Antonin Coutant
email: antonin.coutant@univ-lemans.fr

Yves Aurégan
email: yves.auregan@univ-lemans.fr

Vincent Pagneux
email: vincent.pagneux@univ-lemans.fr

Slow sound laser in lined flow ducts

We consider the propagation of sound in a waveguide with an impedance wall. In the low frequency regime, the first effect of the impedance is to decrease the propagation speed of acoustic waves. Therefore, a flow in the duct can exceed the wave propagation speed at low Mach numbers, making it effectively supersonic. We analyze a setup where the impedance along the wall varies such that the duct is supersonic then subsonic in a finite region and supersonic again. In this specific configuration, the subsonic region act as a resonant cavity, and triggers a laser-like instability. We show that the instability is highly subwavelength. Besides, if the subsonic region is small enough, the instability is static. We also analyze the effect of a shear flow layer near the impedance wall. Although its presence significantly alter the instability, its main properties are maintained. We also point out the analogy between the present instability and a similar one in fluid analogues of black holes known as the black hole laser.

I. INTRODUCTION

Acoustic liners in waveguides offer the interesting possibility to slow down sound waves. In a fluid, the speed of sound c 0 is controlled by both the density of the fluid and its stiffness (the adiabatic bulk modulus). Acoustic liners can be created using tubes mounted flush to the wall of the guide, which lower the effective stiffness of the medium, thereby decreasing the effective propagation speed c eff of sound. This allows one to control and manipulate sound waves in guides. In particular, adding a flow of mean velocity U 0 , an effective supersonic configuration (c eff < U 0 ) can be obtained in the duct at low Mach number (M 0 = U 0 /c 0 < 1). Transsonic configurations, gradually varying from subsonic to supersonic, lead to a rich wave phenomenology [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF] such as amplification and highly non-reciprocal propagation and can even lead to instability.

The existence of instabilities above acoustic materials in the presence of a grazing flow has been experimentally proven [START_REF] Ronneberger | Sound absorption, sound amplification, and flow control in ducts with compliant walls[END_REF][START_REF] Aurégan | Experimental evidence of an instability over an impedance wall in a duct with flow[END_REF][START_REF] Marx | PIV and LDV evidence of hydrodynamic instability over a liner in a duct with flow[END_REF]. Sometimes, it is difficult in computations to distinguish between real and numerical instabilities [START_REF] Li | Time-domain impedance boundary conditions for surfaces with subsonic mean flows[END_REF][START_REF] Burak | Validation of a time-and frequency-domain grazing flow acoustic liner model[END_REF][START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF][START_REF] Xin | Numerical study of acoustic instability in a partly lined flow duct using the full linearized navier-stokes equations[END_REF][START_REF] Pascal | Global linear stability analysis of flow in a lined duct[END_REF][START_REF] Sebastian | Numerical simulation of a compressible channel flow with an acoustic liner[END_REF]. An analysis of the different types of instability that can occur above a material is therefore of importance for a better understanding of the results of experiments or computations. Transonic configurations allow modes with very different wavelengths to interact through the flow and instabilities can occur in cavities created by impedance changes, even if their sizes are very small compared to the acoustic wavelength.

In the absence of flow, two acoustic modes can propagate in the duct at low frequencies. In subsonic lined ducts, there are two additional modes, which are referred to as hydrodynamic modes [START_REF] Rienstra | A classification of duct modes based on surface waves[END_REF]. Moreover, one of them is a negative energy wave. This means that its excitation lowers the total energy of system compared to the mean flow alone [START_REF] Cairns | The role of negative energy waves in some instabilities of parallel flows[END_REF]. Coupling to this negative energy wave through a change in the impedance wall can therefore lead to amplification [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF]. In this work, we study a configuration consisting in a double transition: from supersonic to subsonic to supersonic again, see Fig. 1. In such a configuration, the negative energy wave couples to a resonant cavity (formed by the subsonic region), thereby generating an exponentially growing instability. A peculiarity of the obtained instability, is that it occurs at a much lower frequency than the natural frequencies of the system, such as the frequency associated with the quarter wavelength of the tubes or with the size of the cavity formed by the two transitions. A peculiar feature of transsonic flows has attracted a lot of attention in the last decade: they can provide a laboratory analogue of a black hole [START_REF] Unruh | Experimental black hole evaporation[END_REF]. Interestingly, the instability studied in this work is closely related to the analogue of the Hawking radiation of black hole, and can be seen as a self-amplification of that radiation. This self-amplification is called the "black hole laser" in the analogue gravity community, and was studied in various contexts [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF][START_REF] Steinhauer | Observation of self-amplifying Hawking radiation in an analog black hole laser[END_REF]. The analogue Hawking effect has lead to many experiments in the last decade, in media as diverse as water waves [START_REF] Weinfurtner | Measurement of stimulated Hawking emission in an analogue system[END_REF][START_REF] Euvé | Observation of noise correlated by the Hawking effect in a water tank[END_REF][START_REF] Torres | Rotational superradiant scattering in a vortex flow[END_REF], nonlinear optics [START_REF] Drori | Observation of Stimulated Hawking Radiation in an Optical Analogue[END_REF] or Bose-Einstein condensates [START_REF] Steinhauer | Observation of thermal Hawking radiation and its entanglement in an analogue black hole[END_REF][START_REF] De Nova | Observation of thermal Hawking radiation at the Hawking temperature in an analogue black hole[END_REF], but despite many promising results, the full demonstration of the analogue Hawking radiation and its properties has not been achieved yet. Slow sound offers a promising system for its realization.

Flow

To describe this instability and analyze its main features, we use an effective onedimensional model that was previously derived in [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF]. Moreover, we shall take into account the effect of a shear flow boundary layer near the impedance wall. In a majority of works, the boundary layer is taken to be infinitely thin, leading to the so-called Ingard-Myers boundary condition at the impedance wall [START_REF] Ingard | Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission[END_REF][START_REF] Myers | On the acoustic boundary condition in the presence of flow[END_REF]. This boundary condition has however shown to be problematic, both from the theory [START_REF] Brambley | Fundamental problems with the model of uniform flow over acoustic linings[END_REF] and experimental point of view [START_REF] Renou | Failure of the ingard-myers boundary condition for a lined duct: An experimental investigation[END_REF]. In this work, we will use an improved boundary condition, based on the work of Brambley [START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF][START_REF] Brambley | Surface modes in sheared boundary layers over impedance linings[END_REF]. When using the Ingard-Myers condition, unstable modes can be divided into two categories: static instabilities (purely imaginary frequency, hence non oscillatory) and dynamical instabilities (non zero real part). With the inclusion of the boundary layer correction in the improved boundary condition, static instabilities acquire a non-zero real part, proportional to the ratio of the thickness of the boundary layer with the duct width.

The plan of the paper is as follows. In section II we present the model with the improved boundary condition. Then, we employ a Hamiltonian formalism to identify conserved quantities. In section III we discuss the spectrum of complex eigenfrequencies and separate the discussion of an infinitely thin boundary layer with a finite one. We then present our main conclusions.

II. PROPAGATION IN A LINED DUCT WITH A FLOW HAVING A THIN BOUNDARY LAYER

We consider the propagation of sound waves in a straight two-dimensional duct with a uniform flow U 0 (see Fig. 1). In the following we work with adimensionalized quantities, using the speed of sound c 0 , the density ρ 0 and the height of the duct H, all three are assumed to be constant. This means that the flow velocity is replaced by the (uniform) Mach number M 0 = U 0 /c 0 . Inside the duct, the acoustic velocity field is potential v = ∇φ, and obeys the wave equation

D 2 t φ -∆φ = 0, (1)
where

D t = ∂ t + M 0 ∂ x is the convective derivative.
The lower wall (y = 0) of the waveguide is assumed to be hard. This means that the boundary condition is simply given by a vanishing transverse velocity, that is

(∂ y φ) y=0 = 0. ( 2
)
The upper wall (y = 1) is compliant, and its boundary condition is defined by an impedance: the ratio of the pressure over the transverse acoustic velocity. In the absence of flow in the duct, the normalized impedance reads

Z b = i σ tan(bω) , ( 3
)
where b is the length of the mounted flush tubes, σ is the percentage of open area, and ω the angular frequency. We now work in the low frequency regime, that is when ω is small compared to the tube resonance frequency π/2b. In this regime, tan(b ω) ∼ b ω, and the compliant wall act as a spring of stiffness σb. Without loss of generality, we will now assume that σ = 1. For a nonzero flow inside the duct, although we assume a constant profile (independent of y), we must take into account the thin boundary layer (thickness δ) within which the Mach number decreases from its maximum value M 0 to zero. In a sequence of works [START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF][START_REF] Brambley | Surface modes in sheared boundary layers over impedance linings[END_REF], Brambley derived general boundary conditions that takes into account the boundary layer at order O(δ). For this work in the low frequency limit (ω π/2b), we will use an approximate boundary condition at order O(δ). For a wave of frequency ω and wave number k, that is φ = Re[ φ(y)e -iωt+ikx ], it is given by

(∂ y φ) y=1 = b(ω -M 0 k) 2 + δM 0 k 3 ω φ y=1 . (4)
In Appendix A, we provide details on how it has been obtained and discuss its regime of validity. Notice that by taking δ = 0, one recovers the standard Ingard-Myers condition.

To understand the propagation of waves in a duct of varying impedance, an effective one-dimensional model was derived in [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF]. The effective model is obtained by assuming a short transverse size of the duct compared with relevant wavelengths. As shown in [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF], this allows us to obtain a simple model that displays the same number of propagating modes with the same properties (such as the sign of their energy). However, the model of [START_REF] Aurégan | Slow sound in lined flow ducts[END_REF][START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF] used the Ingard-Myers boundary condition. Here we shall consider instead the more general condition (4).

A. Dispersion relation and propagating modes in the 1D model

We first quickly summarize how the effective model is obtained. First we define

ϕ = 1 0 φ(t, x, y)dy, (5a)
ψ = φ(t, x, 1). ( 5b
)
Integrating the wave equation ( 1) across the duct, we obtain

D 2 t ϕ -∂ 2 x ϕ -(∂ y φ) y=1 = 0. ( 6
)
We now assume that the transverse size of the duct is small enough so that the transverse dependence of the pressure field can be treated in a parabolic approximation, i.e. φ(t, x, y) = φ 1 (t, x) + y 2 φ 2 (t, x). In this approximation, the transverse derivative on the compliant wall can be written as a sum of ϕ and ψ, indeed:

∂ y φ(t, x, 1) = 3ψ -3ϕ. (7)
We can now use this in equations ( 6) and (4). For a single frequency and wave number mode, this leads to the dispersion relation

-(ω -M 0 k) 2 + k 2 + 3 b(ω -M 0 k) 2 + δM 0 k 3 ω -3 + 9 = 0. ( 8
)
In Fig. 2 we represent the dispersion relation, and compare the Ingard-Myers condition with our modified condition (4). We now discuss the different propagating modes at a given frequency. To start we consider the long wavelength limit k → 0 in the absence of flow (M 0 = 0), in which case the dispersion relation becomes

ω 2 = k 2 1 + b . (9)
This means that long wavelength waves have a propagation speed modified by the impedance wall of c eff = 1/ √ 1 + b. Since this value is always lower than the speed of sound in free air, these waves are called "slow sound waves". Based on this, we distinguish two types of flow: effective subsonic ones, when M 0 √ 1 + b < 1, and effective supersonic ones, when M 0 √ 1 + b > 1. We will now discuss the various propagating modes for both types of flows, separating the case of a vanishing boundary layer (Ingard-Myers condition) and a nonzero one (boundary condition (4)).

Let us start by discussing the Ingard-Myers condition. For subsonic flows, see Fig. 2a, there are two frequency ranges separated by a threshold frequency ω max . When ω < ω max there are four solutions. Two of them have long wavelength, and corresponds to acoustic modes k A± Doppler shifted by the flow. The two others are highly dispersive: their group velocity differs significantly from their phase velocity. These two extra modes would not exist in the absence of flow, and for that reason, are referred to as "hydrodynamic modes" [START_REF] Rienstra | A classification of duct modes based on surface waves[END_REF]. A peculiarity of these new modes is that while one (noted k S ) has a positive energy, the other carries (noted k N ) a negative energy. For higher frequencies, ω > ω max , there are only two modes left, one with negative energy, and both propagating with the flow. In effective supersonic flows, see Fig. 2b, there are two propagating modes for all frequencies, both propagating in the direction of the flow. Again, while one has a positive energy, the other has a negative energy.

When using the improved boundary condition of ( 4), a new propagating mode appear. More precisely, there is a new threshold frequency ω min < ω max . In the range ω min < ω < ω max , the discussion is similar as before, but with a fifth mode, noted k B . This modes propagate very slowly (its group velocity being proportional to δ) against the flow. when ω = ω min , this mode merge with k S to give two evanescent modes. For ω < ω min there is therefore only three propagating modes. Although this mode carries a positive energy, at low frequencies it couples to the other ones and alter significantly the laser effect. The asymptotic values when ω min ω ω max of the different wavenumbers are given in Appendix B. In the rest of this work, we are interested in the effects of a change of the wall compliance along the duct. In other words, we will assume that b is a function of x. Such inhomogeneity will couple together different modes sharing the same frequency. Due to the presence of negative energy waves, this can lead to amplification phenomena. In addition, when the change of wall compliance create a resonant cavity, this leads to a temporal instability. Such an instability is the acoustic analogue of the "black hole laser" that have been studied in various analogue models of gravity [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF][START_REF] Finazzi | Black-hole lasers in Bose-Einstein condensates[END_REF][START_REF] Michel | Saturation of black hole lasers in Bose-Einstein condensates[END_REF].

B. Conservation laws

The existence of a conserved energy in ducts is a priori non trivial, even for a conservative impedance as in our case (see e.g. the discussion of Möhring [START_REF] Mohring | Energy conservation, time-reversal invariance and reciprocity in ducts with flow[END_REF]). The question becomes increasingly delicate in the presence of a shear flow boundary layer, due to the existence of critical layers within it [START_REF] Brambley | The critical layer in linear-shear boundary layers over acoustic linings[END_REF][START_REF] Dai | A cavity-by-cavity description of the aeroacoustic instability over a liner with a grazing flow[END_REF]. The effective one-dimensional model and boundary condition (4) discussed in the preceding section presents the advantage to have canonically conserved quantities. To see this, it is valuable to use the Lagrange-Hamilton formalism. This allows us to construct conserved quantities by standard procedures. For instance, the energy is given by the Hamiltonian functional applied on a solution. Our effective model can be obtained from the Lagrangian density

L = 1 2 (D t ϕ) 2 -(∂ x ϕ) 2 -3(ψ -ϕ) 2 + b(D t ψ) 2 + δM 0 (∂ x A)(∂ 2 x ψ) -2α(∂ t A -ψ) . (10)
In this Lagrangian we have introduced an auxiliary field A(t, x) and a Lagrange multiplier α(t, x), which ensures the constraint ∂ t A = ψ. This procedure allows us to generate a correction with a term in 1/ω while having well defined equations in the time domain. Indeed, since the constraint is ∂ t A = ψ, for a stationary state we will have A = iψ/ω. Requiring vanishing variations of the Lagrangian gives the Euler-Lagrange equations1 , which read

D 2 t ϕ -∂ 2 x ϕ = 3(ψ -ϕ), (11a)
-D t bD t ψ + 1 2 δM 0 ∂ 3 x A + α = 3(ψ -ϕ), (11b)
together with the constraint

∂ t A = ψ, (12)
and the equation on the Lagrange multiplier

δM 0 ∂ 3 x ψ -2∂ t α = 0. ( 13
)
Using the constraint, this last equation integrates into

α = 1 2 δM 0 ∂ 3 x A. (14)
Hence the system becomes

D 2 t ϕ -∂ 2 x ϕ = 3(ψ -ϕ), (15a) -D t bD t ψ + δM 0 ∂ 3 x A = 3(ψ -ϕ). (15b)
We now see that these equations indeed give the effective model obtained by combining (4), [START_REF] Li | Time-domain impedance boundary conditions for surfaces with subsonic mean flows[END_REF], and [START_REF] Burak | Validation of a time-and frequency-domain grazing flow acoustic liner model[END_REF]. The Hamiltonian is now obtained by a Legendre transform of the Langrangian. Defining the conjugate momenta

π ϕ = D t ϕ, (16a) π ψ = bD t ψ, (16b) π A = -α, (16c)
the Hamiltonian density is given by

H = π ϕ ∂ t ϕ + π ψ ∂ t ψ + π A ∂ t A -L. (17)
This finally gives

E = 1 2 (∂ t ϕ) 2 + (1 -M 2 0 )(∂ x ϕ) 2 + b(∂ t ψ) 2 + 3(ψ -ϕ) 2 -bM 2 0 (∂ x ψ) 2 -2δM 0 ∂ x A∂ 2 x ψ dx. ( 18
)
In general, it is equally useful to work with a local conservation law of the form2

∂ t E + ∂ x J = 0. ( 19
)
That can be done with the energy, by defining an energy density E e and current J e such that E = E e dx and ∂ t E e +∂ x J e = 0. However, it is slightly simpler to use the conservation of the symplectic norm. The corresponding density is canonically defined as

E s . = -Im (ϕ * π ϕ + ψ * π ψ + A * π A ) = -Im (ϕ * D t ϕ + bψ * D t ψ -A * α) . (20)
Using the equation of motion [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF], we directly show that the corresponding current is given by

J s = Im ϕ * ∂ x ϕ -M 0 ϕ * D t ϕ -M 0 bψ * D t ψ + 1 2 δM 0 ∂ t A * ∂ 2 x A -A * ∂ 2 x ∂ t A -∂ t ∂ x A * ∂ x A . (21)
The local conservation law ∂ t E s + ∂ x J s = 0 implies that any localized solution have a conserved total norm E s dx. Although closely related, this norm is different from the energy. In fact, it is a generalization of the concept of wave action defined as the ratio of the energy over the frequency E/ω (see [START_REF] Vanneste | On wave action and phase in the non-canonical hamiltonian formulation[END_REF][START_REF] Bühler | Waves and mean flows[END_REF] for a general discussion and [START_REF] Coutant | Undulations from amplified low frequency surface waves[END_REF][START_REF] Coutant | The imprint of the analogue Hawking effect in subcritical flows[END_REF] for its use in a similar context). As one can directly verify, for a monochromatic wave, E = ω E s dx. The main difference is that the wave action is defined in a WKB limit, in which case it becomes conserved (it is an adiabatic invariant), while the total norm is always conserved, without any approximation.

We now evaluate the current J s and the density E s for a single frequency wave e -iωt+ikx , assuming b constant. Using the dispersion relation [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF] and the equations of motion [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF], we obtain

E s (ω, k j ) = (ω -M 0 k j ) + 1 9 b(ω -M 0 k j ) - δM 0 k 3 j 2ω 2 -(ω -M 0 k j ) 2 + k 2 j + 3 2 , ( 22
)
where j ∈ {B, S, A-, A+, N }. A direct evaluation of [START_REF] De Nova | Observation of thermal Hawking radiation at the Hawking temperature in an analogue black hole[END_REF] shows that k N is of negative norm, while the four other roots have a positive norm (and the same follows for the energy since E = ω E s dx). To see this more simply, one can notice that in the correction term in δ, one can replace -M 0 k 3 j by (ω -M 0 k j ) 3 /M 2 0 to a good approximation (i.e. in the same limit the boundary condition (4) was obtained -see App. A). Doing so, we see that the sign of E s is that of the intrinsic frequency ω -M 0 k j , which is positive for all roots but k N (we present in App. B asymptotic expressions for the norm and energy). The second important point to notice is that for a single frequency wave, the conservation law [START_REF] Torres | Rotational superradiant scattering in a vortex flow[END_REF] takes the form

J s = v g E s , (23)
where v g = ∂ k ω is the group velocity of the corresponding mode. To derive this identity, the idea is to apply the conservation law [START_REF] Torres | Rotational superradiant scattering in a vortex flow[END_REF] to a tight wave packet ϕ = f (t, x)e -iωt+ikx , where f is a slowly varying envelope, which depends essentially on the variable t -x/v g . Since J s and E s depend on t and x only through f , the above identity [START_REF] Ingard | Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmission[END_REF] follows. This equality shows that for a negative energy wave (E s < 0), the wave propagate in the opposite direction as the energy current.

III. LASER EFFECT

We now consider a configuration with two transsonic transitions. The flow is supersonic, except in a region of size L (in the units of transverse height) where it is subsonic. In this case, modes can be trapped in the central region, triggering the black hole laser instability. As illustrated in Fig. 3, the core of the mechanism is the coupling between a resonant cavity (0 < x < L) and the negative energy wave. When the cavity modes radiate energy through the negative energy wave, it results in an increase of energy in the cavity. In general there is a competition between the coupling to the negative energy wave, which tends to increase the cavity energy, and the coupling to the acoustic mode k A+ , which takes energy away. On a transsonic transition, the coupling to k A+ is generally quite smaller than to k N (see e.g. Fig. 7 in [START_REF] Aurégan | Slow sound in a duct, effective transonic flows, and analog black holes[END_REF]) and hence the cavity is unstable. When taking into account a finite boundary layer, the presence of an extra mode provides an additional source of damping. As we shall see in section III B, the presence of the boundary layer generally reduces the laser instability.

This instability is encoded in a set of complex eigenfrequencies of positive imaginary parts [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF], hence leading to a growing behavior. To find them, we look for stationary solutions of the form

ϕ(t, x) = ϕ ω (x)e -iωt , (24a) ψ(t, x) = ψ ω (x)e -iωt , (24b)
which obey the stationary equations

-(ω + iM 0 ∂ x ) 2 ϕ -∂ 2 x ϕ = 3(ψ -ϕ), (25a)
(ω + iM 0 ∂ x )b(ω + iM 0 ∂ x )ψ + iδM 0 ω ∂ 3 x ψ = 3(ψ -ϕ). (25b)
We now look for ω ∈ C such that there exist a purely outgoing solution. To properly define this, we proceed in the following way. When Im(ω) > 0, we select solutions such that ϕ and ψ are both decaying for x → ±∞. In other words, unstable modes are spatially localized. We then obtain the condition for Im(ω) < 0 by analytic continuation in the lower half plane. This procedure, identical to that of [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF], is equivalent to the Briggs-Bers prescription [START_REF] Briggs | Electron-stream interaction with plasmas[END_REF][START_REF] Bers | Space-time evolution of plasma instabilities -absolute and convective[END_REF][START_REF] Crighton | Fluid loading with mean flow. i. response of an elastic plate localized excitation[END_REF]. Indeed, we verify that when starting with Im(ω) → +∞, the sign of Im(k ω ) do not change while lowering Im(ω). This is shown in Fig. 4. Moreover, when Im(ω) → 0, the identity

k(ω r + i ) = k(ω r ) + i /v g + O( 2 ) ( 26
)
allows us to show that the sign of Im(k ω ) is the same as the group velocity. In addition, the spectrum is invariant under the discrete symmetry This comes from the fact that the boundary conditions we consider are unchanged under complex conjugation. Then, taking the complex conjugate of the mode equation [START_REF] Brambley | Fundamental problems with the model of uniform flow over acoustic linings[END_REF] gives a one-to-one mapping between a solution for ω and -ω * . This symmetry naturally divides the spectrum into two classes: purely imaginary frequencies, which are maximally symmetric and encode a static instability, and complex frequencies with non-zero real part, which come in pairs (ω, -ω * ) and correspond to a dynamical instability 3 . Keeping that symmetry in mind, we will in the following focus on the part of the spectrum Re(ω) > 0. We now analyse the set of complex eigen-frequencies for two abrupt changes in the wall compliance separated by a distance L. For this we assume that the length of the tubes vary along the wall as

ω → -ω * . (27)
A+ k B E N < 0 E cav. > 0 E A+,B > 0 k B k Lev k N k A+ k N k A+ k B k A- k S k B k Rev k N k A+
b(x) =      b O (x < 0), b C (0 < x < L), b O (L < x). (28)
The values are chosen such that the flow is supersonic on the outer region (M 0 √ 1 + b O > 1) but subsonic in the central region (M 0 √ 1 + b C < 1). We consider separately the case of a vanishing boundary layer (Ingard-Myers condition) and a nonzero one (boundary condition (4)). In each region, the solutions of the mode equation ( 25) are superpositions of exponentials e ikx where k satisfies the dispersion relation [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF]. To satisfy outgoing boundary conditions, we look for solutions of the form

ϕ ω (x) =          a L e ik Lev x J(ω, k Lev ) (x < 0), a N e ik N x J(ω, k N ) + a A+ e ik A+ x J(ω, k A+ ) + a R e ik Rev x J(ω, k Rev ) (L < x), (29)
where k Lev (resp. k Rev ) is the evanescent mode on the left (resp. right). At both interfaces, x = 0 and x = L, we must use proper continuity conditions. Because the boundary layer correction (see equation ( 4)) changes the order of the equation, going from 4 to 5, one must carefully discuss the Ingard-Myers condition and Brambley condition separately. For δ = 0, the mode equations [START_REF] Brambley | Fundamental problems with the model of uniform flow over acoustic linings[END_REF] imply that the fields ϕ, ∂ x ϕ, ψ, and b(ω + iM 0 ∂ x )ψ are continuous. Notice that the last condition ensures that the current is conserved across the interfaces. These conditions give a linear relation between the coefficients a j in equation [START_REF] Finazzi | Black-hole lasers in Bose-Einstein condensates[END_REF]. We numerically evaluate the determinant of the corresponding system, and look for its zeros. We display the results for the spectrum on Fig. 5, and the eigen-modes on Fig. 6.

When the size L of the subsonic region increases, more and more unstable modes appear. Each new mode appear by following two steps. It first comes out of the origin (ω = 0) to be purely imaginary: it is a static instability. It then comes back to the origin and is converted into a complex frequency (both real and imaginary parts are nonzero), hence becoming a dynamical instability. This laser instability has several peculiar properties which we underline now: • The studied configuration is not always unstable. If the trapping region is 'too small,' no unstable mode exist. For the parameters of Fig. 5, we see that this is the case for L 0.8. Moreover, it is noticeable that the purely imaginary resonance that precedes the static instability for small L tends to a finite value in the limit L → 0 (this is due to the presence of a branch point at this value of ω ∈ C).

• There are generically both static and dynamical unstable modes. However, the dominating one can be either one, depending on the parameters of the duct configuration. This is due to interference effects, which generate an oscillating behavior of the imaginary part of the complex frequency when varying external parameters (see section IV. C in [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF]). Notice that these interferences would be reduced if the impedance values are different on both supersonic regions.

• When Re(ω) > ω max we no longer see any unstable mode. This a priori nontrivial, since negative energy waves still exist pass this threshold frequency. The reason is that when ω > ω max , the negative energy waves can transfer energy only to the mode k A+ , but the latter is essentially decoupled from the other ones. This means that the transmission is almost perfect at both interfaces, which prevents the unstable mechanism.

As we mentioned in the introduction, a similar laser instability ('the black hole laser') has been studied in Bose-Einstein condensates, in theory [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF][START_REF] Finazzi | Black-hole lasers in Bose-Einstein condensates[END_REF][START_REF] Michel | Saturation of black hole lasers in Bose-Einstein condensates[END_REF][START_REF] Finazzi | Instability of the superfluid flow as black-hole lasing effect[END_REF] but also observed in an experiment [START_REF] Steinhauer | Observation of self-amplifying Hawking radiation in an analog black hole laser[END_REF]. The above properties of the acoustic black hole laser are very close to the one identified in Bose-Einstein condensates. This is somewhat surprising, since in the latter, it is the negative energy wave that is trapped in the cavity (compare with Fig. 3). This is due to a different nature of the dispersion relation in both media: in condensates, acoustic waves propagate faster when the wavelength decreases, while they propagate slower in acoustic liners (see Fig. 2). However, that the two configurations share many similarities was anticipated in [START_REF] Coutant | Black hole radiation with short distance dispersion, an analytical S-matrix approach[END_REF] using a semiclassical symmetry argument. We now briefly discuss the influence of the Mach number on the laser effect. In Fig. 7, we have shown the evolution of unstable modes when varying M 0 for two values of L. First, we recall that if M 0 < 1/ √ 1 + b O , the flow is everywhere subsonic. On the contrary, when M 0 > 1/ √ 1 + b C , the flow is everywhere supersonic. The instability can a priori exist in these regimes, since both negative and positive energy waves are present. However, it is in general much less strong. This is because transsonic flows couples these waves together much more efficiently. For instance, in Fig. 7 we see that everywhere subsonic flows alternate between being stable and unstable, but the growth rate is significantly smaller that in the doubly transsonic case. In the latter regime, we observe that at low Mach, the instability is dynamical, and it becomes static at higher values of M 0 .

B. Complex spectrum for a finite boundary layer

We compute the spectrum again, but using a finite boundary layer. Since there is an extra solution k B of the dispersion relation [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF], purely outgoing modes now read

ϕ ω (x) =          a L e ik Lev x J(ω, k Lev ) + a B e ik B x J(ω, k B ) (x < 0), a N e ik N x J(ω, k N ) + a A+ e ik A+ x J(ω, k A+ ) + a R e ik Rev x J(ω, k Rev ) (L < x). (30)
Since the order of the equation has changed, so did the continuity conditions at the interfaces. Inspection of the mode equation [START_REF] Brambley | Fundamental problems with the model of uniform flow over acoustic linings[END_REF] shows that ϕ, ∂ x ϕ, ψ, and ∂ x ψ, and b(ω

+ iM 0 ∂ x )ψ + δ ω ∂ 2
x ψ are continuous. Again, the last condition corresponds to the continuity of the current [START_REF] Steinhauer | Observation of thermal Hawking radiation and its entanglement in an analogue black hole[END_REF].

The results for the spectrum and its evolution when varying L are exposed in Figs. 8 and9. The first important difference is that static instabilities are much less frequent. The trajectory of purely imaginary frequency travelling up and down found in the Ingard-Myers case (Fig. 5) is replaced by a complex frequency coming from the lower half plane, crossing the real line at about ω ∼ ω min , reaching a maximum imaginary part before travelling down to ω = 0 (see the blue line in Fig. 8). When this mode is unstable (Im(ω) > 0), it satisfies Re(ω) ω min . Since ω min is a cut-off frequency induced by the boundary layer (ω min = O(δ)), such instability is a low frequency one. When the boundary layer thickness is tiny enough, this mode can merge with its partner with respect to [START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF] and stick to the imaginary axis, becoming a fully static instability (see Fig. 11). Upon approaching ω = 0, they will split again into a pair with Re(ω) = 0. In addition, the maxima reached by the imaginary part of ω are quite smaller. This means that a thin boundary layer tends to stabilize the system. The second main difference is in the region Re(ω) > ω min . There is now a series of resonances, with a rather small frequency gap. The explanation is that the boundary layer mode k B and the negative energy mode k N form a Fabry-Perot like cavity: a wave can travel from right to left with wave number k B , be converted to a mode of wavenumber k N to travel back right. Eigenmodes of this effective cavity are then characterized by having a finite number of wavelength fitting inside the two interfaces (Bohr-Sommerfeld condition). To confirm that this is the case, we compare the real parts of these resonances to the solutions of the condition

L(k B (ω) + k N (ω)) = 2nπ + ν (n ∈ Z), (31)
where ν is sum of the phase shifts induced at each interfaces. In the short wavelength limit, ν is twice the well-known Airy phase shift of π/2. In general, it stays of order 1. The result is shown in Fig. 11. We see that the condition captures well the set of Fabry-Perot like resonances. One can notice that the condition predicts a slightly larger frequency gap. This is due to the fact that ν is not constant, but slowly varies with ω. We have shown the first, third and fifth mode (see remark in Fig. 8). On the right: real parts. On the left: imaginary parts.

When varying L, the resonances come closer together. From time to time one resonance will come out of the set and migrates to the upper plane, becoming an instability (see trajectories in Fig. 8). This mechanism replaces the appearance of dynamical instabilities (Re(ω) = 0) from purely imaginary frequency merging in the Ingard-Myers case. Finally, in Fig. 12, we show the profiles of unstable modes.

IV. CONCLUSION

In this work we have studied a particular configuration of lined ducts, where the flow changes from effectively supersonic (c eff < U 0 ) to subsonic (c eff > U 0 ) to supersonic again. We have shown that these configurations are subject to a temporal instability similar to the "black hole laser instability" [START_REF] Coutant | Black hole lasers, a mode analysis[END_REF]. In addition, we have studied the effect of different boundary conditions taking into account the boundary shear flow layer near the impedance wall: an infinitely small layer (Ingard-Myers condition) and a finite but small 8). On the left: fifth mode for L = 4 (cyan curve in Fig. 8). The grey shading delimitates the subsonic region.

layer (modified Brambley condition ( 4)).

This laser instability has several remarkable properties. When using the Ingard-Myers condition, unstable modes divide into two classes: static instabilities, associated with a purely imaginary frequency, and dynamical instabilities with a complex frequency (see Fig. 5). When taking into account the boundary layer's thickness, we found that static instabilities acquire a finite real part of order O(δ). Only for very small δ can one recover static instabilities with Re(ω) = 0 (see Fig. 10). Moreover, the presence of a boundary layer tends to the growth rates of unstable modes compared to the Ingard-Myers case.

The stability of acoustic liner with grazing flow is still much debated, and in particular the role of boundary layers [START_REF] Brambley | Surface modes in sheared boundary layers over impedance linings[END_REF][START_REF] Rienstra | Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall[END_REF]. The present work shows that varying impedances can lead to much stronger instabilities. This is particularly true when transitions from effectively subsonic to supersonic are involved (see Fig. 7) because they couple together all propagating modes present at a given frequency. impedance wall, but there will be inevitably be another one near the hard wall. This second boundary layer can quite possibly give rise to more propagating modes (as one could see by using again the boundary condition (4) with b = 0), but we assume here that such extra modes would not couple significantly from the ones present in our model. We believe that a precise understanding of these two effects go beyond the scope of this paper, but it would be interesting to understand how they could affect the laser instability, and more generally the scattering over a spatially varying impedance.

To gain more confidence in the effective boundary condition used in this work, we compared the dispersion relation obtained with it (equation ( 8)) with the one obtained by directly solving the Pridmore-Brown equation. In order to avoid complication due to critical layers, we restricted ourselves to ω > 0 and k < 0. The result is shown in Fig. 13. Although the differences are quantitatively significant, the qualitative behavior and main features are fairly reproduced. It is instructive to look at the asymptotics of the solutions of the dispersion relation for small δ. The first thing to notice is that ω min = O(δ), while ω max is independent of the boundary layer, i.e. O(1). Thus, in the limit of small δ, there is a separation of scale. Let us consider ω min ω ω max in a subsonic flow, in which case there are five real solutions. The first four solutions are weakly dependent on δ (see Fig. 2a) and can be estimated using the Ingard-Myers boundary condition (i.e. δ = 0 in (4)). Moreover, using ω ω max , we can solve the dispersion relation at first order in ω/ω max . In that limit, two modes scale as O(ω). These are the acoustic modes, which reads

k A-∼ √ 1 + b M 0 √ 1 + b -1 ω, (B1a)
k A+ ∼ √ 1 + b M 0 √ 1 + b + 1 ω. (B1b)
Because of the flow, there are also two hydrodynamic modes k S , k N . Using the Ingard-Myers boundary condition, their first order expressions are of the form:

k S ∼ -k Z + ω v Z g , (B2a)
k N ∼ k Z + ω v Z g . (B2b)
The value k Z at zero frequency is simply obtained and reads

k Z = 3 -3(1 + b)M 2 0 bM 2 0 (1 -M 2 0 ) . ( B3
)
A more tedious calculation leads to

v Z g = M 0 (1 -M 2 0 )(1 -M 2 0 (1 + b)) 1 -2M 2 0 + M 4 0 (1 + b) . ( B4
)
The fifth root only exist for finite boundary layer and scales as 1/δ. Its asymptotic expression for δ → 0 is given by

k B ∼ - M 0 b δ ω. (B5)
It is also interesting to obtain an asymptotic expression for ω min in the limit of small δ.

For this, we interpret the fact that ±k Z is not a solution of the dispersion relation [START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF] when δ = 0 as an avoided crossing. When δ = 0, the line of k S ∼ -k Z + ω/v Z g and ω = 0 (the degenerated line of the root k B for δ = 0) cross at k = -k Z and ω = 0. Calling D(ω, k) the left-hand side of equation ( 8), we approximate the dispersion relation near the crossing by

ωD(ω, k) δ=0 ∼ ∂ k D Z ω k + k Z - ω v Z g , (B6)
where ∂ k D Z is short for ∂ k D(ω = 0, k = -k Z ). Now, when δ is small but non-zero, the dispersion relation near the crossing becomes

ωD(ω, k) ∼ -∂ ω D Z ω + δk M b v Z g (k + k Z ) -ω -δM k 3 Z (1 -M 2 )k 2 Z + a , (B7)
where the last term is obtained by taking the limit of ωD(ω, k) at the would-be crossing. Using this we obtain two approximate branches ω ± (k). The minimum value of ω + gives us ω min . Doing so we have

ω min = δ M 2 b 6 b + 3(1 -M 2 (1 + b)) b(1 -M 2 ) + O(δ 2 ). ( B8
)
Using the same asymptotics, we can also evaluate the norm density (20) associated with each root of the dispersion relation. Using equation ( 22) in the limit δ → 0 and ω/ω max 1, we find

E s [k A± ] ∼ (1 + b)(ω -M 0 k A± ), (B9a) ∼ (1 + b)(ω -M 0 k A± ), (B9b)
for the acoustic modes,

E s [k S/N ] ∼ E s [∓k Z ] ∼ ± 3 -3(1 + b)M 2 0 b(1 -M 2 0 ) 1 + (1 -M 2 0 ) 2 bM 4 0 , (B10)
for the hydrodynamic modes, and

E s [k B ] ∼ b 5 M 6 0 (1 -M 2 0 ) 2 ω 3 18δ 4 , (B11)
for the boundary layer mode. We recall that the energy density of each mode is directly obtained by E e = ωE s . In these asymptotic expressions, we recover the important result that all modes have a positive energy except for k N , which corresponds to a negative energy wave.

Figure 1 :

 1 Figure 1: Configuration of the problem.

Figure 2 :

 2 Figure 2: Dispersion relation obtained in the one-dimensional model, i.e. equation (8). The solid blue line is the dispersion relation with the finite boundary layer correction (4), and the dashed red line is the dispersion relation in the Ingard-Myers limit (δ = 0). We have used M 0 = 0.3 and δ = 0.005, and b = 6 on the left and b = 15 on the right. [Note that in black and white, the curves k A+ and k N with the two boundary conditions on the right side are indistinguishable.]

Figure 3 :

 3 Figure 3: Schematic representation of the laser mechanism. Black arrows and lines carry a positive energy while red carry a negative energy. The mechanism keeps the total energy E N + E cav. + E A+ + E B constant.

Figure 4 :

 4 Figure 4: Evolution of Im(k ω ) when Im(ω) changes from large positive values to 0 for the modes in a supersonic region. We have taken M 0 = 0.3, b = 15, δ = 0.005, and Re(ω) = 0.03. Using the Briggs-Bers criterion, the sign of Im(k ω ) gives the direction of propagation of the corresponding mode. (We have verified that the signs of Im(k ω ) don't change when increasing Im(ω) further.)

Figure 5 :Figure 6 :

 56 Figure 5: Trajectory of unstable modes when L varies from 0 to 8 with M 0 = 0.3, b C = 6, b O = 15. On the left: solid (resp. dashed) lines show the evolution of the real (resp. imaginary) parts of ω as a function of L. On the right: trajectories in the complex plane of the first pair of modes.

Figure 7 :

 7 Figure 7: Evolution of unstable modes when varying M 0 . We have chosen b C = 6, b O = 15. On the left: L = 2.5, on the right: L = 4.5. Solid lines are real parts, and dashed lines are imaginary parts. The black dashed lines shows the values of M 0 belowwhich the flow is everywhere subsonic (M 0 < 0.25) and above which it is everywhere supersonic (M 0 > 0.38).

3 Figure 8 :

 38 Figure 8: Trajectory of unstable modes when L varies from 0 to 8 with M 0 = 0.3, b C = 6, b O = 15, and δ = 0.005. We have shown the first, third and fifth mode (the second and fourth are stable or have a much smaller imaginary part, hence are subdominant for the laser instability). The dashed black lines mark Re(ω) = ω min and Re(ω) = ω max .

Figure 9 :

 9 Figure9: Evolution of complex ω as a function of L. We have shown the first, third and fifth mode (see remark in Fig.8). On the right: real parts. On the left: imaginary parts.

Figure 10 :

 10 Figure 10: Trajectories of the first unstable mode for L from 0.6 to 3.5 and for different values of the boundary layer thickness δ.

Figure 11 :

 11 Figure 11: Color map of ln | det(a L , a B , a N , a A+ , a S )| with M 0 = 0.3, b C = 6, b O = 15, δ = 0.005 and L = 4. The white crosses shows the (real) frequencies satisfiying the Bohr-Sommerfeld condition (31), which we have used with ν = 0. The dashed black lines mark Re(ω) = ω min and Re(ω) = ω max .

Figure 12 :

 12 Figure 12: Profile of unstable modes with M 0 = 0.3, b C = 6, b O = 15. On the right: first mode for L = 4 (blue curve in Fig.8). On the left: fifth mode for L = 4 (cyan curve in Fig.8). The grey shading delimitates the subsonic region.

Figure 13 :

 13 Figure 13: Dispersion relation in the ω > 0 and k < 0 range. The solid blue line is obtained using the one-dimensional model with effective boundary condition (equation (8)), while the dashed black line is obtained by direct integration of the Pridmore-Brown equation. We have used b = 6, M 0 = 0.3, δ = 0.005 and a Mach profile given by M (y) = M 0 (1 -y 1/δ ).

At this level, we considered localized time dependent solutions (finite energy), and hence all boundary terms arising by integration by parts vanish.

Notice that unlike equation[START_REF] Euvé | Observation of noise correlated by the Hawking effect in a water tank[END_REF], the local conservation law[START_REF] Torres | Rotational superradiant scattering in a vortex flow[END_REF] does not rely on any assumption on boundary terms.

This distinction is similar to the spectrum of PT symmetric Hamiltonian, which are purely real only when they are maximally symmetric[START_REF] Bender | Introduction to PT -symmetric quantum theory[END_REF]. In fact, the symmetry[START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF] comes from the symmetry (ω, k) → (-ω, -k) of the dispersion relation[START_REF] Gabard | A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow[END_REF], which can be seen as a local version of PT symmetry.

Appendix A: Low frequency Brambley condition

In this appendix, we briefly explain how the effective boundary condition (4) has been obtained from the work of [START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF][START_REF] Brambley | Surface modes in sheared boundary layers over impedance linings[END_REF]. This condition corresponds to the "short wavelength modified Myers" derived by Brambley (see section III.B in [START_REF] Brambley | Well-posed boundary condition for acoustic liners in straight ducts with flow[END_REF] and 4 in [START_REF] Brambley | Surface modes in sheared boundary layers over impedance linings[END_REF]), but is further simplified by discarding mass, momentum and kinetic energy deficits of the boundary layer (encoded in I 0 ). To derive that boundary condition, it is assumed that the Mach profile, which we denote M (y), is constant equal to M 0 except in a region of size O(δ) near the impedance wall (δ will be defined more precisely later on). The normalized mean density ρ 0 (y) is also allowed to vary away from its bulk value in the boundary layer. Under this assumption, Brambley has shown that, to first order in δ, the effect of the boundary layer is equivalent to a modified boundary condition:

where Z b is the impedance of the wall supposed to depend only on the frequency. I 0 and I 1 are integrals of the boundary layer profiles, given by

Notice that both are of order O(δ) since M (y) ∼ M 0 and ρ 0 (y) ∼ 1 outside the boundary layer. We now approximate this boundary condition in the limit of low frequency, more precisely assuming ω |k| only in the terms of order δ. The reason is that for small δ (lower than 1% in this work), the O(δ) correction of the Ingard-Myers condition is always small unless ω |k|. In this limit, while the integral I 0 stay bounded, I 1 dominates and becomes

where the effective boundary layer size δ (in units of the transverse size of the duct) is given by

Notice that this expression is exact for a piecewise linear velocity profile. Taking the limit ω |k| in Brambley's condition (A1), as well as the low frequency impedance Z b ∼ i/(bω), we obtain the effective boundary condition (4) used in the core of the paper.

At this level, we would like to make a few remarks concerning the validity of our approximate boundary condition [START_REF] Aurégan | Experimental evidence of an instability over an impedance wall in a duct with flow[END_REF]. First-of-all, the derivation used by Brambley ignores the possible presence of a critical layer within the boundary layer. This is technically illicit in the range 0 < ω/k < M 0 , which corresponds to the hydrodynamic continuum. Some arguments were presented by Brambley, Darau and Rienstra suggesting that the critical layer could be neglected under reasonable assumptions [START_REF] Brambley | The critical layer in linear-shear boundary layers over acoustic linings[END_REF], but the fact that the impedance changes along the wall could on the contrary excite modes in the hydrodynamic continuum [START_REF] Dai | A cavity-by-cavity description of the aeroacoustic instability over a liner with a grazing flow[END_REF]. Moreover, we discussed here only the boundary layer on the side of the


Julien Bichon
email: julien.bichon@uca.fr

Agustín García
email: aigarcia@famaf.unc.edu.ar

Agust Ín

Garc Ía

Hopf-Galois structures on ambiskew polynomial rings

published or not. The documents may come

Introduction

Let R be an algebra over a field k and consider the following data:

(1) A pair of commuting algebra automorphisms τ, ω ∈ Aut R; set σ := τ ω = ωτ .

(2) A σ-central element c; that is c ∈ Z σ (R) := {x ∈ R : xr = σ(r)x, r ∈ R}.

(3) A scalar ξ ∈ k * . The associated generalized ambiskew polynomial algebra A = A(R, X, Y, τ, ω, c, ξ), see [START_REF] Jordan | Simple ambiskew polynomial rings[END_REF], is the quotient of the free product R * k X, Y by the relations

Xr = τ (r)X, Y r = ω(r)Y, XY -ξY X = c,
for any r ∈ R.

When c is assumed to be central (so that σ = id or ω = τ -1 ) then A = A(R, X, Y, τ, τ -1 , c, ξ) is called an ambiskew polynomial algebra [START_REF]Down-up algebras and ambiskew polynomial rings[END_REF]; we set A(R, X, Y, τ, c, ξ) := A(R, X, Y, τ, τ -1 c, ξ).

A common question is about the ring-theoretical properties such as simplicity, or the representation theory, of the resulting algebra A, see for example [START_REF] Jordan | A simple localization of the quantized Weyl algebra[END_REF], [START_REF] Jordan | Simple ambiskew polynomial rings[END_REF], a useful summary of known results being given in [4, Section 6].

Another interesting question is to find necessary and sufficient conditions on these data so that A preserves some extra structure on R. When R is a Hopf algebra, Brown-Macaulay [START_REF] Brown | Ambiskew Hopf algebras[END_REF] discuss the case in which the Hopf structure on R extends to a Hopf structure on the ambiskew polynomial Hopf algebra A R = A(R, X, Y, τ, c, ξ) via the formula

∆(X) = X ⊗ 1 + g ⊗ X, ∆(Y ) = Y ⊗ 1 + h ⊗ Y (1.1)
for some group-like elements g, h ∈ G(R); here, c is central 1 . This generalized previous results from [START_REF] Hartwig | Hopf structures on ambiskew polynomial rings[END_REF] in the case when R is commutative. As an example, U q (sl 2 ) can be constructed from kZ by this process.

In this article, we generalize these results in two directions: (A) We discuss the extension of the Hopf algebra structure on R to a generalized ambiskew polynomial algebra A. (B) We further extend this to the case when R is a Hopf-Galois object over some Hopf algebra. Hopf-Galois objects are natural generalizations of Hopf algebras with a Galois-theoretic flavour, introduced in [START_REF] Kreimer | Hopf algebras and Galois extensions of an algebra[END_REF], and have proved to be fundamental tools in analysing tensor categories of comodules [START_REF] Ulbrich | Fibre functors of finite-dimensional comodules[END_REF][START_REF] Schauenburg | Hopf bigalois extensions[END_REF] as well as in recent classification questions for pointed Hopf algebras [START_REF] Andruskiewitsch | Lifting via cocycle deformation[END_REF][START_REF] Angiono | Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations[END_REF].

While our first goal was to extend the results of [START_REF] Brown | Ambiskew Hopf algebras[END_REF] to Hopf-Galois objects in the case of ambiskew polynomial rings, our analysis evidenced that the proper context for doing that was that of generalized ambiskew polynomial rings, leading to the results in item (A).

The punchline for item (B) is that while having to deal simultaneously with R and the Hopf algebra coacting on it can look difficult, it turns out that, fortunately, results of Grunspan [START_REF] Grunspan | Quantum torsors[END_REF] and Schauenburg [START_REF]Quantum torsors with fewer axioms[END_REF] ensure that there is exists a definition of Hopf-Galois objects that does not use any Hopf algebra, in terms of an algebra map µ : R → R ⊗ R op ⊗ R, subject to certain axioms, see Section 2. We start with a Hopf-Galois algebra R, and discuss the ingredients with necessary and sufficient conditions to extend the Hopf-Galois structure of R to A(R, X, Y, σ, c, ξ) via the formula

µ(X) = X ⊗ 1 ⊗ 1 -g ⊗ g -1 X ⊗ 1 + g ⊗ g -1 ⊗ X, µ(Y ) = Y ⊗ 1 ⊗ 1 -h ⊗ h -1 Y ⊗ 1 + h ⊗ h -1 ⊗ Y, (1.2)
for some quasi-central group-like elements g, h ∈ G 0 (R), see Definition 2. [START_REF] Brown | Ambiskew Hopf algebras[END_REF].

Given a Hopf-Galois algebra (R, µ) as above, there exists a canonical Hopf algebra H(R) such that R is an H(R)-Galois object [START_REF] Grunspan | Quantum torsors[END_REF][START_REF]Quantum torsors with fewer axioms[END_REF]. As for the first item (A), the need of a generalization of the results in [START_REF] Brown | Ambiskew Hopf algebras[END_REF] becomes evident when one considers the Hopf algebra associated to the Hopf-Galois algebra A R = A(R, X, Y, τ, c, ξ) extending a Hopf-Galois algebra R: this Hopf algebra is an extension of H(R) not in terms of an ambiskew polynomial algebra but of a generalized ambiskew polynomial algebra A. See Corollary 5.5 and Subsection 6.5 for an example.

The paper is organized as follows. In Section 2 we recall the definition of Hopf-Galois algebras (R, µ) and their realization as Hopf-Galois objects over a Hopf algebra H(R); we discuss the concepts of group-like and skew-primitive elements in this setting. Section 3 is devoted to the answer of item (A) above: in Theorem 3.1 we find necessary and sufficient conditions so that a generalized ambiskew polynomial algebra A R extends the structure of a Hopf algebra R in a way such that the added variables are skew-primitive; we recover some of the results in [START_REF] Brown | Ambiskew Hopf algebras[END_REF] as a corollary. Next, in Section 4 we find necessary and sufficient conditions so that a generalized ambiskew polynomial algebra extends the structure of a Hopf-Galois algebra R as in (1.2) and we provide an answer for item (B) in Theorem 4.1. In Section 5 we show in Theorem 5.3 that the Hopf algebra associated to a Hopf-Galois ambiskew polynomial algebra A R is a Hopf ambiskew polynomial algebra, associated to H(R). We conclude the article with a series of examples in Section 6.

Notation and conventions. We work over a field k. All algebras, vector spaces and unadorned tensor products ⊗ are assumed to be defined over k. We assume that the reader has familiarity with Hopf algebra theory, for which the book [START_REF] Montgomery | Hopf algebras and their actions on rings[END_REF] is a convenient reference. We use standard notations, in particular ∆, ε and S stand respectively for the comultiplication, counit and antipode of a Hopf algebra, and Sweedler's notation

h → h (1) ⊗ h (2) , resp. m → m (0) ⊗ m (1)
for the comultiplication, resp. the coaction on a (right) comodule M , of a Hopf algebra H; h ∈ H, m ∈ M Acknowledgements. The authors were supported by CONICET, FONCyT PICT-2015-2845, Secyt (UNC), the MathAmSud project GR2HOPF, and the Emergence Program of the I-SITE project CAP 20-25 of Clermont Auvergne University.

Hopf-Galois algebras

2.1. Basic definitions. The definition below was introduced by Grunspan [START_REF] Grunspan | Quantum torsors[END_REF] with more axioms, and simplified later by Schauenburg [START_REF]Quantum torsors with fewer axioms[END_REF], under the name quantum torsor. In view of Theorem 2.8 given in the next subsection, it seems natural to call such a structure a Hopf-Galois algebra.

Definition 2.1. A Hopf-Galois algebra is a non-zero algebra R together with an algebra map

µ : R -→ R ⊗ R op ⊗ R such that (µ ⊗ id R ⊗ id R ) • µ = (id R ⊗ id R ⊗ µ) • µ, (2.1) (m ⊗ id R ) • µ = η ⊗ id R , (id R ⊗ m) • µ = id R ⊗ η, (2.2)
where m : R ⊗ R → R and η : k → R denote the respective multiplication and unit of R.

We shall write

µ(r) = r (1) ⊗ r (2) ⊗ r (3) , r ∈ R. Hence, for any r ∈ R, r ⊗ 1 = r (1) ⊗ r (2) r (3) , 1 ⊗ r = r (1) r (2) ⊗ r (3) ;
and we may set

r (1) ⊗ r (2) ⊗ r (3) ⊗ r (4) ⊗ r (5) := µ(r (1) ) ⊗ r (2) ⊗ r (3) = r (1) ⊗ r (2) ⊗ µ(r (3) ).
Example 2.2. The basic example is a Hopf algebra H with

µ(x) = x (1) ⊗ S(x (2) ) ⊗ x (3) , x ∈ R.
We denote by R(H) the resulting Hopf-Galois algebra. This defines a functor from the category of Hopf algebras to the category of Hopf-Galois algebras (a morphism of Hopf-Galois algebras being an algebra map commuting with µ in the obvious way).

A Hopf-Galois algebra arises from a Hopf algebra as above if and only if there exists an algebra map α : R → k. Indeed, starting with R and such an α, the Hopf algebra H whose counit is α and with the other structure maps defined by

∆(x) = α(x (2) )x (1) ⊗ x (3) , S(x) = α(x (1) x (3) )x (2) is such that R = R(H).
Example 2.3. A more significant example is the Weyl algebra

A 1 (k) = k x, y | xy -yx = 1
with Hopf-Galois structure defined by

µ(x) = x ⊗ 1 ⊗ 1 -1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x, µ(y) = y ⊗ 1 ⊗ 1 -1 ⊗ y ⊗ 1 + 1 ⊗ 1 ⊗ y.
More examples will be discussed in the final section. We now introduce group-like elements and skew-primitives in Hopf-Galois algebras.

Definition 2.4. A group-like element in a Hopf

-Galois algebra R is an invertible element g ∈ R such that µ(g) = g ⊗ g -1 ⊗ g. (2.3)
We denote by G 0 (R) the set of group-like elements in R; g ∈ G 0 (R) is quasi-central if there exists a character α on G 0 (R) such that xg = α(x)gx, for any x ∈ G 0 (R).

If g ∈ G 0 (R), then we write g • r := grg -1 . Remark 2.5. It should be noticed that, contrary to the Hopf algebra case, if g ∈ R is grouplike, then λ g is grouplike for any nonzero scalar λ ∈ k. If we consider G 0 (R) acting on R by conjugation, then g and λ g define the same operator. Definition 2.6. Let R be a Hopf-Galois algebra and let g, h ∈ G 0 (R). The set P 0 g,h (R) of (g, h)-skew primitive elements is the collection of those x ∈ R such that

µ(x) = x ⊗ h -1 ⊗ h -g ⊗ g -1 xh -1 ⊗ h + g ⊗ g -1 ⊗ x. Remark 2.7. If x ∈ P 0 g,h (R), then h -1 x ∈ P 0 h -1 g,1 (R) and xg -1 ∈ P 0 1,hg -1 (R). Notice also that g, h ∈ P 0 g,h (R). Indeed, µ(h) = h ⊗ h -1 ⊗ h = h ⊗ h -1 ⊗ h -g ⊗ g -1 hh -1 ⊗ h + g ⊗ g -1 ⊗ h.
and similarly for g.

2.2.

From Hopf-Galois algebras to Hopf algebras. We recall that a (right) Hopf-Galois object R over a Hopf algebra H is a (right) comodule algebra R over H with trivial coinvariants R co H = k and such that the Galois map

can : R ⊗ R → R ⊗ H, can(r ⊗ s) = rs (0) ⊗ s (1) , r, s ∈ R is bijective.
The following result relates what we called Hopf-Galois algebras and classical Hopf-Galois objects.

Theorem 2.8 (Grunspan-Schauenburg, [START_REF] Grunspan | Quantum torsors[END_REF][START_REF]Quantum torsors with fewer axioms[END_REF]). Let R be an algebra. The following assertions are equivalent.

(1) There exists an algebra map µ : R -→ R⊗R op ⊗R making R into a Hopf-Galois algebra.

(2) R is a right Hopf-Galois object over some Hopf algebra H(R).

Therefore the notion of Hopf-Galois algebra provides a definition of Hopf-Galois object which is Hopf algebra free.

The constuction of H(R) in the theorem is as follows. Let R = (R, µ) be a Hopf-Galois algebra. Then R is a right Hopf-Galois object for the Hopf algebra H(R) defined as the subalgebra (2.4)

H(R) = x ⊗ y : xy (1) ⊗ y (2) ⊗ y (3) = 1 ⊗ x ⊗ y ⊆ R op ⊗ R with coalgebra structure (2.5) ∆(x ⊗ y) = x ⊗ y (1) ⊗ y (2) ⊗ y (3) , ε(x ⊗ y) = xy.
The coaction is given by µ, as µ : R → R ⊗ H(R).

Remark 2.9. Here, we follow Schauenburg in [15, §3] and we write a generic element x i ⊗y i ∈ R op ⊗ R as a single tensor x ⊗ y, "in the spirit of Sweedler's notation". Definition (2.4) and formula (2.5) should be interpreted in this sense.

Conversely, let H be a Hopf algebra and let R be a right Hopf-Galois object, with coaction r → ρ(r) = r (0) ⊗ r (1) , r ∈ R. Then R is a Hopf-Galois algebra with

µ(r) = r (0) ⊗ can -1 (1 ⊗ r (1) ).
We now use the construction H(R) to relate the group-likes and skew-primitives of the previous subsection with ordinary group-likes and skew-primitives in a Hopf algebra. Lemma 2.10. Let R be a Hopf-Galois algebra.

(1) There is a short exact sequence of groups

1 → k * → G 0 (R) ϕ -→ G(H(R)) → 1, where ϕ(g) = g -1 ⊗ g. (2) For any g ∈ G 0 (R), there is a linear short exact sequence 0 → k → P 0 g,1 (R) ψ -→ P ϕ(g),1 (H(R)) → 0, where ψ(x) = g -1 ⊗ x -g -1 x ⊗ 1.
Proof. (1) On the one hand, we see that

k * ⊂ G 0 (R), as in Remark 2.5. If g ∈ G 0 (R), then it is clear that g -1 ⊗ g ∈ G(H(R)). Moreover, ϕ(λ) = 1 ⊗ 1 for any λ ∈ k * . Now, if u = x ⊗ y = i x i ⊗ y i ∈ G(H(R)), then ε(x ⊗ y) = xy = i x i y i = 1 and ∆(x ⊗ y) = x ⊗ y ⊗ x ⊗ y = i,j x i ⊗ y i ⊗ x j ⊗ y j .
On the other hand, by definition

∆(x ⊗ y) = x ⊗ y (1) ⊗ y (2) ⊗ y (3) .
Hence, by multiplying the factors in the middle, we get

x ⊗ 1 ⊗ y = i x i ⊗ 1 ⊗ y i = i,j x i ⊗ y i x j ⊗ y j .
If we assume that {x i } i is linearly independent, we obtain, for any i, that 1 ⊗ y i = j y i x j ⊗ y j and hence 1 ⊗ µ(y i ) = j y i x j ⊗ µ(y j ).

Multiplying the left handed tensors, this gives

µ(y i ) = j y i x j y j(1) ⊗ y j(2) ⊗ y j(3) = j y i ⊗ x j ⊗ y j = y i ⊗ u.
Applying the same reasoning to

u -1 = k x ′ k ⊗ y ′ k yields µ(y ′ k ) = y ′ k ⊗ u -1 for any k. Hence for any i, k, we have µ(y i y ′ k ) = y i y ′ k ⊗ 1 ⊗ 1 and µ(y ′ k y i ) = y ′ k y i ⊗ 1 ⊗ 1, so that there exists scalars λ i,k , µ k,i such that y i y ′ k = λ i,k and y ′ k y i = µ k,i .
There exists j, l such that y j y ′ l = 0, so we see from the previous identities that λ j,l = µ l,j and that y ′ l is invertible in R. Hence for any i we have

y i = λ i,l (y ′ l ) -1 . We get u = ( i λ i,l x i ) ⊗ (y ′ l ) -1 = g -1 ⊗ g for g = y ′ l -1
, and we then see that g ∈ G 0 (R).

(2) We know that k ⊂ P 0 g,1 (R), see Remark 2.7. Also, it is easy to check that g

-1 ⊗ x - g -1 x ⊗ 1 ∈ P ϕ(g),1 (H(R)) if x ∈ P 0 1,g (R), and that Ker(ψ) = k. Now, let u = x⊗y = x i ⊗y i ∈ P ϕ(g),1 (H(R)). We assume that {y i } is linearly independent. We have xy (1) ⊗ y (2) ⊗ y (3) = 1 ⊗ x ⊗ y and (2.6) x ⊗ y (1) ⊗ y (2) ⊗ y (3) = x ⊗ y ⊗ 1 ⊗ 1 + g -1 ⊗ g ⊗ x ⊗ y
so that multiplying the factors in the middle as before:

x ⊗ 1 ⊗ y = x ⊗ y ⊗ 1 + g -1 ⊗ gx ⊗ y, i.e. i x i ⊗ 1 ⊗ y i = i x i ⊗ y i ⊗ 1 + i g -1 ⊗ gx i ⊗ y i .
If 1 ∈ Span{y i }, we get that u = i x i ⊗ y i = 0. Otherwise, there exists for any i, a scalar λ i such that

x i ⊗ 1 = g -1 ⊗ gx i + λ i u with λ k = 0 for some k. This gives u = g -1 ⊗ z -g -1 z ⊗ 1 for z = -λ -1 k gx k . Going back to the expression (2.6), we get g -1 ⊗µ(z)-g -1 z⊗1⊗1⊗1 = g -1 ⊗z⊗1⊗1-g -1 z⊗1⊗1⊗1+g -1 ⊗g⊗g -1 ⊗z-g -1 ⊗g⊗g -1 z⊗1.
This shows that z ∈ P 0 g,1 (R), and we have u = ψ(z).

Ambiskew Hopf algebras

In this section we assume that R is a Hopf algebra and consider the generalized ambiskew algebra A = A(R, E, F, τ, ω, c, ξ) as in the introduction. We look for necessary and sufficient conditions to extend the comultiplication in R to A via the formulas

∆(E) = E ⊗ 1 + g ⊗ E, ∆(F ) = F ⊗ 1 + h ⊗ F, (3.1)
for some (necessarily group-like) elements g, h ∈ G(R).

When the element c is assumed to be central, or equivalently when ω = τ -1 (so σ = id), this question has been addressed in [START_REF] Brown | Ambiskew Hopf algebras[END_REF]. Our result reads as follows.

Theorem 3.1. Let H be a Hopf algebra, let τ, ω ∈ Aut H be such that τ ω = ωτ and define σ = τ ω. Let c ∈ Z σ (H) be σ-central and ξ ∈ k * . Then the generalized ambiskew polynomial algebra A(H, E, F, τ, ω, c, ξ) has a Hopf algebra structure extending that of H and such that (3.1) holds for some g, h ∈ G(H) if and only if

• there are characters α, β of H, with α * β = β * α and H,E,F,τ,ω,c,ξ) is a Hopf algebra as in the statement. Then, by comparing ∆(Xr) and ∆(τ (r)X) we obtain that

τ (r) = α(r (1) )r (2) , ω(r) = β(r (1) )r (2) . (3.2) α(r (1) )r (2) = g • r (1) α(r (2) ), β(r (1) )r (2) = h • r (1) β(r (2) ) r ∈ H, (3.3) • the group-like elements g, h are central in G(H) and α(h) = β(g) -1 = ξ, • the σ-central element c is (gh, 1)-skew primitive. Proof. Assume A = A(
τ (r (1) ) ⊗ r (2) = τ (r) (1) ⊗ τ (r) (2) = g • r (1) ⊗ τ (r (2) ). Hence, if α = ε • τ : H → k we get that α(r (1) )r (2) = g • r (1) α(r (2)
) and τ (r) = α(r (1) )r [START_REF] Angiono | Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations[END_REF] . Similarly, by comparing ∆(Y r) and ∆(ω(r)Y ) we obtain that ω(r) = β(r (1) )r (2) for β = ε • ω and both identities in (3.3) hold. Also, since τ ω = ωτ , it follows that α * β = β * ω. Now, observe that (3.3) applied to r = h gives g • h = h, that is gh = hg. We use this to see that

∆(EF -ξF E) =(EF -ξF E) ⊗ 1 + gh ⊗ (EF -ξF E) + (α(h) -ξ)hE ⊗ F + (1 -ξβ(g))gF ⊗ E = ∆(c) ∈ H ⊗ H and thus, since A is a free R-module with basis {X a Y b : a, b ≥ 0}, we have α(h) = β(g) -1 = ξ and ∆(c) = c ⊗ 1 + gh ⊗ c, so c = EF -ξF E is a σ-central (gh, 1)
-skew primitive element. The previous computations allow as well to prove converse statement. Remark 3.2. Notice that g • c = gσ(g) -1 c = α(g) -1 β(g) -1 c = ξα(g) -1 c. We can thus compute τ (c) in two ways and get

ξ(α(g) -1 -α(g))c = α(c)(1 -gh).
So either α(g) = ±1 (and α(c) = 0 or 1 = gh) or c = α(ghc)

1-α(g) 2 (1gh). Similarly, if we compute ω(c), we get that

ξ -1 (β(h) -1 -β(h))c = β(c)(1 -gh), that is β(h) = ±1 (and β(c) = 0 or 1 = gh) or c = β(ghc)
1-β(h) 2 (1gh). Theorem 3.1 extends the result in [START_REF] Brown | Ambiskew Hopf algebras[END_REF] as mentioned; we recall this here as a corollary. ,E,F,τ,c,ξ) = A(R,E,F,τ,c,ξ) has a Hopf algebra structure extending that of H and such that (3.1) holds for some g, h ∈ G(H) if and only if there are

Corollary 3.3. [4] Assume c is central (i.e. ω = τ -1 so σ = id). Then the ambiskew polynomial algebra A(R
• a character χ of R, • central group-like elements g, h ∈ Z(G(R)) such that gh ∈ Z(R) and χ(g) = χ(h) = ξ,
• a central (gh, 1)-skew primitive element c, such that the following2 holds:

χ(r (1) )r (2) = g • r (1) χ(r (2) ), r ∈ R.
In this setting,

τ (r) = χ(r (1) )r (2) = g • r (1) χ(r (2) ).
Moreover, either • ξ = ±1; (and gh = 1 or χ(c) = 0),

or else • ξ = ±1 and c = λ(1 -gh), for λ = χ(c) 1 -ξ 2 ∈ k. ( 3

Ambiskew Hopf-Galois algebras

Let (R, µ) be a Hopf-Galois algebra and let us fix A = A(R, X, Y, τ, ω, c, ξ) an associated generalized ambiskew polynomial algebra. We investigate when A is again a Hopf-Galois algebra, extending R.

τ (r) (1) ⊗ τ (r) (2) ⊗ τ (r) (3) = τ (r (1) ) ⊗ r (2) ⊗ r (3) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ τ (r (3) ), (4.1) ω(r) (1) ⊗ ω(r) (2) ⊗ ω(r) (3) = ω(r (1) ) ⊗ r (2) ⊗ r (3) = ω(r (1) ) ⊗ ω(r (2) ) ⊗ ω(r (3) ). (4.2) (2) g, h ∈ G 0 (R) are quasi-central group-like elements such that, for any r ∈ R, g • r (1) ⊗ g • r (2) ⊗ r (3) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ r (3) , (4.3) h • r (1) ⊗ h • r (2) ⊗ r (3) = ω(r (1) ) ⊗ ω(r (2) ) ⊗ r (3) . (4.4) (3) ξ ∈ k * is a scalar such that τ (h) = ξ h, ω(g) = ξ -1 g. (4) c ∈ Z σ (R) ∩ P gh,1 (R), for σ := τ ω; that is c is a σ-central skew-primitive element: µ(c) = c ⊗ 1 ⊗ 1 -gh ⊗ (gh) -1 c ⊗ 1 + gh ⊗ (gh) -1 ⊗ c
and cr = σ(r)rc, for all r ∈ R. Then there is a unique Hopf-Galois algebra structure on A = A(R, X, Y, τ, ω, c, ξ) extending µ and such that µ

(X) = X ⊗ 1 ⊗ 1 -g ⊗ g -1 X ⊗ 1 + g ⊗ g -1 ⊗ X, µ(Y ) = Y ⊗ 1 ⊗ 1 -h ⊗ h -1 Y ⊗ 1 + h ⊗ h -1 ⊗ Y. (4.5)
Conversely, given commmuting τ, ω ∈ Aut(R), c ∈ Z σ (R) for σ = τ ω and ξ ∈ k * , if (4.5) defines a Hopf-Galois algebra structure on A(R, X, Y, τ, ω, c, ξ) extending µ, for some g, h ∈ G 0 (R), then conditions (1) to (4) are satisfied.

Proof. We assume there is an extension µ :

A → A ⊗ A op ⊗ A and elements g, h ∈ G(R) such that (4.5) holds. Recall the notation x • r := xrx -1 , for x ∈ G(R), r ∈ R.
♦The commutation rule. On the one hand, we have, in

A ⊗ A op ⊗ A, µ(Xr) = µ(τ (r)X) = (τ (r) (1) ⊗ τ (r) (2) ⊗ τ (r) (3) )µ(X).
On the other,

µ(Xr) = Xr (1) ⊗ r (2) ⊗ r (3) -gr (1) ⊗ r (2) g -1 X ⊗ r (3) + gr (1) ⊗ r (2) g -1 ⊗ Xr (3) = τ (r (1) )X ⊗ r (2) ⊗ r (3) -(g • r (1) )g ⊗ g -1 Xτ -1 (g • r (2) ) ⊗ r (3) + (g • r (1) )g ⊗ g -1 (g • r (2) ) ⊗ τ (r (3) )X = (τ (r (1) ) ⊗ r (2) ⊗ r (3) )(X ⊗ 1 ⊗ 1) -(g • r (1) ⊗ τ -1 (g • r (2) ) ⊗ r (3) )(g ⊗ g -1 X ⊗ 1) + (g • r (1) ⊗ g • r (2) ⊗ τ (r (3) ))(g ⊗ g -1 ⊗ X).
Since A is a free R-module with basis {X a Y b : a, b ∈ N}, we get the following identities:

τ (r) (1) ⊗ τ (r) (2) ⊗ τ (r) (3) = τ (r (1) ) ⊗ r (2) ⊗ r (3) = g • r (1) ⊗ g • r (2) ⊗ τ (r (3) ) g • r (1) ⊗ g • r (2) ⊗ r (3) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ r (3)
We may combine these identities to show that µ • τ = τ ⊗3 • µ:

τ (r (1) ) ⊗ r (2) ⊗ r (3) = g • r (1) ⊗ g • r (2) ⊗ τ (r (3) ) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ τ (r (3) )
and thus we see that our identities are then equivalent to (4.1) and (4.3). Similarly, from the identity Y r = ω(r)Y we obtain (4.2) and (4.4). ♦The group characters. We make the following Claim. There are characters α, β, γ, δ ∈ G 0 (R) such that for any x ∈ G 0 (R),

τ (x) = α(x)x ω(x) = β(x)x, (4.6) τ (x) = γ(x)g • x, ω(x) = δ(x)h • x. (4.7)
Hence, α(h)β(g) = γ(h)δ(g) and both g, h are quasi-central group-like elements in G 0 (R) with

gh(hg) -1 ∈ k * ⊂ Z(R), (gh) ⊗ (gh) -1 = (hg) ⊗ (hg) -1 . (4.8)
As for the proof, notice that combining (2.2) and (4.1) we get

τ (x)x -1 ⊗ x = 1 ⊗ τ (x),
x ∈ G 0 (R).

Hence τ (x) is a scalar multiple of x, and we get a character α ∈ G 0 (R). Similarly for ω; we call the corresponding character β and (4.6) follows. Now, we see from (4.3) that there is a character χ ∈ G 0 (R) such that g • x = χ(x)x; which shows that g is a quasi-central group-like element. Hence τ

(x) = γ(x)g • x, for γ(x) = α(x)χ(x) -1 . Analogously, h ∈ G(R) is quasi-central and ω(x) = δ(x)h • x, for some δ ∈ G 0 ; (4.7) follows.
Finally, we see that gh = α(h)γ(h) -1 hg, (4.9) whence gh(hg) -1 ∈ k × and (4.8) follows.

♦The bracket rule. Recall that, for any t ∈ G 0 (R), Xt = α(t) tX and Y t = β(t) tY . Also, observe that both XY and Y X are σ-central, σ = τ ω, in A R and hence XYξ Y X ∈ Z σ (R). Now,

µ(XY ) =XY ⊗ 1 ⊗ 1 -Xh ⊗ h -1 Y ⊗ 1 + Xh ⊗ h -1 ⊗ Y -gY ⊗ g -1 X ⊗ 1 + gh ⊗ h -1 Y g -1 X ⊗ 1 -gh ⊗ h -1 g -1 X ⊗ Y + gY ⊗ g -1 ⊗ X -gh ⊗ h -1 Y g -1 ⊗ X + gh ⊗ h -1 g -1 ⊗ XY =XY ⊗ 1 ⊗ 1 -α(h) hX ⊗ h -1 Y ⊗ 1 + α(h) hX ⊗ h -1 ⊗ Y -gY ⊗ g -1 X ⊗ 1 + β(g -1 ) gh ⊗ (gh) -1 Y X ⊗ 1 -gh ⊗ (gh) -1 X ⊗ Y + gY ⊗ g -1 ⊗ X -β(g -1 ) gh ⊗ (gh) -1 Y ⊗ X + gh ⊗ (gh) -1 ⊗ XY.
On the other hand, using (4.8),

µ(Y X) =Y X ⊗ 1 ⊗ 1 -β(g)gY ⊗ g -1 X ⊗ 1 + β(g)gY ⊗ g -1 ⊗ X -hX ⊗ h -1 Y ⊗ 1 + α(h -1 )gh ⊗ (gh) -1 XY ⊗ 1 -gh ⊗ (gh) -1 Y ⊗ X + hX ⊗ h -1 ⊗ Y -α(h -1 )gh ⊗ (gh) -1 X ⊗ Y + gh ⊗ (gh) -1 ⊗ Y X.
As we request XYξ Y X ∈ R, we see that

α(h) = ξ, β(g) = ξ -1 , (4.10)
and there is a σ-central skew-primitive element c ∈ Z σ (R) ∩ P gh,1 (R), so that XYξ Y X = c. As a result, τ (h) = ξ h, ω(g) = ξ -1 g.

♦On the converse. The previous computations show that given a Hopf-Galois algebra (R, µ), if there is a quintuple (τ, ω, g, h, c, ξ) satisfying conditions (1) to (4), then (4.5) defines a Hopf-Galois structure on A that extends µ.

This ends the proof of the theorem.

We write down the case ω = τ -1 for completeness. We start with a remark that will be of importance further on. Remark 4.2. We point out that even in the case ω = τ -1 we do not have gh = hg as in the Hopf algebra case, cf. Theorem 3.1 and Corollary 3.3. Instead, we have that g, h commute up to a scalar, see (4.8). In particular,

gh • r = α(h)γ(h) -1 r, r ∈ R. (4.11)
Corollary 4.3. Let R = (R, µ) be a Hopf-Galois algebra. Assume given a quintuple (τ, g, h, c, ξ) where

(1) τ ∈ Aut(R) is an algebra automorphism satisfying, for any r ∈ R,

τ (r) (1) ⊗ τ (r) (2) ⊗ τ (r) (3) = τ (r (1) ) ⊗ r (2) ⊗ r (3) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ τ (r (3) ). (4.12) (2) g, h ∈ G 0 (R) are quasi-central group-like elements such that, for any r ∈ R, g • r (1) ⊗ g • r (2) ⊗ r (3) = τ (r (1) ) ⊗ τ (r (2) ) ⊗ r (3) , (4.13) h • r (1) ⊗ h • r (2) ⊗ r (3) = τ -1 (r (1) ) ⊗ τ -1 (r (2) ) ⊗ r (3) . (4.14) (3) ξ ∈ k * is a scalar such that τ (g) = ξ g and τ (h) = ξ h. (4) c ∈ Z(R) ∩ P gh,1 (R) is a central skew-primitive element, i.e. µ(c) = c ⊗ 1 ⊗ 1 -gh ⊗ (gh) -1 c ⊗ 1 + gh ⊗ (gh) -1 ⊗ c.
Then there is a unique Hopf-Galois algebra structure on A = A(R, X, Y, τ, c, ξ) extending µ and such that (4.5) holds.

Conversely, given τ ∈ Aut(R), c ∈ Z(R) and ξ ∈ k * , if (4.5) defines a Hopf-Galois algebra structure on A(R, X, Y, τ, c, ξ) extending µ, for some g, h ∈ G 0 (R), then conditions (1) to (4) are satisfied.

Proof. This is Theorem 4.1 for ω = τ -1 . In particular, notice that the four conditions (4.1)-(4.4) become just three, as we obtain (4.12) and (4.13), together with the additional (4.14). For the remaining identity in (4.2), namely

τ -1 (r (1) ) ⊗ r (2) ⊗ r (3) = τ -1 (r (1) ) ⊗ τ -1 (r (2) ) ⊗ τ -1 (r (3) ),
we remark that it follows by applying τ -2 ⊗ τ -1 ⊗ τ -1 to the corresponding identity in (4.1), as µ • τ -1 = (τ -1 ) ⊗3 • µ .

The Hopf algebra associated to an ambiskew Hopf-Galois algebra

Let (R, µ) be a Hopf-Galois algebra. Recall that this is equivalent to a structure of Hopf-Galois object over certain Hopf algebra H(R), by Theorem 2.8. In this section, we study the corresponding Hopf algebra H(A R ), where R,X,Y,τ,ω,c,ξ) is the generalized ambiskew polynomial algebra that extends (R, µ) as in Theorem 4.1. Recall that σ = τ ω = ωτ .

A R = A(
We start with a technical lemma. We shall stick to the notation r ⊗ s := i r i ⊗ s i ∈ H(R) for a generic element in H(R), see Remark 2.9.

Lemma 5.1. Let (R, µ R ) and A R be as above. Then [START_REF] Andruskiewitsch | Lifting via cocycle deformation[END_REF]

H(R) ⊂ H(A R ). (2) (g ⊗ g -1 )(h ⊗ h -1 ) = (h ⊗ h -1 )(g ⊗ g -1 ). (3) If r ⊗ s ∈ H(R), then τ (r) ⊗ τ (s) = r ⊗ s, ω(r) ⊗ ω(s) = r ⊗ s. (4) If r ⊗ s ∈ H(R), then g • r ⊗ τ (s) ∈ H(R), h • r ⊗ ω(s) ∈ H(R).
In particular, (gh • r) ⊗ σ(s) ∈ H(R).

Proof.

(1) This is clear, as µ A|R = µ R . ( 2) is (4.8).

(3) Recall that r ⊗ s ∈ H(R) if and only if

rs (1) ⊗ s (2) ⊗ s (3) = 1 ⊗ r ⊗ s.
If we apply τ ⊗3 to this equality, as τ (1) = 1, then we get

1 ⊗ τ (r) ⊗ τ (s) = τ (rs (1) ) ⊗ τ (s (2) ) ⊗ τ (s (3) ) (4.1) = τ (rs (1) ) ⊗ s (2) ⊗ s (3) = 1 ⊗ r ⊗ s.
The result for ω is analogous. ( 4) By (4.1) and ( 4.3):

(g • r)τ (s) (1) ⊗ τ (s) (2) ⊗ τ (s) (3) = (g • r)(g • s (1) ) ⊗ g • s (2) ⊗ τ (s (3) ) = g • 1 ⊗ g • r ⊗ τ (s). Thus g • r ⊗ τ (s) ∈ H(R).
The other case is similar.

We use Lemma 5.1 to define algebra automorphisms τ ′ , ω ′ ∈ Aut H(R) via the formulas

τ ′ (r ⊗ s) = g • r ⊗ τ (s), ω ′ (r ⊗ s) = h • r ⊗ ω(s), r ⊗ s ∈ H(R). (5.1) Notice that τ ′ ω ′ = ω ′ τ ′ as g, h are quasi-central and thus gh • r = hg • r, r ∈ R. Remark 5.2. If τ ω = id, that is if gh • r ⊗ s = r ⊗ s for all r ⊗ s ∈ H(R), then gh ∈ Z(G 0 (R)) and gh = hg. Proof. Indeed, let t ∈ G 0 (R) and fix r ⊗ s = t ⊗ t -1 ∈ H(R). Then (gh • t) ⊗ t -1 = t ⊗ t -1 which implies gh • t = t, i.e. gh ∈ Z(G 0 (R)). If t = h, then it follows that gh = hg.
Theorem 5.3. Let (R, µ) be a Hopf-Galois algebra, let (τ, ω, g, h, c, ξ) be a sextuple satisfying the conditions of Theorem 4.1 and let A R = A(R, X, Y, τ, ω, c, ξ) be the associated Hopf-Galois algebra.

Consider τ ′ , ω ′ ∈ Aut H(R) as in (5.1), the group-like elements g -1 ⊗ g, h -1 ⊗ h ∈ G(H(R)), the scalar λ ∈ k * such that gh = λhg, and the elements c ′ ∈ H(R) and ξ ′ ∈ k * defined by

c ′ = (gh) -1 ⊗ c -(gh) -1 c ⊗ 1, ξ ′ = ξλ -1 .
Then A(H(R), E, F, τ ′ , ω ′ , c ′ , ξ ′ ) has a Hopf algebra structure as in Theorem 3.1, and we have a Hopf algebra isomorphism

H(A R ) ≃ A(H(R), E, F, τ ′ , ω ′ , c ′ , ξ ′ ).
Proof.

Step 1. We first have to show that the conditions in Theorem 3.1 are satisfied. Define α, β :

H(R) → k by α(r ⊗ s) = (g • r)τ (s) and β(r ⊗ s) = (h • r)ω(s).
It is clear that α and β are characters on H(R). We have

α((r ⊗ s) (1) )(r ⊗ s) (2) = α(r ⊗ s (1) )s (2) ⊗ s (3) = (g • r)τ (s (1) )s (2 ⊗ s (3) (4.1) = (g • r)τ (s (1) )τ (s (2 ) ⊗ τ (s (3) ) = g • r ⊗ τ (s) = τ ′ (r ⊗ s)
so that (3.2) holds for τ ′ , and for ω ′ . Similarly

g -1 ⊗ g • (r ⊗ s) (1) α((r ⊗ s) (2) ) = g -1 ⊗ g • (r ⊗ s (1) )α(s (2) ⊗ s (3) ) = (g • r ⊗ g • s (1) )g • s (2) τ (s (3) ) (4.3) = (g • r ⊗ τ (s (1) ))τ (s (2 )τ (s (3) ) = g • r ⊗ τ (s) = τ ′ (r ⊗ s)
and (3.3) holds for τ ′ , and for ω ′ as well.

It is clear that τ ′ and ω ′ commute since τ and ω commute and g and h commute up to a scalar, so the characters α and β convolution commute.

We have

α(h -1 ⊗ h) = g • h -1 τ (h) = λ -1 h -1 ξh = ξ ′ and similarly β(g -1 ⊗ g) = ξ ′-1 . It is an immediate verification that c ′ ∈ H(R) and that c ′ is ((gh) -1 ⊗ gh, 1)-primitive, with (gh) -1 ⊗ gh = (g -1 ⊗ g)(h -1 ⊗ h). We have finally, for σ ′ = τ ′ ω ′ , c ′ (r ⊗ s) = r(gh) -1 ⊗ cs -r(gh) -1 c ⊗ s = (gh) -1 (gh) • r ⊗ σ(s)c -(gh) -1 (gh • r)c ⊗ s = (gh) -1 (gh) • r ⊗ σ(s)c -(gh) -1 cσ -1 (gh • r) ⊗ s = (gh) -1 (gh) • r ⊗ σ(s)c -(gh) -1 c(gh • r) ⊗ σ(s) (by Lemma 5.1) = ((gh) • r ⊗ σ(s))c ′ = σ ′ (r ⊗ s)c ′
and hence c ′ is σ ′ -central.

Step 2. We now check that A R = A(R, X, Y, τ, ω, c, ξ) has a natural A(H(R), E, F, τ ′ , ω ′ , c ′ , ξ ′ )comodule algebra structure. We start with preliminary computations. We have

(g ⊗ E + X ⊗ 1 ⊗ 1)r (1) ⊗ r (2) ⊗ r (3) = gr (1) ⊗ E(r (2) ⊗ r (3)) ) + Xr (1) ⊗ r (2) ⊗ r (3) = gr (1) ⊗ g • r (2) ⊗ τ (r (3) )E + τ (r (1) )X ⊗ r (2) ⊗ r (3) (4.1) = (g • r (1) )g ⊗ g • r (2) ⊗ τ (r (3) )E) + τ (r) (1) X ⊗ τ (r) (2) ⊗ τ (r) (3) (4.3) = τ (r (1) )g ⊗ τ (r (2) ) ⊗ τ (r (3) )E + τ (r) (1) X ⊗ τ (r) (2) ⊗ τ (r) (3) (4.1) = τ (r) (1) g ⊗ τ (r) (2) ⊗ τ (r) (3) E + τ (r) (1) X ⊗ τ (r) (2) ⊗ τ (r) (3) = τ (r) (1) ⊗ τ (r) (2) ⊗ τ (r (3) ) (g ⊗ E + X ⊗ 1 ⊗ 1)
and similarly

(h ⊗ F + Y ⊗ 1 ⊗ 1) r (1) ⊗ r (2) ⊗ r (3) = ω(r) (1) ⊗ ω(r) (2) ⊗ ω(r (3) ) (h ⊗ F + Y ⊗ 1 ⊗ 1) .
We also have

(g ⊗ E + X ⊗ 1 ⊗ 1) (h ⊗ F + Y ⊗ 1 ⊗ 1) -ξ (h ⊗ F + Y ⊗ 1 ⊗ 1) (g ⊗ E + X ⊗ 1 ⊗ 1) = gh ⊗ (EF -ξ ′ F E) + (XY -ξY X) ⊗ 1 ⊗ 1 + (gY -ξY g) ⊗ E + (Xh -ξhX) ⊗ F = gh ⊗ c ′ + c ⊗ 1 ⊗ 1 = gh ⊗ (gh) -1 ⊗ c -gh ⊗ (gh) -1 c ⊗ 1 + c ⊗ 1 ⊗ 1 = c (1) ⊗ c (2) ⊗ c (3)
These computations show that we have an algebra map

A R -→ A R ⊗ A(H(R), E, F, τ ′ , ω ′ , c ′ , ξ ′ ), defined by R ∋ r -→ r (1) ⊗ r (2) ⊗ r (3) , X -→ g ⊗ E + X ⊗ 1 ⊗ 1, Y -→ h ⊗ F + Y ⊗ 1 ⊗ 1
which is easily seen to furnish the expected comodule algebra structure.

Step 3. We now prove that the previous comodule algebra structure makes A R a Hopf-Galois object over A(H(R)) = A(H(R), E, F, τ ′ , ω ′ , c ′ , ξ ′ ). For this, we claim that there exists an algebra map

θ : A(H(R)) -→ A op R ⊗ A R r ⊗ s, E, F -→ r ⊗ s, E ′ := g -1 ⊗ X -g -1 X ⊗ 1, F ′ := h -1 ⊗ Y -h -1 Y ⊗ 1 such that r ⊗ s -→ r ⊗ s and E -→ E ′ := g -1 ⊗ X -g -1 X ⊗ 1, F -→ F ′ := h -1 ⊗ Y -h -1 Y ⊗ 1.
To prove the claim, we have to check a number of identities. We have, for r ⊗ s ∈ H(R),

E ′ (r ⊗ s) = rg -1 ⊗ Xs -rg -1 X ⊗ s = rg -1 ⊗ τ (s)X -rg -1 X ⊗ s = (g • r ⊗ τ (s))(g -1 ⊗ X) -(τ -1 (g • r) ⊗ s)(g -1 X ⊗ 1) = (g • r ⊗ τ (s))(g -1 ⊗ X -g -1 X ⊗ 1) = τ ′ (r ⊗ s)E ′
where the third equality holds since g • r ⊗ τ (s) = τ -1 (g • r) ⊗ s, by Lemma 5.1. Similarly F ′ (r ⊗ s) = ω ′ (r ⊗ s)F ′ . Finally we see that:

E ′ F ′ = h -1 g -1 ⊗ XY -h -1 Y g -1 ⊗ X -h -1 g -1 X ⊗ Y + h -1 Y g -1 X ⊗ 1 = (gh) -1 ⊗ XY -ξ(gh) -1 Y ⊗ X -(gh) -1 X ⊗ Y + ξ(gh) -1 Y X ⊗ 1. F ′ E ′ = λ -1 (gh) -1 ⊗ Y X -λ -1 ξ -1 (gh) -1 X ⊗ Y -λ -1 (gh) -1 Y ⊗ X + λ -1 ξ -1 (gh) -1 XY ⊗ 1.
Hence we have

E ′ F ′ -ξ ′ F ′ E ′ = (gh) -1 ⊗ (XY -ξY X) -(gh) -1 (XY -ξY X) ⊗ 1 = (gh) -1 ⊗ c -(gh) -1 c ⊗ 1 = c ′ .
We have thus proved that the announced algebra map is well-defined. It is now an immediate verification that the composite map

A R ⊗ A(H(R)) id⊗θ -→ A R ⊗ A R ⊗ A R mult⊗id -→ A R ⊗ A R
is inverse to the Galois map in Section 2, so we conclude that A R is a Hopf-Galois object over A(H(R)).

Step 4. To finish the proof, first notice that the elements E ′ = g -1 ⊗ Xg -1 X ⊗ 1 and

F ′ = h -1 ⊗ Y -h -1 Y ⊗ 1 from
the previous step belong to H(A R ), and that the computations done in this previous step ensure the existence of a Hopf algebra map

f : A(H(R)) -→ H(R),
given by

r ⊗ s -→ r ⊗ s, E -→ E ′ , F -→ F ′ .
It is obvious that f commutes with the respective coactions on A R , so using the bijective canonical maps of our two Hopf-Galois objects, we conclude that f is an isomorphism.

While the group-like and skew-primitive elements in a Hopf-Galois algebra R can be described from those of H(R) thanks to Lemma 2.10, the proof of the previous theorem also provides the description of the automorphisms in Theorem 4.1 in terms of characters of H(R):

Corollary 5.4. Let (R, µ) be Hopf-Galois algebra, let (τ, ω, g, h, c, ξ) be a sextuple satisfying the conditions of Theorem 4.1. Then there are convolution commuting characters α, β :

H(R) → k such that for r ∈ R, τ (r) = α(r (2) ⊗ r (3) )g • r (1) , ω(r) = β(r (2) ⊗ r (3) )h • r (1) and for r ⊗ s ∈ H(R), α((r ⊗ s) (1) )(r ⊗ s) (2) = g -1 ⊗ g • (r ⊗ s) (1) α((r ⊗ s) (2) ), β((r ⊗ s) (1) )(r ⊗ s) (2) = h -1 ⊗ h • (r ⊗ s) (1) β((r ⊗ s) (2) ).
Proof. We have, for any r ∈ R, by the construction of τ ′ , ) ). Multiplying everything, we get, by the definition of α in the previous proof the announced identity for τ and α, and similarly for ω and β. The final identities have been proved in the previous proof as well.

r (1) ⊗ τ ′ (r (2) ⊗ r (3) ) = r (1) ⊗ g • r (2) ⊗ τ (r (3) ) hence g • r (1) ⊗ τ ′ (r (2) ⊗ r (3) ) = g • r (1) ⊗ g • r (2) ⊗ τ (r ( 3
Notice that when ω = τ -1 , that is when A R = A(R, X, Y, τ, c, ξ) is an ambiskew polynomial algebra, for some τ ∈ Aut R, then we do not necessarily have ω ′ = τ ′ -1 , as we get

τ ′ ω ′ (g -1 ⊗ g) = gh • g -1 ⊗ g = λg -1 ⊗ g.
Thus it becomes evident that we need to work with generalized ambiskew polynomial algebras rather than plain ambiskew polynomial algebras. We resume this situation in the next corollary, see §6.5 for a concrete example.

Corollary 5.5. Let (R, µ) be a Hopf-Galois algebra and assume that A R = A(R, X, Y, τ, c, ξ) is a Hopf-Galois algebra. Then the corresponding Hopf algebra H(A R ) is a generalized ambiskew polynomial algebra A(H(R), τ ′ , ω ′ , c ′ , ξ ′ ), where τ ′ , ω ′ ∈ Aut H(R) are commuting automorphisms not necessarily inverse to each other.

(x (1) , y (1) )π(x (2) y (2) , z) = π(y (1) , z (1) )π(x, y (2) z (2) ).
The algebra π H has H as underlying vector space, and product defined by

x.y = π(x (1) , y (1) )x (2) y (2) .
It follows from standard cocycle identities [START_REF] Doi | Braided bialgebras and quadratic bialgebras[END_REF] that π H is a Hopf-Galois algebra with

µ : π H -→ π H ⊗ π H op ⊗ π H x -→ π -1 (S(x (3) ), x (4) )x (1) ⊗ S(x (2) ) ⊗ x (5)
and that the map

H -→ H( π H) ⊂ π H op ⊗ π H x -→ π -1 (S(x (2) ), x (3) )S(x (1) ) ⊗ x (4) )
is a Hopf algebra isomorphism. We thus see from Lemma 2.10 that

G 0 ( π H) = {λg, λ ∈ k * , g ∈ G(H)} and for g, h ∈ G(H), P 0 g,h ( π H) = {λ1 + x, λ ∈ k, x ∈ P g,h (
H)}. Using this description and Corollary 5.4, we then see that the sextuples (τ, ω, g, h, c, ξ) as in Theorem 4.1 can be chosen as follows:

(1) g, h are central group-like elements in H;

(2) the automorphisms τ, ω ∈ Aut π H are of the form

τ (x) = α(x (2) )g • x (1) , ω(x) = β(x (2) )h • x (1)
for some convolution commuting algebra maps α, β : H → k that also satisfy

α(x (2) )x (1) = α(x (1) )g • x (2) , β(x (2) )x (1) = β(x (1) )h • x (2)
(these last identities are in H) and α(h) = β(g) -1 , ξ = α(h)λ, where λ is such that gh = λhg in π H.

(3) c is σ-central for σ = τ ω, and c = γ1 + x, for γ ∈ k and x ∈ P gh,1 (H); 6.3. The example R = k[T ]. We now examine the case when R = R(k[T ]). The analysis of the previous subsection applies: we have G 0 (R) = k * and P 0 1,1 (R) = k + kT and for the sextuples (τ, ω, g, h, c, ξ) of Theorem 4.1, the automorphisms τ , ω are of the form τ (T ) = T +α1, ω(T ) = T + β1 for some α, β ∈ k. Moreover we must have β = -α if c = 0 because of the σ-centrality of c.

At the end, after suitable changes in the generators, we find that a Hopf-Galois algebra A = A(R, τ, ω, g, h, c, ξ) as in Theorem 4.1 belongs to one of the following classes.

(1) A is the enveloping Hopf algebra of the Lie algebra sl 2 :

A = U (sl 2 ) = k T, X, Y | [X, T ] = X, [Y, T ] = -Y, [X, Y ] = T .
(2) A is the enveloping Hopf algebra of the 3-dimensional Heisenberg Lie algebra h:

A = U (h) = k T, X, Y | [X, T ] = 0 = [Y, T ], [X, Y ] = T .
(3) A = U β , β ∈ k, is the enveloping algebra of a solvable 3-dimensional Lie algebra:

A = U β = k T, X, Y | [X, T ] = X, [Y, T ] = βY, [X, Y ] = 0 . (4) A = k[T, X, Y ] is abelian.
(5) A is a Hopf-Galois algebra and not a Hopf algebra, with

A = U α,λ = k T, X, Y | [X, T ] = αX, [Y, T ] = -αY, [X, Y ] = 1 + λT , α ∈ {0, 1}, λ ∈ k.
Moreover, the Hopf algebra H(U α,λ ) is As c is σ-central, it follows that in this case β = α -1 (hence α n = α m = ξ). Thus, a Hopf-Galois algebra A = A(R, τ, ω, g, h, c, ξ) as in Theorem 4.1, with g = K n and h = K m , belongs to one of the following classes.

(1) For each α, β with ξ = α m = β -n :

A = k K, X, Y | XK = αKX, Y K = βKY, XY -ξ Y X = 0 .
(2) For each α with ξ = α m = α n and γ, µ ∈ k:

A = k K, X, Y | XK = αKX, Y K = α -1 KY, XY -ξ Y X = γ + µ(1 -K n+m ) .
In the last class, we recognize the Hopf algebra U q (sl 2 ) when n = m = -1, γ = 0, µ = 1 q-q -1 and ξ = q 2 = α -1 . On the other hand, when γ = 0, we obtain the U q (sl 2 )-cleft objects A (a) as computed in [START_REF] Günther | Crossed products for pointed Hopf algebras[END_REF]Lemma 16], for a = γ + µ and n = m = -1, µ = 1 q-q -1 as before.

6.5. The example R = k T ±1 1 , T ±1 2 |T 1 T 2 = qT 2 T 1 . Here q ∈ k * and R = π H is a twisted group algebra of the group algebra

H = kZ 2 = k K ±1 1 , K ±1 2 |K 1 K 2 = K 2 K 1 and π such that π(K 1 ,K 2 )
π(K 2 ,K 1 ) = q. Subsection 6.2 applies to give the complete description of all the generalized ambiskew polynomial algebras that are Hopf-Galois as in Theorem 4.1. We focus on giving an example to illustrate Corollary 5.5, see §6.5.1 below. In particular, we assume that q = 1. We choose g = K 1 , h = K 2 ∈ G(H), so gh = q hg in R, that is, λ = q. Now, we get that τ (T 1 ) = α(K 1 )T 1 , τ (T 2 ) = qα(K 2 )T 2 , ω(T 1 ) = q -1 β(K 1 )T 1 , ω(T 2 ) = β(K 2 )T 2 , for some algebra maps α, β : H → k such that α(K 2 ) = ξq -1 = β(K 1 ) -1 .

In particular, σ(T 1 ) = ξ -1 α(K 1 )T 1 and σ(T 2 ) = ξβ(K 2 )T 2 . Now c = γ + µ(1 -T 1 T 2 ). If c = 0, then being σ-central amounts to 6.6. The example R = A 1 (k). Here R = k U, V |U V -V U = 1 is a cleft object for the polvnomial Hopf algebra H = k[u, v], with G(H) = {1} and P 1,1 (H) = ku + kv. As above, we obtain that the automorphisms τ, ω are determined by algebra maps α, β : H → k, as τ (U ) = U + α(u), ω(U ) = U + β(u), similarly for τ (V ), ω(V ). We have that c = γ + c 1 U + c 2 V , for some γ, c 1 , c 2 ∈ k. The fact that c is σ = τ ω-central determines that c 1 = α(v) + β(v) and c 2 = -α(u)β(u). Therefore A R = A(R, τ, ω, g, h, c, ξ) with g = h = 1 is determined by scalars γ, α u , α v , β u , β v ∈ k and relations

[U, V ] = 1, [X, U ] = α u X, [X, V ] = α v X, [Y, U ] = β u Y, [Y, V ] = β v Y [X, Y ] = γ + (α v + β v )U -(α u + β u )V.
If c = γ + (α v + β v )U -(α u + β u )V = 0, then this is a Hopf algebra. When c = 0, if we identify H(R) with

H = k[u, v] via 1 ⊗ U -U ⊗ 1 u, 1 ⊗ V -V ⊗ 1
v then for the corresponding Hopf algebra H(A R ) it follows that ξ ′ = ξ, τ ′ = τ and ω ′ = ω, in such a way that H(A R ) = A(H, τ, ω, c ′ , ξ) for c ′ = (α v + β v )u -(α u + β u )v.

. 4 ) 3 . 4 .

 434 Remark Conditions(3.4) are stated in [4, §3.1], in this setting they follow by Remark 3.2.

Theorem 4 . 1 .

 41 Let R = (R, µ) be a Hopf-Galois algebra. Assume given a sextuple (τ, ω, g, h, c, ξ) where (1) τ, ω ∈ Aut(R) are commuting algebra automorphisms satisfying, for any r ∈ R,

6 . 4 .

 64 (a) U (sl 2 ) if α = 1 and λ = 0; (b) U -1 if α = 1 and λ = 0; (c) U (h) if α = 0 and λ = 0; (d) k[T, X, Y ] if α = 0 and λ = 0. The example R = k[T, T -1 ] = kZ. In this part, we set R = R(H), for H = kZ = k K ±1 .As above, we choose two (central) group-like elements g = K n , h = K m , m, n ∈ Z. We fix two characters α, β : H → k, which are determined by nonzero scalars α := α(K), β := β(K), satisfying ξ = α m = β -n (as λ = 1). If c = 0, then there is γ, µ ∈ k such that c = γ + µ(1gh).

  Baby example: R = k. It is immediate that Theorem 4.1 produces only two examples: the Hopf algebra k[X, Y ] with primitive X, Y , and the Weyl algebra A 1 (k) as in Example 2.3.6.2. Almost general example: cleft Galois objects. Before going into other specific examples, we examine the case of cleft Galois objects, i.e. those obtained by deforming the multiplication of a Hopf algebra by a Hopf 2-cocycle. This is the most studied type of Hopf-Galois objects, but there exist some that are not of this type, see[START_REF] Bichon | The representation category of the quantum group of a non-degenerate bilinear form[END_REF].Let H be a Hopf algebra and let π : H ⊗ H → k be a 2-cocycle on H, i.e. π is convolution invertible, π(x, 1) = ε(x) = π(1, x) for any x ∈ H, and we have, for any x, y, z ∈ H, π

	6. Examples
	6.1.

In loc.cit. the authors choose the notation (X+, X-) and (g+, g-) instead of (X, Y ) and (g, h), respectively; and use h instead of c, σ instead of τ .

In the notation from[START_REF] Brown | Ambiskew Hopf algebras[END_REF], this condition reads τ l χ = ad l (y+) • τ r χ and then τ = τ l χ .

(1) If µ = 0, then ξ = α(K 1 ) = β(K 2 ) -1 . Thus, ω = τ -1 and hence σ = id.

(2) If µ = 0 and γ = -µ, then α(K 1 ) = q -1 ξ and β(K 2 ) = qξ -1 . Hence σ = id.

(3) Otherwise, q = 1 and this is a contradiction.

Next we analyze cases (1) and (2). 6.5.1. Case [START_REF] Andruskiewitsch | Lifting via cocycle deformation[END_REF]. Fix µ = 0, γ = 1, so τ ω = id and c = 1. This will provide an example for Corollary 5.5. We have

Let A R = A(R, X, Y, τ, c, ξ) be the corresponding Hopf-Galois algebra. This is generated by

2 , X and Y with relations

In this case,

Finally, τ ′ , ω ′ ∈ Aut H(R) are the automorphisms given by

In other words, under the identification above, we have that

In particular, we see that τ ′ and ω ′ are not inverse to each other and the Hopf algebra

2 , E and F with relations

For the Hopf algebra

In this case, ω ′ τ ′ = id. The Hopf algebra is now generated by K ±1 1 , K ±1 2 , E and F with relations