}(h_{-z})\\&\quad -q^4X_{-}(h_{z+})-q^4K^{-2}X_{+}(h_{+z}) +K^{-2}X_{z}\big (q^4h_{++}-q^6h_{--}\big ) = 0, \end{aligned}$$

which is equivalent to (4.4). Hence, assuming the above relation to hold true, a solution to the torsion free equations, which is also compatible with h, is given by (4.8)–(4.16). The free parameters in this solution are $\gamma _{+-}, \, \gamma _{-+}, \, \gamma _{zz}, \, \rho _{+-}$ and $\rho _{--}^*=\rho _{--}, \, \rho _{++}^*=\rho _{++}$. $\square$

Although the general q-affine Levi–Civita connection on $\Omega ^1(S^{3}_{q})$ may be written down, the expressions are rather lengthy and not particularly illuminating. However, let us explicitly write down a Levi–Civita connection in the particular case of a diagonal metric of the form

$$\begin{aligned}&h_{--} = h, \quad h_{++}=q^2h, \quad h_{zz}=h_z, \quad h_{ab}=0\quad \text { if }a\ne b, \end{aligned}$$

with h and $h_z$ invertible elements of $S^{3}_{q}$; note that this choice clearly satisfies (4.4) in Proposition 4.2. Using the solution given by (4.8)–(4.16) in the proof of Proposition 4.4 one finds

$$\begin{aligned}&\nabla _{X_{+}}\omega _a = \omega _bh^{bc}\big (\tfrac{1}{2}X_{+}(h_{ca})+K(\gamma _{ca})\big ),\\&\nabla _{X_{-}}\omega _a = \omega _bh^{bc}\big (\tfrac{1}{2}X_{-}(h_{ca})+K(\gamma _{ac}^*)\big ),\\&\nabla _{X_{z}}\omega _a= \omega _bh^{bc}\big (\tfrac{1}{2}X_{z}(h_{ca})+K^2(\rho _{ca})\big ), \end{aligned}$$

with

$$\begin{aligned}&\gamma _{++} = \tfrac{1}{2}q^2K^{-1}X_{+}(h)+q^2\gamma _{+-}^*,\\&\gamma _{--} = -\tfrac{1}{2}K^{-1}X_{+}(h)+q^{-2}\gamma _{+-}^*,\\&\gamma _{+z} = (1+q^2)K^{-1}(h)+\tfrac{1}{2}q^{-2}K^{-1}X_{z}(h),\\&\gamma _{z+} = q^4K^{-1}(\rho _{-+}),\\&\gamma _{-z} = q^{-4}K(\rho _{-+}),\\&\gamma _{z-} =q^4K^{-1}(\rho _{--})-q^2(1+q^2)K^{-1}(h)-\tfrac{1}{2}q^4K^{-3}X_{z}(h), \end{aligned}$$

and

$$\begin{aligned}&\rho _{z+} = q^4K^{-1}(\gamma _{zz})+q^4K^{-2}X_{+}(h_z),\\&\rho _{+z} = \rho _{z+}^*= q^4K(\gamma _{zz}^*)-q^4X_{-}(h_z),\\&\rho _{z-} = \tfrac{1}{2}q^{-4}K^{-2}X_{-}(h_z)+q^{-4}K^{-1}(\gamma _{zz}^*),\\&\rho _{-z} = \rho _{z-}^*= -\tfrac{1}{2}q^{-4}X_{+}(h_z)+q^{-4}K(\gamma _{zz}),\\&\rho _{++} = q^{10}\rho _{--}+q^4h_z -\tfrac{1}{2}q^4(1+q^2)(1+q^4)\big (K^2(h)+K^{-2}(h)\big ). \end{aligned}$$

Furthermore, setting $\gamma _{+-}=\rho _{-+}=\gamma _{zz}= \rho _{--}=0$ one obtains

$$\begin{aligned}&\nabla _+\omega _+ = \omega _+h^{-1}X_{+}(h),\\&\nabla _+\omega _- = \omega _z\frac{h_z^{-1}}{1-q^{-2}} \Big (\big (1-\tfrac{1}{2}q^4\big )K^2(h)-\tfrac{1}{2}q^4K^{-2}(h)\Big ),\\&\nabla _+\omega _z = \omega _+q^{-2}h^{-1}\Big (K^2(h_z)+\tfrac{1}{1-q^{-2}}\big (\left( q^2-\tfrac{1}{2}q^6\right) h-\tfrac{1}{2}q^6K^4(h)\big )\Big ) +\omega _z\tfrac{1}{2}h_z^{-1}X_{+}(h_z),\\&\nabla _-\omega _+ = \omega _z +\omega _z\frac{h_z^{-1}}{1-q^{-2}}\Big (\big (q^2-\tfrac{1}{2}q^6\big )K^2(h)-\tfrac{1}{2}q^6K^{-2}(h)\Big ),\\&\nabla _-\omega _- = \omega _-h^{-1}X_-(h),\\&\nabla _-\omega _z = \omega _-\frac{h^{-1}}{1-q^{-2}}\Big (\left( 1-\tfrac{1}{2}q^4\right) h-\tfrac{1}{2}q^4K^4(h)\Big ) +\omega _z\tfrac{1}{2}h_z^{-1}X_{-}(h_z),\\&\nabla _z\omega _+ = \omega _z\tfrac{1}{2}q^4h_z^{-1}X_{+}(h_z)+\omega _+q^2h^{-1}K^2(h_z) +\omega _+\frac{h^{-1}}{1-q^{-2}}\Big (\left( 1-\tfrac{1}{2}q^8\right) h-\tfrac{1}{2}q^8K^4(h)\Big ),\\&\nabla _z\omega _- = \omega _z\tfrac{1}{2}h_z^{-1}q^{-4}X_{-}(h_z) +\omega _-\frac{h^{-1}}{1-q^{-2}}\big (\tfrac{1}{2}h-\tfrac{1}{2}K^4(h)\big ),\\&\nabla _z\omega _z = \omega _z\tfrac{1}{2}h_z^{-1}X_{z}(h_z) -\omega _+\tfrac{1}{2}q^2h^{-1}K^2X_{-}(h_z) -\omega _-\tfrac{1}{2}q^{-4}h^{-1}K^2X_{+}(h_z), \end{aligned}$$

giving a q-affine Levi–Civita connection on $\Omega ^1(S^{3}_{q})$ with respect to the Hermitian form h.

5 The Quantum 2-Sphere

The noncommutative (standard) Podleś sphere $S^2_q$ [20] can be considered as a subalgebra of $S^{3}_{q}$ by identifying the generators $B_0,B_+,B_-$ of $S^{2}_{q}$ as

$$\begin{aligned} B_0= cc^*\quad B_+= ca^*\quad B_-= ac^*= B_+^*, \end{aligned}$$

satisfying then the relations

$$\begin{aligned}&B_-\,B_0= q^{2}\, B_0\,B_-\quad B_+\,B_0= q^{-2}\, B_0\,B_+, \\&B_-\,B_+= q^2\, B_0\,\big ( \mathbb {1}- q^2\, B_0\big ) \quad B_+\,B_-= B_0\,\big ( \mathbb {1}- B_0\big ) . \end{aligned}$$

These elements generate the fix-point algebra of the right U(1)-action

$$\begin{aligned} \alpha _z(a) =az\quad \alpha _z(a^*)=a^*\bar{z}\quad \alpha _z(c) =cz\quad \alpha _z(c^*)=c^*\bar{z} \end{aligned}$$

(5.1)

for $z\in U(1)$ and $a\in S^{3}_{q}$, related to the U(1)-Hopf-fibration $S^{2}_{q}\hookrightarrow S^{3}_{q}$. Equivalently, the sphere $S^{2}_{q}$ is the invariant subalgebra of $S^{3}_{q}$ for the left action of K: $S^{2}_{q}= \{f \in S^{3}_{q}, \, K \triangleright f = f \}$. Then, the left action of the $X_a$ does not preserve the algebra $S^{2}_{q}$ (since their left action does not commute with that of K): one readily computes,

$$\begin{aligned}&X_{+}\triangleright B_0= q a^*c^*\quad X_{-}\triangleright B_0= -q^{-1} ca \quad X_{z}\triangleright B_0= 0,\\&X_{+}\triangleright B_+= q (a^*)^2 \quad X_{-}\triangleright B_+= c^2 \quad X_{z}\triangleright B_+= 0,\\&X_{+}\triangleright B_-= q^{2} (c^*)^2 \quad X_{-}\triangleright B_-= -q^{-1} (a)^2 \quad X_{z}\triangleright B_-= 0. \end{aligned}$$

Note, however, that the right action of $X_a$ leaves $S^{2}_{q}$ invariant; i.e. $f\triangleleft X_a\in S^{2}_{q}$ for $f\in S^{2}_{q}$ and $a=\pm ,z$. This is shown explicitly in Eq. (A.1) in the “Appendix”.

5.1 A Left Covariant Calculus

Since the element $X_{z}$ acts trivially (on the left) on $S^{2}_{q}$, the differential (2.3) when restricted to $f\in S^{2}_{q}$ becomes

$$\begin{aligned} d f = (X_{-}\triangleright f) \,\omega _{-} + (X_{+}\triangleright f) \,\omega _{+} . \end{aligned}$$

(5.2)

Moreover, when acting on $S^{2}_{q}$ both $X_{+}$ and $X_{-}$ are usual derivations since K and then $\sigma _z$ are the identity on $S^{2}_{q}$. Classically, the form (5.2) of the differential that uses left invariant vector fields and forms can be seen as identifying the cotangent bundle of $S^2$ with the direct sum of the line bundles of ‘charge’ $\pm 2$, that is $\Omega ^{1}(S^2) \simeq \mathcal {L}_{-2} \omega _{-} \oplus \mathcal {L}_{+2} \omega _{+}$. This identification can be used also for the quantum sphere $S^{2}_{q}$ with the line bundles defined as in (5.4).

In particular from (5.2) one finds

$$\begin{aligned} dB_+&= q \, (a^*)^{2} \, \omega _{+}+ c^{2} \, \omega _{-}, \\ dB_-&= - q^{2}\, (c^*)^{2} \, \omega _{+}- q^{-1} \, a^{2} \, \omega _{-}, \\ dB_0&= c^*a^*\, \omega _{+}-q^{-1} c a \, \omega _{-}, \end{aligned}$$

which can be inverted to yield

$$\begin{aligned} \omega _{+}&= q^{-1} a^2 \, dB_+-q^2 c^2 \, dB_-+ (1+q^2) ac \, dB_0, \\ \omega _{-}&= (c^*)^{2} \, dB_+- q (a^*)^{2} \, dB_-- (1+q^2) c^*a^*\, dB_0, \end{aligned}$$

implying that the differential in (5.2) can be expressed as

$$\begin{aligned} d f&= \big ( q^{-1} (X_{+}\triangleright f) \, a^2 + (X_{-}\triangleright f) \, (c^*)^{2}\big )dB_+\nonumber \\&\quad - \big ( q^2 (X_{+}\triangleright f) \, c^2 + q (X_{-}\triangleright f) \, (a^*)^{2}\big ) dB_-\nonumber \\&\quad + (1+q^2)\big ( (X_{+}\triangleright f) \, ac - (X_{-}\triangleright f) \, c^*a^*\big )dB_0. \end{aligned}$$

(5.3)

In spite of the fact that $X_{\pm }\triangleright f \notin S^{2}_{q}$, from the commutation relations $K X_{\pm }= q^{\mp } X_{\mp }K$ one infer that all coefficients are in $S^{2}_{q}$. For instance: $K \triangleleft \big ((X_{+}\triangleright f) \, a^2\big ) = \big ((K X_{+}\triangleright f) \, K \triangleleft a^2\big ) = ( q X_{+}\triangleright f) \, q^{-1} \triangleleft a^2 = (X_{+}\triangleright f) \, a^2$, and similarly for the other terms.

5.2 Connections on Projective Modules over $S^{2}_{q}$

The definition of q-affine connections applies equally well to the subalgebra $S^{2}_{q}$. The right actions of $X_{\pm },X_{z}$ preserve $S^{2}_{q}$ [cf. (A.1) in the “Appendix”], and thus restrict to twisted derivations on $S^{2}_{q}$. However, even classically it is not possible to find two vector fields that span the tangent space of $S^2$ at each point. This is a consequence of the fact that the module of vector fields on $S^2$ is not a free module and one needs at least three vector fields to generate the module of vector fields. Analogously, for the quantum 2-sphere, even though the right actions of $X_{\pm },X_{z}$ are related, as shown in (A.2), there is no global way of writing e.g $X_{z}$ as a $S^{2}_{q}$-linear combination of $X_\pm$. Hence, for a q-affine connection $\nabla$ on a $S^{2}_{q}$-module M we still need three operators, that is a map

$$\begin{aligned} \nabla : \mathbb {C}\left\langle X_+,X_-,X_z \right\rangle \times M\rightarrow M \end{aligned}$$

satisfying the conditions of Definition 3.2. Moreover, even if a q-affine connection is not $S^{2}_{q}$-linear in its first argument, one expects a relation among the covariant derivatives, although this needs not be immediately implied by the relation (A.2) on the derivations. In this section, we construct q-affine connections on a class of projective modules over $S^{2}_{q}$.

The quantum Peter–Weyl theorem for $S^{3}_{q}$ results into an explicit (vector space) decomposition of the algebra $S^{3}_{q}$, that is $S^{3}_{q}= \oplus _{n\in \mathbb {Z}}\mathcal {L}_n$, with

$$\begin{aligned} \mathcal {L}_n = \{f\in S^{3}_{q}:\alpha _z(f) = \bar{z}^nf\}, \end{aligned}$$

(5.4)

for the U(1) action $\alpha _z$ in (5.1). Equivalently, $\mathcal {L}_n =\left\{ f \in S^{3}_{q}\, , \, K \triangleright f = q^{-\frac{n}{2}}f\right\}$. It follows that $\mathcal {L}_0 = S^{2}_{q}$, as well as $\mathcal {L}_n\mathcal {L}_m\subseteq \mathcal {L}_{n+m}$. Clearly, the right action of $\mathcal {U}_q(\text {su}(2))$ leaves each $\mathcal {L}_n$ invariant. On the other hand, for the left action one has $X_\pm \triangleright \mathcal {L}_n \subset \mathcal {L}_{n \mp 2}$.

It is easy to see that $\mathcal {L}_n$ is a $S^{2}_{q}$-bimodule. For $f, g \in S^{2}_{q}$ and $\psi _n\in \mathcal {L}_n$,

$$\begin{aligned} \alpha _z(f \psi _n g) = \alpha _z(f) \alpha _z(\psi _n) \alpha _z(g) = \bar{z}^n (f \psi _n g), \end{aligned}$$

which says that $\mathcal {L}_n$ is a $S^{2}_{q}$-bimodule. As a right (or equivalently left) module, each $\mathcal {L}_n$ can be realised as a finitely generated projective $S^{2}_{q}$-module as we now briefly recall (cf. [11, 15, 17]).

For $n\ge 0$ and $\mu =0,1,\ldots ,n$, let $(\Psi _n)_\mu ,(\Phi _n)_\mu \in S^{3}_{q}$ be given as

$$\begin{aligned}&(\Phi _n)_\mu = \sqrt{\alpha _{n\mu }}c^{n-\mu }a^{\mu } \quad (\Psi _n)_\mu = \sqrt{\beta _{n\mu }}(c^*)^{\mu }(a^*)^{n-\mu } \end{aligned}$$

with

$$\begin{aligned}&\alpha _{n\mu }=\prod _{k=0}^{n-\mu -1}\frac{1-q^{2(n-k)}}{1-q^{2(k+1)}}\quad \beta _{n\mu }=q^{2\mu }\prod _{k=0}^{\mu -1}\frac{1-q^{-2(n-k)}}{1-q^{-2(k+1)}}. \end{aligned}$$

It is straight-forward to check that

$$\begin{aligned} \sum _{\mu =0}^{n}(\Phi _n)_{\mu }^*(\Phi _n)_{\mu }= \sum _{\mu =0}^{n}(\Psi _n)_{\mu }^*(\Psi _n)_{\mu }= \mathbb {1}, \end{aligned}$$

implying that

$$\begin{aligned}&{(p_{n})^{\mu }}_{\nu } = (\Psi _n)_\mu (\Psi _n)^*_{\nu } =\sqrt{\beta _{n\mu }\beta _{n\nu }}(c^*)^{\mu }(a^*)^{n-\mu }a^{n-\nu }c^{\nu },\\&{(p_{-n})^{\mu }}_{\nu } = (\Phi _n)_\mu (\Phi _n)^*_{\nu } =\sqrt{\alpha _{n\mu }\alpha _{n\nu }}c^{n-\mu }a^\mu (a^*)^\nu (c^*)^{n-\nu }, \end{aligned}$$

satisfy $p_n^2=p_n$ and $p_{-n}^2=p_{-n}$. Moreover, it is easy to see that the entries ${(p_{n})^{\mu }}_{\nu }$ and ${(p_{-n})^{\mu }}_{\nu }\in S^{2}_{q}$, which implies that one has finitely generated projective $S^{2}_{q}$-modules

$$\begin{aligned} M_n = {\left\{ \begin{array}{ll} p_n(S^{2}_{q})^{n+1}&{}\text {if }n\ge 0,\\ p_{-|n|}(S^{2}_{q})^{|n|+1}&{}\text {if }n<0. \end{array}\right. } \end{aligned}$$

These modules $M_n$ are isomorphic as right $S^{2}_{q}$-modules to $\mathcal {L}_n$ for each $n\in \mathbb {Z}$.

Now, let $\{e_\mu \}_{\mu =0}^n$ be a basis of $(S^{2}_{q})^{n+1}$. Given an invertible Hermitian form h on $(S^2_q)^{n+1}$, the proof of Proposition 3.6 (repeated verbatimly for the algebra $S^{2}_{q}$), gives a q-affine connection on $(S^{2}_{q})^{n+1}$ compatible with h as

$$\begin{aligned}&\widetilde{\nabla }_{X_{+}}e_\mu = e_{\nu }\Gamma ^\nu _{+\mu } = e_{\nu }h^{\nu \rho }\big (\tfrac{1}{2}h_{\rho \mu }\triangleleft X_{+}+a_{\rho \mu }\triangleleft K\big ),\\&\widetilde{\nabla }_{X_{-}}e_\mu = e_{\nu }\Gamma ^\nu _{-\mu } = e_{\nu }h^{\nu \rho }\big (\tfrac{1}{2}h_{\rho \mu }\triangleleft X_{-}+a_{\mu \rho }^*\triangleleft K\big ),\\&\widetilde{\nabla }_{X_{z}}e_{\mu } = e_\nu \Gamma ^\nu _{z\mu } = e_\nu h^{\nu \rho }\big (\tfrac{1}{2}h_{\rho \mu }\triangleleft X_{z}+b_{\rho \mu }\triangleleft K^2\big ), \end{aligned}$$

for arbitrary $a_{\mu \nu },b_{\mu \nu }\in S^{2}_{q}$ such that $b_{\mu \nu }^*=b_{\nu \mu }$. If $n\ge 0$ then $\hat{e}_\mu =e_{\nu }{(p_n)^\nu _\mu }$ are generators of $M_n=p_n(S^{2}_{q})^{n+1}$ and Proposition 3.7 applied (mutatis mutandis) to $S^2_q$ implies that $\nabla =p_n\circ \widetilde{\nabla }$ is a q-affine connection on $M_n$ with

$$\begin{aligned} \nabla _{X_{+}}\hat{e}_{\mu }&= p_n\big (\widetilde{\nabla }_{X_{+}}e_\nu {(p_n)^\nu }_{\mu }\big ) = p_n(\widetilde{\nabla }_{X_{+}}e_\nu )\big ({(p_n)^\nu }_{\mu }\triangleleft K^2\big )+\hat{e}_{\nu }\big ({(p_n)^\nu }_{\mu }\triangleleft X_{+}\big )\\&= \hat{e}_{\gamma }h^{\gamma \rho }q^{2(\mu -\nu )}\big (\tfrac{1}{2}h_{\rho \nu }\triangleleft X_{+}+a_{\rho \nu }\triangleleft K\big ){(p_n)^\nu }_{\mu } +\hat{e}_{\nu }\big ({(p_n)^\nu }_{\mu }\triangleleft X_{+}\big ),\\ \nabla _{X_{-}}\hat{e}_{\mu }&= \hat{e}_{\gamma }h^{\gamma \rho }q^{2(\mu -\nu )}\big (\tfrac{1}{2}h_{\rho \nu }\triangleleft X_{-}+a^*_{\nu \rho }\triangleleft K\big ){(p_n)^\nu }_{\mu } +\hat{e}_{\nu }\big ({(p_n)^\nu }_{\mu }\triangleleft X_{-}\big ),\\ \nabla _{X_{z}}\hat{e}_{\mu }&= \hat{e}_{\gamma }h^{\gamma \rho }q^{4(\mu -\nu )}\big (\tfrac{1}{2}h_{\rho \nu }\triangleleft X_{z}+b_{\rho \nu }\triangleleft K^2\big ){(p_n)^\nu }_{\mu } +\hat{e}_{\nu }\big ({(p_n)^\nu }_{\mu }\triangleleft X_{z}\big ), \end{aligned}$$

using that ${(p_n)^\mu }_{\nu }\triangleleft K=q^{\nu -\mu }{(p_n)^\mu }_{\nu }$. Moreover, if $p_n$ is orthogonal with respect to h, then $\nabla$ is compatible with the restriction of h to $M_n$. A similar construction goes for $n<0$.

6 Further Comments: Sketching a Generalization

As final section of comments we sketch a way to generalise (some of) the constructions above for any Hopf algebra with a left covariant differential calculus and corresponding quantum tangent space [23]. While referring to [16,  14.1] for details, we recall that a first order differential calculus $(\Gamma , d)$ over the Hopf algebra $(H, \Delta , S, \varepsilon )$ is called left-covariant if there is a linear map $\Delta _\Gamma : \Gamma \rightarrow H \otimes \Gamma$ such that, for all $f,g \in H$ it holds that

$$\begin{aligned} \Delta _\Gamma (f \, dg) = \Delta (f) (\mathrm{id} \otimes d)) \Delta (g) . \end{aligned}$$

An element $\rho \in \Gamma$ is called left-invariant if $\Delta _\Gamma (\rho ) = 1 \otimes \rho$ and we let ${}_\mathrm{inv}\Gamma$ denote the vector space of invariant elements. There is then a corresponding quantum tangent space $T_\Gamma \subset H^{\circ }$ (the dual Hopf algebra) with a unique bilinear form $\left\langle \cdot , \cdot \right\rangle : T_\Gamma \times \Gamma \rightarrow \mathbb {C}$ such that

$$\begin{aligned} \left\langle X, f \, d g \right\rangle = \varepsilon (f) X (g) , \end{aligned}$$

for $g, f \in H$, and $X\in T_\Gamma$. The vector spaces ${}_\mathrm{inv}\Gamma$ and $T_\Gamma$ form a non-degenerate dual pair with respect to this bilinear form. Also, the pairing induces a left action as in (2.1),

$$\begin{aligned} X \triangleleft f = f_{(1)}\left\langle X,f_{(2)} \right\rangle \end{aligned}$$

for $X\in T_\Gamma$ and $f\in H$. Furthermore, one has dual bases $\{X_a, \, a = 1, 2, \dots , n \}$ of $T_\Gamma$ and $\{\omega _a, \, a = 1, 2, \dots , n\}$ of ${}_\mathrm{inv}\Gamma$ and a family of functionals $\{\sigma ^a_b, \, a, b = 1, 2, \dots , n \}$ such that

$$\begin{aligned}&d f = \sum \limits _a (X_a \triangleleft f ) \, \omega _a, \nonumber \\&X_a \triangleleft (f g ) = f X_a \triangleleft (g ) + X_b \triangleleft (f) \, \sigma ^b_a \triangleleft (g) . \end{aligned}$$

(6.1)

In the dual Hopf algebra $H^{\circ }$ we have

$$\begin{aligned} \Delta {\sigma ^a_b}&= \sigma ^a_c \otimes \sigma ^c_b , \quad S(X_a) = - X_b S(\sigma ^b_a) . \end{aligned}$$

With compatible $*$-structures, using the second expression and requiring (2.2) one computes:

$$\begin{aligned} X_a \triangleleft f^* = - \sigma ^b_a \triangleleft (X^\dagger _b \triangleleft f)^* . \end{aligned}$$

(6.2)

By way of illustration let us consider the trivial right module $M=H$ with Hermitian form $h(m_1, m_2) = m_1^* m_2$. The analogue of the condition (3.2) in Definition 3.2 is read from (6.1) as

$$\begin{aligned} \nabla _{X_a} \triangleleft (m f)&= m \, X_a \triangleleft (f) + \big (\nabla _{X_b} \triangleleft (m)\big ) \, \sigma ^b_a \triangleleft (f) . \end{aligned}$$

(6.3)

In turn, the compatibility with the Hermitian form reads:

$$\begin{aligned} X_a\big (h(m_1,m_2)\big ) = h\big (m_1,\nabla _{X_a}m_2\big ) - \sigma ^b_a \triangleleft \big (h(\nabla _{X^\dagger _b}m_1,m_2)\big ) . \end{aligned}$$

(6.4)

Indeed, using (6.1) and (6.2), we compute

$$\begin{aligned} X_a\big (h(m_1,m_2)\big ) = X_a \triangleleft (m_1^* m_2)&= m_1^* X_a \triangleleft (m_2) + X_b \triangleleft (m_1^*) \, \sigma ^b_a \triangleleft (m_2) \\&= m_1^* X_a \triangleleft (m_2) - \sigma ^c_b \triangleleft (X^\dagger _c \triangleleft m_1)^*\, \sigma ^b_a \triangleleft (m_2) \\&= m_1^* X_a \triangleleft (m_2) - \sigma ^c_a \triangleleft \big (X^\dagger _c \triangleleft m_1)^* \, (m_2)\big ) \end{aligned}$$

from which (6.4) follows.

Equations (6.3) and (6.4) can be the starting point for a theory of affine connections on a quantum group with a quantum tangent space. For a torsion freeness condition one would need (twisted) commutation relations among the elements of $T_\Gamma$. In general these commutation relations could be involved; in particular they do not need to be quadratic as in the classical case or in the example in (2.6)–(2.4). Details should await a different time.

Appendix A: The Calculus on the Sphere \(S^2_q\) via the Right Action

As we have seen in Sect. 5 the left action of the $X_a$ does not preserve the algebra $S^{2}_{q}$, since their left action does not commute with that of K defining the fibration. On the other hand, the right action of $X_a$ does preserve the algebra $S^{2}_{q}$ since the action does commute with the left one of K. Let us denote $Y_a=X_a$ for the right action. Then, it is easy to check that

$$\begin{aligned}&B_0\triangleleft Y_{+}= q^{-1}B_-\quad B_0\triangleleft Y_{-}= -q^{-1}B_+\quad B_0\triangleleft Y_{z}= 0, \nonumber \\&B_+\triangleleft Y_{+}= q\mathbb {1}-q(1+q^2)B_0\quad B_+\triangleleft Y_{-}= 0 \quad B_+\triangleleft Y_{z}= -q^2(1+q^2)B_+,\nonumber \\&B_-\triangleleft Y_{+}= 0 \quad B_-\triangleleft Y_{-}= -q^{-1}\mathbb {1}+ q^{-1}(1+q^2)B_0\quad B_-\triangleleft Y_{z}= (1+q^{-2})B_-. \nonumber \\ \end{aligned}$$

(A.1)

Note that when restricted to $S^{2}_{q}$ the $Y_a$ are not independent. A long but straightforward computation shows that they are indeed related as

$$\begin{aligned} \begin{aligned}&\big ( (f \triangleleft Y_{+}) B_+\, q + (f \triangleleft Y_{-}) B_-\, q^{-1} \big )(1+q^2) + (f \triangleleft Y_{z}) \left( 1-2 \frac{1+q^2}{1+q^4} B_0\right) \\&= (f \triangleleft Y_{z}^2) \, q^{-2} \left( \frac{1-q^2}{1+q^4} \, ( 2 q^4 + q^2 + 1 ) B_0- (1-q^6) B_0^2\right) \\&\quad + (f \triangleleft K^4) \, q^{-2} (1+q^2) \big ( (q^4 -1) B_0+ (1-q^6) B_0^2\big ), \end{aligned} \end{aligned}$$

(A.2)

for $f\in S^{2}_{q}$. This is checked on a vector space basis for the algebra $S^{2}_{q}$, a basis which can be taken as $X(m) (B_0)^n$ for $m\in \mathbb {Z}$, $n \in \mathbb {N}$ with $X(m) = (B_+)^m$ for $m\ge 0$ and $X(m) = (B_-)^{-m}$ for $m < 0$ (cf. [19]). From the expression in (5.3) one writes the differential d on $S^{2}_{q}$ in terms of the right acting operators $Y_a$.

Lemma A.1

For $f\in S^{2}_{q},$ the differential in (5.2) can be written as

$$\begin{aligned} d f = (f \triangleleft V_+) \, dB_++ (f\triangleleft V_-) \,\, dB_-+ (f \triangleleft V_0) \, dB_0, \end{aligned}$$

(A.3)

where

$$\begin{aligned} V_+&= Y_{+}\big ( 1 - q^{-2}(1+q^2) B_0\big ) q^{-1} - Y_{z}\,\, B_-\frac{q^{-2}(1+q^6)}{1+q^4} + Y_{z}^2 \,\, B_-\frac{1-q^2}{(1+q^2)(1+q^4)}, \\~~\\ V_-&= - Y_{-}\big ( 1 - q^2(1+q^2) B_0\big ) q + Y_{z}\,\, B_+\frac{q^{-2}(1+q^6)}{1+q^4} - Y_{z}^2 \,\, B_+\frac{1-q^2}{(1+q^2)(1+q^4)}, \\~~\\ V_0&= \big ( Y_{+}\,\, B_+\, q^{-1} - Y_{-}\,\, B_-\, q \big ) (1+q^2) + Y_{z}\,\, B_0\frac{(1-q^4)(1+q^6)}{1+q^4} - Y_{z}^2 \,\, B_0\frac{1-q^2}{1+q^4} . \end{aligned}$$

Proof

By acting on the vector space basis $X(m)(B_0)^n$ (as introduced previously), one explicitly checks the equality of (5.2) and (A.3) via a tedious but straightforward computation. $\square$

Remark A.2

When $q=1$ the derivative (A.3) reduces to

$$\begin{aligned} \begin{aligned} d f&= 2 \big ( ( f \triangleleft Y_{+}) \,\, B_+- (f \triangleleft Y_{-}) \,\, B_-\big ) dB_0\\&\quad + \big ( ( f \triangleleft Y_{+}) \,\, (1-2 B_0) - ( f \triangleleft Y_{z}) \,\, B_-\big ) dB_+\\&\quad + \big ( - ( f \triangleleft Y_{-}) \,\, (1-2 B_0) + (f \triangleleft Y_{z}) \,\, B_+\big ) dB_-. \end{aligned} \end{aligned}$$

(A.4)

Classically, the vector field $X_a$ are the left invariant vector fields on $S^3 = SU(2)$ with dual left invariant forms $\omega _a$. Thus they do not project to vector fields on the base space $S^2$ with commuting coordinates $(B_+, B_-, B_0)$ and relation $B_+B_-= B_0(1-B_0)$: $X_a \triangleright f$ is not a function on $S^2$ even when f is. On the other hand, the vector fields $Y_a$ are the right invariant vector fields on SU(2) and thus they project to vector fields on $S^2$, where they are not independent any longer and are related by

$$\begin{aligned} 2 ( B_+Y_{+}+ B_-Y_{-}) + (1-2 B_0) Y_{z}= 0 , \end{aligned}$$

which is just the relation to which (A.2) reduces when $q=1$.

By changing coordinates $B_0 = \tfrac{1}{2}(1-x)$ so that the radius condition for $S^2$ is written as $r^2 = 4 B_+B_-+ x^2$, the exterior derivative operator in (A.4) becomes

$$\begin{aligned} d f = \partial _x f \, d x + \partial _+ f \, d B_++ \partial _- f \, d B_-- (\Delta f) \, ( x\, dx + 2 B_-\, dB_++ 2 B_+\, dB_-) \end{aligned}$$

where $\Delta = x \, \partial _x + B_+\, \partial _+ + B_-\, \partial _-$ is the Euler (dilatation) vector field. One then computes $d r^2 = 2 ( 1 - r^2 ) ( x\, dx + 2 B_-\, dB_++ 2 B_+\, d B_-)$, which vanishes when restricting to $S^2$: $r^2-1=0$.