Alexandre Monnin

Keywords: non-knowledge, unknowledge, ontology, digitality, distance, proximity, absence, presence

Digitality, (un)knowledge and the ontological character of nonknowledge.

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Although we have no word for it, establishing an appropriate degree of "middle connectivity" to the world is such a basic feature of the human condition that doing it successfully has been lifted into the rarefied reaches of sainthood and enlightenment; failing to accomplish it, identified as a cause of paralytic anxiety. Brian Cantwell Smith The relationship between knowledge, non-knowledge and digitality is a complex one still waiting to be fully explored. As evidenced in this volume, efforts to shed some light on "nonknowledge" open up new directions of research especially relevant as we'll see in a world that's becoming everyday more digitized. On the other hand, as such the opposition between knowledge and non-knowledge tends to obscure the very fact that digitization has also "remedied" knowledge lending it the character of a commodity instead of a norm (as it used to be previously considered, despite the disagreement philosophers and epistemologists could entertain among themselves). Hence, one is required not only to situate non-knowledge vis-à-vis knowledge but also knowledge vis-à-vis digitization. Such will be our attempt in this paper.

Knowledge, digitality and unknowledge

Knowledge and digitality: epistemic issues

Knowledge both admits of a vast number of characterizations and comes in different flavors. While it is possible to hold shared views on the purview of knowledge while disagreeing at the same time on its exact definition disagreement may still loom on the horizon. Whether tacit or practical knowledge for instance refer to a phenomenon that can be subsumed under one Heading along with scientific knowledge or knowledge as traditionally conceived by epistemologists is a question that remains largely open to debate.

For that reason, at first glance it would seem pretty much illusory to contrast a unified concept of knowledge to non-knowledge. Yet, without such a unified concept, the opposition seemingly becomes moot at best. This is where digitization comes in. Digitization and knowledge entertain a complex and very paradoxical relationship. Going back to the concept "knowledge economy"1 , made possible by the advances of digitization, one immediately sees this relation for what it is: a relation of commodification. "Knowledge", in "knowledge economy", no longer denotes to any norm or domain of knowledge (which it merely connotes) but rather betokens a broad assimilation to a commodity essentially cultivated in order to sustain growth. Both the normative and pluralistic aspects have thence simply vanished.

While paradoxical this evolution shouldn't come as a surprise for it may very well characterize digitization as such. As a consequence, our claim in this paper will be that digitality has both overplayed and downplayed salient aspects of knowledge to the point that we might on initial approximation think of this evolution as bringing knowledge nearer to non-knowledge (as we shall see however, as we progressively move away from epistemic questions the drive to introduce an additional category will be more and more compelling).

Overplayed since models and abstractions have become non only a sign of the portability of conceptual knowledge but a way to perform assemblages that induce new realities instead of defering one way or another to an existing world (let's not be mistaken: deferring to the world includes taking into account the way it is transformed by our own activity -especially at the time of the Anthropocene). Google's PageRank algorithm is a good example of that since it construes incoming "links" as votes (never votes of defiance!) in its willingness to redefine the Web by using measures of authority, pretending to remain neutral whereas its own presence modifies the very topology of the thing it was supposed to independently measure. Downplayed since the commodification of knowledge, made possible due to the lack of regards towards the traditional norms of knowledge (in a sense "anything goes" in the knowledge economy as long as its goals are achieved), resulted in more and more data, metadata, documents, and so on and so forth -what we'd call "knowledge traces" -being produced and made available with unforeseen consequences well-worth examining.

Innovation is better served or so it seems by people who have little regard to the nooks and crannies of everyday life, assured as they are of the well-foundedness of their mission to transform it. Of course, why on earth transform that which is not fully apprehended and make sure it is replaced (or modified) by something genuinely new and relevant? The answer to the second objection is only induction through enumeration should provide an answer and it it wellknown to be insufficient. Let's just put it aside basically because we have to live with similar "uncertainties". The answer to the first objection is much less straightforward. Digital technologies produce new assemblages while at the same time claiming to operationalize preexisting realities (intelligence, authority, vote, trust, etc.). Changing the meaning of those concept is not an explicit goal, rather such realities are taken for granted and whether or not the ensuing operationalization and these new assemblages turns them into something wholly different, even in logical contradiction to what they previously stood for, is no one's business.

Assemblages and performation have always marched hand in hand since assemblages do by definition perform a specific effect2 . Whence the legitimate fear that focusing on assemblages and their performances might obfuscate any reference (and deference) to the world. But here it is rather the lack of care displayed by innovators which is misleading, leading to abstain from paying attention to the nooks and crannies of the assemblages produced (the results of the aforementioned operationalization). We thus go from "unknowledge" to "unknowledge" -here we introduce this concept in order to characterize a specific contrast to knowledge akin to a lack of willingness to defer to the world still unabashedly regarded as full-fledged knowledge.

Unknowledge is very well illustrated by this quote from Phil Agre about AI:

As a practical matter, the purpose of AI is to build computer systems whose operation can be narrated using intentional vocabulary. Innovations frequently involve techniques that bring new vocabulary into the field: reasoning, planning, learning, choosing, strategizing, and so on. Whether the resulting systems are really exhibiting these qualities is hard to say, and AI people generally treat the question as an annoying irrelevance. What matters practically is not the vague issue of what the words "really mean" but the seemingly precise issue of how they can be defined in formal terms that permit suitably narratable systems to be designed. If you disapprove of the way that we formalize the concept of reasoning or planning or learning, they are likely to say, then you are welcome to invent another way to formalize it (Agre 1997)." 3

Non-knowledge and unknowledge: an ontological characterization

Let's go back to non-knowledge. With unknowledge in sight, what can we now say about nonknowledge? We here follow Brian Cantwell Smith who contends that content, a technical term in analytic philosophy used to designate the basis of knowledge and action, can be either conceptual or non-conceptual. Conceptual content involves positing a world consisting of objects, properties and relation which amounts to carving reality into discretized individuals (seen as the bearers of properties and in relations with one another). Non-conceptual content, while still representational, doesn't register the world likewise but rather in terms of unindividuated "features" that precedes the advent or objects or individualswhich, for Smith, is essentially an ethical matter (a matter of "mattering" as he puts it). The picture offered by non-conceptual content is essentially a "subobjective" 4 one. Whereas non-conceptual content depicts the world in overwhelming details, fit for situated and local encounters, conceptual content and objectivity in general strip those same details so as to make it possible to refer over a long distance (to things long gone and buried in the past, not yet born in a distant future, or too shrouded in vagueness to do otherwise) 5 .

With objects and ontology predicated on ethics, what remains metaphysically essential is to give room to reference making. That is, to articulate the causally effective local encounters with the world with non-causal long-distance reference. In other words, what is valued here is less one overarching metaphysical category (the One, the transcendental a priori, ideas, the body, and so on and so forth.) than enough room to fit local proximal connections and distal reference: it is essential (...) and also an anchor of common sense, that the multi-various parts of the world do not march in lockstep together. The world is fundamentally characterized by an underlying flex or slop -a kind of slack or "play" that allows some bits to move about or adjust without much influencing, and without being much influenced by, other bits. (...) The world's flex and slop is so obvious that it is a little hard to talk about. As a contrast, therefore, imagine a world quite unlike ours, consisting, as suggested of nothing but an endless series of interlocked gears. Suppose, to make this precise, that every gear is constructed so as to mesh with one or more immediate neighbors, and that the entire gear universe is interconnected, but in such a way that it is still possible for them all to be turned (...) so that it does not lock up. Suppose, too, that the gears are perfect: no friction, no play between the teeth, and shaped so that rotating one at an 3 We have have suggested [START_REF] Monnin | L'ingénierie philosophique de Rudolf Carnap: De l'IA au Web sémantique[END_REF] that the roots of formalization thus conceived lie in Rudolph Carnap's concept of "explanation" to which scholars have turned their attention in recent years. See especially [START_REF] Carus | Carnap and Twentieth-Century Thought Explication as Enlightenment[END_REF], (Richardson 2013). 4 See [START_REF] Lowe | Registration Marks: Metaphors for Subobjectivity[END_REF] whose subtitle is composed of three texts by Adrian Cussin, Brian Cantwell Smith and Bruno Latour (currently being translated by the author). Perhaps the best way to summarize this is by an analogy. I sometimes think of objects, properties, and relations (i.e., conceptual, material ontology) as the long-distance trucks and interstate highway systems of intentional, normative life. They are undeniably essential to the overall integration of life's practices -critical, given finite resources, for us to integrate the vast and open-ended terrain of experience into a single, cohesive, objective world. But the cost of packaging up objects for portability and long-distance travel is that they are thereby insulated from the extraordinarily finegrained richness of particular, indigenous life-insulated from the ineffable richness of the very lives they sustain. (Cantwell Smith, s. d.), p. 37. even speed causes the others to rotate evenly as well, though at a potentially different speed. The gear world would lack slop. Effects would not dissipate. If one gear were to move by even a tiny amount, every other gear in the universe, no matter how far flung, would instantly and proportionally be affected." (...) If the flex were too little (...) the world would lock up like the gear world, and everything would be correlated with everything else. Such a world would be too rigid, too straight, too stuffy intentionality would be neither possible nor necessary. If the flex were too great, on the other hand, it would have the opposite problem: things would be too loose, everything would be random, and effect-transcending coordination would be impossible. Imagine (...) an infinite space randomly occupied by an indefinitely large number of particles, all of which drift aimlessly around, none of which ever interact (Cantwell Smith 1998, p. 199-200). 6 At first glance, unknowledge appears to threaten long-distance reference since according to the definition we adopted it no longer defers to the world preoccupied as it is by its own selfcentered efficiency. But such a criticism would be mistaken if left at that. While unknowledge denotes a peculiar lack of knowledge it is also defined by what is produces: namely, "knowledge traces". In other words, it adds gears where there were none, where space gave enough room to the "world's flex and slop", filling in preexisting gaps with no special care for long distance reference, favoring the multiplication of interlocked gears and short-distance communication to simulate continuity over long distance reference. The strategy adopted is rather one of generalized padding where gears can be endlessly introduced and correlated with one another. Therefore, both unknowledge and non-knowledge also raise ontological -rather than purely epistemic -issues of absence and presence, distance and proximity.

Non-knowledge and (un)knowledge

We will address the way these ontological issues manifest themselves by looking at the followings examples, all three drawn from digital cultures and technologies.

The Internet of Things

The Internet of Things (IoT) stands as one of the main promises of digital innovation for the coming years. Out of the numerous features of IoT is the possibility to interact with concrete 6 Compare with William James who put a great emphasis on the fact that "Not all the parts of the world are united mechanically, for some can move without the others moving", (Some problems of philosophy in [START_REF] James | [END_REF], p. 1046. Latour's project in An Inquiry into the Modes of Existence [START_REF] Latour | Enquête sur les modes d'existence : Une anthropologie des Modernes[END_REF]) has been summarized the following way: "What is at stake: to take seriously the first proposition, to civilize the moderns until they do not successfully "make room" [for] others. The inquiry indicates that the function of its metaphysics is simply to make a place", commentary signed by the GECo (Groupe d'Etudes Constructivistes) on the online version of [START_REF] Latour | Enquête sur les modes d'existence : Une anthropologie des Modernes[END_REF]), available at http://modesofexistence.org/inquiry/#a=SET+DOC+LEADER&c[leading]=DOC&c[slave]=TEXT&i [id]=#doc-257&i[column]=DOC&s=0&q=make+room. Giving room to modes of existence (whether modern or non-modern) is the new diplomatic goal of metaphysics. Of peculiar interest here is the fact that modes of existence themselves are all described by certain kinds of continuity and discontinuity ("hiatuses"). This is very much in tune with Cantwell Smith so much so in fact that it generalizes it in a pluralistic fashion. While a systematic comparative study of Brian Cantwell Smith's and Bruno Latour's positions hasn't been undertaken it would be definitely be a task wellworth embarking on. physical objects out there in the "real-world". The perspective to go "beyond" the screen and to re-territorialize computation is a very powerful and appealing one indeed. At long last computation shall escape the realm of the "virtual" and reach out to concrete objects. Or such goes the saying.

What is fascinating with the IoT is how much of a thorough attempt to circumvent the boundaries of objects through technical means it proves to be. One may wonder whether it is sufficient to attach an apparatus to a patch of reality in order to reality get a good grip on its identity conditions. Probably not. But maybe this is beside the point. At least attaching and RFID chip or any other prop will make it possible to localize the object, potentially throughout time, even allowing to grasp it less through deictic encounter -this object here and now -and rather in an abstract way as a space-time arrow, something we can barely conceive of, less so encounter directly.

Remember though, abstraction is no part of the sales pitch for connected objects but rather their concreteness, proximity and availability through direct encounter. While we may still conceive of objects as abstractions what connected objects continuously produce are ever more traces, making it possible not to refer in a non-causal way but rather to reach out in a causal way by artifactualizing the path to the object, adding enough gears to fill in the gap between agent and objet (or between objects) to make that happen.

Eventually, there precisely lies the paradox behind the whole enterprise. The boundaries of connected objects are defined and "maintained" through technology and the active production of knowledge-traces. While discussing the importance of "boundary-maintaining 7 Donna Haraway quotes Brian Cantwell Smith in support of her argument: "You have to stop being what you were when you start paying attention to the work it takes to maintain your clear distinctions". That perfectly sums up the situation we found ourselves in: just as devices are introduced that should help to circumscribe as precisely as possible the boundaries of objects (allowing for direct interaction and localization over time), the latter nevertheless expand in unprecedented ways.

Just ask yourself "what exactly are the boundaries of a connected object?". "It is truly wholly present as I interact with it?" The answer is no, of course, and not only that but being wholly present would simply defeat the very purpose of the IoT. Connected objects as their very name betokens communicate with servers on the cloud and/or other objects in their vicinity (sometimes even using Blockchains -public inviolable ledgers!). They also display behavior dictated by the algorithms using for the programming, sometimes going as far as betraying their original purpose as exampled in the recent Volswagen scandal. For all these reasons, not only are the boundaries not maintained or even expanded -they literally shatter! In a nutshell, the IoT is an attempt to substitute digital devices for abstractions and place upon their shoulders the onus of providing identity conditions to objects. The morals of the story ends up being an ontological one: it is not possible to replace abstraction, causally ineffective reference making, distance and absence with actual technical devices, causally effective localization, proximity and presence. Or expect paradoxes like the one we've just encountered. 7 [START_REF] Haraway | Modest Witness: Feminism and Technoscience[END_REF], p. 67. Smith's quotes comes from a discussion that took place in 1994 after he presented a paper entitled "Coming Apart at the Seams: the Role of Computation in a Successor Metaphysics" at Stanford University.

Love and felicity

The second example has been studies by Eva Illouz8 in her inquiry on love and the new digital life that shapes our most intimate relationships. Illouz' diagnosis romantic encounter become increasingly saturated by knowledge practices -that thanks to the generalization of online profiles, and the metadata they contain, knowledge play a role that has gone awry, obliterating, as she puts it, other types of knowledge and reshuffling the boundaries between proximity and distance (an ontological feat of no little consequences as we've seen), among other things.

Nowhere else is the subtle dialectic between absence and presence, distance and proximity, more at play than in the phenomenon called "love". It is not surprising then that unbalancing this relation with knowledge (even under the guise of unknowledge) should put it at risk. Before intimacy grew to become a norm we never knew that much about our love interests. Additionally, before the advent of digital cultures, social networks, online profiles and the likes, we never knew that much about our potential love interests ahead of encountering them.

In the wake of what we said about the IoT, we might want to assess that the boundaries personhood in a relationship have by and large been displaced. While profiles do seem to provide accurate (if coarse) knowledge, making visible "who" we are by maintaining the boundaries of our identity, in fact what happens is that they delegate (outsource, really) what was previously left to chance and hazardous encounters to algorithms that calculate our best match.

Of course, pretending that love owes nothing to chance is not entirely new. Sociology, for one, is the discipline that literally saw as its mission (as opposed to novel writing for instance) to shed light on the social dynamics underneath the most intimate and private phenomena, including lovers felt attraction. It held dear and strived to uncover the unseen determinants at play behind the curtain. And it was correct in his own right of course. There's no denying that love might not escapes determinism.

Yet, we should nonetheless pay heed to a paramount difference between these two cases. While sociologists did provide statistical conclusions in favor of their claims, no one ever (mis)took them as spiritual advisers. In a sense, so much so has happened with the advent of social networks and dating websites. Filling in innumerable fields on a daily basis means people become both providers and consumers of the (un)knowledge thus produced. Whereas sociologists' take on love used to be discussed mainly among peers or the educated readership with an interest in the discipline it may be said to have now infused many if not most of our daily transactions.

Then again, such a move might be readily welcomed. Aren't relationships, now that we can mimic the behaviors and functional possibilities of connected objects (especially the localization bit and the availability of "leaky" knowledge traces), all the better for that? After all, no talk about the identity conditions of objects will likely dispel the belief that cheating is cheating and that mobiles phones (undoubtedly the IoT's first citizen) do provide an efficient way to learn the truth. Must we eventually backtrack on the idea of unknowledge if deferring to the world (apparently) means deferring to such simple truths?

The point rather is that deferring to the world might precisely mean something else, at least as far as love is concerned. In [START_REF] Latour | Enquête sur les modes d'existence : Une anthropologie des Modernes[END_REF], Bruno Latour suggests that we adopt a pluralistic view on metaphysics so as to give space to phenomena that are amenable each to specific felicity or infelicity conditions. Going back to Agre's remark, we may begin to understand why digitality is by no means harmless. Digital tools do promise transparency. It is all too easy then to treat love as demanding it. Paying attention to the felicity conditions of love should however advice otherwise. Indeed, the latter may lie less in the search for truth (or knowledge) than in love's own subsistence; a matter of delicate, finespun, dialectic between proximity and distance, presence and absence, knowledge and non-knowledge, put at risk when (un)knowledge takes over in its "profusing transparency" (talk of "transparency" bears witness to an interesting choice of words as the immediate danger is either to be blinded by the abundance of digital traces of all kinds or to treat them indeed as transparent intermediaries). However, in order to properly understand the key role played by non-knowledge with regards to love one has to overcome unknowledge first.

Art: caring for the artists

Let us dig further by taking one final example that will hopefully prolong the previous discussion.

Since 2014 I have had the luck and opportunity to work as the architect of the digital platform of Lafayette Anticipation9 , the Galeries Lafayette's foundation for contemporary art. The challenge was (and remains) to foster interest in contemporary art beyond the small circles of aficionados (be them critics or patrons). Contemporary art notoriously challenges the logic of the exhibition and as a symbol of the excesses of capitalistic markets is subject to criticism from both the general public and intellectuals. The aim of the project soon became to dispense knowledge about contemporary creative process, loaded as it is with uncertainties, so as to share an understanding of art in the making, a risky process that generates its own criteria of judgement (far from presupposing that is should always be successful).

That is how the idea emerged of developing an archive that would centralize both data and documents (traces, really) generated by the foundation's various members in their daily activities, from the treasurer to the head of production. Pioneering laboratory studies published in the 80's have completely modified our understanding of the scientific process by insisting on the hidden contribution of a great many actors previously left in the dark by epistemologists. The fulcrum of their description underscored the importance of practice, distributed action and hidden (dirty) work. Shifting the narrative from geniuses inhabiting ivory towers to a more realistic depiction of science did upset some scientists although in fairness deconstructing the robustness of scientific results had never been on the agenda of the ethnographers who produced these descriptions.

Neither is it our intention to ultimately "deconstruct" the figure of the artist or the work of art. The ideas behind the platform were thereupon discussed at a very early stage of development in the summer 2015, in Paris, during a two days session involving about 40 artists, designers, architects, critics, philosophers and curators 10 . Reactions among the artists were mixed, some expressing fears that unveiling their "secrets" of creations would be tantamount to yielding to a broader demand for transparency and mimic the extractivist logic exhibited by the likes of Google or Facebook.

Even though scientists admittedly found no significant hindrance in ethnographers work, artists had every right to express their fears. And for the exact same reason mentioned above in the preceding section: transparency shouldn't put artists at risk. The goal of the foundation had always been to assist them and cater to their (or their work's) needs.

The willingness to open a new window on the artist labor did not entail a primacy of transparency over creation. Not unlike love, art has its own conditions of felicity. Subsistence, then may adequately translate into being able to listen to the call of the work of art (to speak Etienne Souriau's language 11 ). And that might imply to obfuscate part of the creative process if needs be. Immediately comes the question "does it amount to lying?". Whenever truth is equated with transparency, with little to no regards to the phenomenon at stake, the answer is yes. By contrast, when subsistence, understood as the continuation of the phenomenon at stake, takes priority, the answer shall be a clear "no", knowledge then being subservient to care. Going back to the ethnography, this is reminiscent of "ethnographic refusal", a decision not to write about a subject matter to avoid putting it at risk, being exploitive or unhelpful (among the many traps that await researchers) 12 .

Latour himself expresses the need for a "crooked language" in politics: nothing is more important for this inquiry than to find the difference between truth and falsity in politics. If there is one area where our inheritance has to be revisited, it is surely that of the hopes placed in politics and its capacity for extension. What will we have to do to situate appropriately crooked speaking once again at the center of our civility as the only means to collect the collective, and above all to universalize it? Does the Circle give us a thread like Ariadne's that will let us speak here again of the rational and the irrational but in a well-curved way, that is, in its own language, provided that we don't seek to judge it with the help of a different touchstone? We need this thread, for how could we stand up straight on the agora, with no hope of help from any Science and yet without giving up on reason, about controversial issues that have taken on the dimensions of the planet and in the heat of a crowd that now numbers in the billions? (Latour 2012) 13 Contrary to Latour, we would not restrict such a crooked language to politics. Or rather, to put it more adequately, this kind of language can be seen as the political answer provided to a broader issue. Throughout the discussion of the last three examples we have come to give precedence to subsistence over truth with respect to non-knowledge 14 . Subsistence requires care 15 and a 11 See (Souriau 2009) and [START_REF] Souriau | L'Ombre de Dieu[END_REF]. This is tacitly acknowledged in a recent tweet published on the AIME account: "It's one hypothesis of AIME that beings of [POL] are so fragile that their mode of existence may disappear entirely through hospitable middle-ground, whether in politics, love or the arts. Unknowledge, by contrast, unable as it is to defer to the world even as it conveys trite truths, striving to unbalance the middle-ground, sorely lacks this aspect. As Phil Agre puts it, "a reformed technical practice [should] employ the tools of critical inquiry to engage in a richer and more animated conversation with the world" 16 . For this conversation with the world to be genuinely fruitful non-knowledge shall not be overlooked.

Conclusion

The aforementioned crooked language is no enemy of reason yet neither is it to be understood in term of truth or falsity as science does. As we have seen, non-knowledge unlike knowledge (and to a lesser extent unknowledge, which not only is a degraded norm but also has an ontological dimension) is less of an epistemic value than an ontological middle ground allowing for the subsistence of a multiplicity of generic phenomena according to their own requirements (much akin to Latour's modes of existence). William James himself noted that "the same thing (...) can belong to many systems, as when a man is connected with other objects by heat, by gravitation, by love, and by knowledge [my emphasis] 17 ." Tellingly, knowledge in his enumeration was but one among many such systems. James also noticed our relentless propensity to add what he called new "systems of concatenation": "We ourselves are constantly adding to the connection of things, organizing labor-unions, establishing postal consular, mercantile, railroad, telegraphs, colonial, and other systems that bind us and things together in ever wider reticulations." 18 It is somewhat ironic that we only have a negative expression like "non-knowledge" at our disposal to refer to the multiplicity of these systems of concatenations minus one... Such is, today, the overwhelming weight of unknowledge: no longer a norm but rather a system of concatenations that not only overshadows and twists others but eventually jeopardizes their conditions of subsistence.

I would like to thank the CDC for the kind invitation I was offered to be part of the semester dedicated to non-knowledge as a fellow in November 2015. I hereby express my special gratitude to Martina Leeker. lack of care.", https://twitter.com/AIMEproject/status/756786152548409344. That the generalization is not made outside [POL] is a testament of Latour's rather complex relationship to care. 16 [START_REF] Agre | The Soul Gained and Lost[END_REF]. 17 [START_REF] James | [END_REF](James ), p. 1048. 18 . 18 Ibidem.

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  note on ethnographic refusal with a bibliography was recently published online: 13 Online notes available at http://modesofexistence.org/inquiry/?lang=en#b[chapter]=#29&b[subhead ing]=#541&a=SET+TEXT+LEADER&c[leading]=TEXT&c[slave]=VOC&s=0&q=nothing+is+mor e+important+for+this+inquiry+than+to+find+the+difference 14 "Add some transparency, some truth (still in the sense of Double Click), and you still get only dissolution, stampede, the dispersal of that very agora in which the fate of all categories is judged." http://modesofexistence.org/inquiry/?lang=en#b[chapter]=#13&b[subheading]=#211&a=SET+TEXT +LEADER&c[leading]=TEXT&c[slave]=VOC&s=0&q=agora 15

See the contribution of Christoph Wulf in this volume.

(Centre de sociologie de l'innovation 2013)

Her presentation at the CDC during the winter semester dedicated to non-and entitled "Knowing way too much… Love, Therapy, Technology" is available online: https://vimeo.com/153692828

https://lafayetteanticipation.squarespace.com/accueil

https://lafayetteanticipation.squarespace.com/lafayette-re-source


Nicolas Suas-David

Vinayak N Kulkarni
email: vinayak@iitg.ernet.in

Abdessamad Benidar

Samir Kassi

Robert Georges

Line Shape in a Free-jet Hypersonic Expansion Investigated by Cavity Ring-down Spectroscopy and Computational Fluid Dynamics

Experiments are carried out for spectroscopic studies using hypersonic jet of carbon monoxide seeded in argon as a carrier gas. Probing of this jet using cavity ring-down spectroscopy revealed a double peak structure for various absorption lines. Flow field simulation using computational fluid dynamics is used to understand the shape of such lines integrated over line of sight. Absorption contribution from warmer non-isentropic part of the jet, owing to its transverse velocity variation, is found responsible for those line shapes.

Introduction

Experimental facilities with supersonic or hypersonic free expansion of gases are widely used as tools for molecular spectroscopic studies. Such jets are characterized by supersaturated gaseous conditions with low rotational temperatures which simplify the structure of molecular spectra. Therefore, steady expanded jets or pulsed jets are conventionally used to attain desired levels of cooling. Understanding the line shape for a particular transition has been one of the major objectives of free-jet based spectroscopic studies. Among such fundamental studies, Veeken and Reuss considered free expansion of ammonia initially to determine the rotational temperature and onwards to estimate the number density [1]. There, anomalous line shape with double peak structure was attributed to condensation occurring preferentially in the allegedly denser axial region. Similar line shape with central shallow dip was also noticed by Mizugai et al. [2]. Snels and Baldacchini [3] conducted experiments with ammonia using diode laser and found non-Gaussian line shape with central dip for rotation-vibration lines of 2 band. Clustering at the axis was invoked for those line shapes showing dependence with probing distance from the nozzle exit and the stagnation pressure of the jet. In a similar study, apparent double peak line structure was attributed to clustering by Bajaj et al. [4]. Possibility of clustering was also reported in the supersonic jet of N 2 O [5] and acetylene [6].

While one of the explanations for anomalous line shapes is the formation of clusters in the neighborhood of jet axis, other explanations are also available in the literature. Gaveau et al. numerically predicted shape and width of absorption lines in case of CO-He free-jet for P(1) and P(2) lines [7]. In their study, shape of the absorption lines was found to be dependent on the variation of density and temperature in the jet. However the effect of jet boundary was not considered in that study. Campbell et al. found that 'Doppler dephasing' of rotational lines is responsible for the peculiar line shape observed in their microwave study of OCS and rare gas/OCS complexes [8]. This explanation was found non-unique by Lovas and Suenram for the supersonic jet of OCS seeded in various rare gases [9]. Besides, 'doubling' of line shape was accounted for loss of density due to clustering of OCS with carrier gas. Nevertheless, in their collaborative efforts, Campbell and Lovas finally concluded that density and temperature gradients along the laser beam are responsible for double peaked lines [10]. Ramos et al. disregarded condensation of nitrogen and attributed the observed double peak structure of lines to inhomogeneous line broadening due to strong temperature and density gradients within the silent zone, along the line of sight [11].

Thus diversity can be seen in the literature for explaining line shape with two peaks. In some of the reported findings, it has been noticed that, the observations used for justification of clustering do not exclusively conclude the clustering. Besides, flow field gradients can also be led responsible for such observations. However, it has also been felt that the studies, which account for flow field along the laser beam for the anomalous line shape, need detailed investigations to mark specific region of the flow and the flow property responsible for such line shape. In view of this, spectroscopic measurements are carried out for hypersonic jet of carbon monoxide and argon using cavity ring-down spectroscopy (CRDS). These studies are continuation of the earlier findings of Louviot et al. [12] where as well double peak structure of absorption lines had been observed for carbon monoxide in CRDS based measurements. Therefore present investigations specifically deal with analysis of absorption line shapes for line of sight spectroscopic measurements. Numerical simulations are also carried out to get the simulated lines. Efforts are then extended to correlate the experimental and simulated lines with structure of the free-jet. Details of the experimental setup, numerical simulations and reasoning for the anomalous line shape are given in the following sections.

Experimental and numerical methodology

Experiments are carried out in an experimental facility having a High Enthalpy Source (HES) connected to a vacuum chamber [12,13]. In the present studies, carbon monoxide (Air Liquide, 99.998% purity) is used as the seeding gas (5% by mass) and argon (Air Liquide, 99.998% purity) as the carrier gas (95% by mass). Mass flow rate of this mixture is regulated while flowing through HES where it is heated by passing 80 A current through graphite. A circular nozzle of 2 mm diameter is used as a throat for expansion of the gas in the vacuum chamber. This chamber is evacuated to the pressure of 11 1 Pa so as to establish flow field known as a Campargue-type expansion [START_REF] Beijerinck | Campargue-type supersonic beam sources: Absolute intensities, skimmer transmission and scaling laws for mono-atomic gases He[END_REF]. Expansion of jet is probed using high sensitive cw-CRD spectrometer. Telecom grade fibered Distributed Feedback (DFB) diode lasers of power 10-20 mW are used to act as light source. Each of these lasers covers about 7 nm of wavelength ( 30 cm -1 ). For this, their temperatures are adjusted between -10 to 60°C with the help of in-house developed proportional integral differential (PID) stabilizer. From the total light emitted by the laser, 10% is transferred to wavemeter (Burleigh WA-1600) and 90% is introduced into the optical cavity. This high finesse cavity (F > 200 000) consists of two high reflectivity (>99.9987%), plano-concave (1000 mm radius) mirrors which are separated by about 800 mm. Mode matching between laser beam and TEM 00 mode of cavity is achieved using single lens and two steering mirrors. Cavity length is modulated by mounting the output mirror on a piezoelectrical transducer in order to set resonance with laser frequency.

At the resonance, acousto-optical modulator (AOM) gets switched off, owing to which photons evolve freely inside the cavity. Ring-down time is then obtained using an exponential fit of the intensity measured by a photodiode. Spectroscopic measurements are carried out at locations which are 5 mm, 20 mm and 40 mm downstream of the nozzle exit. These locations would be referred here onwards as X1, X2 and X3 respectively. It should be noted here that, the location of Mach disk for present experimental conditions is about 16 cm from nozzle exit. Therefore these axial probing stations are about 1/32 th , 1/8 th and 1/4 th of the Mach disk location. These line of sight measurements are always ensured in the direction perpendicular to the jet axis. Experimental data is recorded using a LabVIEW based data acquisition system for the spectral range of carbon monoxide as [6220-6320] cm -1 corresponding to the 3 0 vibrational band.

Two numerical techniques, computational fluid dynamics (CFD) and method of characteristics (MOC), are employed to predict flow properties in the jet. These predictions form inputs for obtaining two synthetic absorption lines. Among these numerical techniques, initially, commercial CFD solver Fluent of ANSYS-15 [START_REF]ANSYS FLUENT Release 15.0 User Manual[END_REF] is used to visualize and understand the structure of hypersonic free-jet. This viscous flow solver considers mass, momentum and energy conservation equations to simulate the flow from reservoir till 50 mm downstream of the orifice. Beyond this distance, the Knudsen number (K n ), which can be defined as the ratio between the mean free path of the expanding molecules and the nozzle diameter, becomes too high. As a result of this, Fluent solver remains invalid due to break down of continuum assumption necessary for using Navier-Stokes equation. It has also been noticed from the experimental conditions that the Knudsen number for present case varies between 0.2 (core) and 0.6 (shear layer / jet boundary) which is still far from the free molecular regime (K n > 10). Thus, using this computational technique, it is possible to analyze the flow in all parts of the jet which would later be helpful to construct the absorption line along entire line of sight. In view of this, two dimensional (2D) axi-symmetric viscous simulations are carried using mixture properties and assumption of perfect gas. Density based solver and Sutherland's law for viscosity are opted for these computations. Reservoir conditions (P 0 =133.32 kPa and T 0 =1300K) are provided at the inlet of the fluid domain. Supersonic flow boundary condition with chamber pressure is implemented at the outlet. Converged steady state results are considered for further analysis. After viscous flow simulations using CFD, MOC based fit has been initially used to get the axial variation of Mach number. Axial temperature of the flow is then recovered using law of energy conservation for adiabatic expansion. Other flow variables are also obtained from isentropic relations.

Then Eq. 1 is used to estimate off-axial variation of density in the silent zone [START_REF] Ashkenas | The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnels[END_REF].

(

Here, is off axial density evaluated from the axial counterpart [ and angle made by velocity vector with jet axis . Value of , given by Ashkenas and Sherman [START_REF] Ashkenas | The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnels[END_REF] is used here. This equation is the best fit for MOC based prediction hence it is valid only in the isentropic core of jet [START_REF] Ashkenas | The Structure and Utilization of Supersonic Free Jets in Low Density Wind Tunnels[END_REF]. Therefore, boundary of isentropic core is mandatory input for this approach to construct the computational domain.

Results obtained from CFD simulations are considered here to provide this valuable input. Thus obtained off-axial density and known reservoir density are used to evaluate local temperature using isentropic relation. Mach number of any location is then obtained from isentropic relation with the help of local and stagnation temperatures. Velocity is computed using Mach number and temperature.

Flow field information received from CFD based 2D axi-symmetric simulations is used to build the entire three dimensional (3D) jet. Besides, only 3D inviscid core is constructed using similar information given by MOC. These transformations help to obtain the absorption line shape for entire line (

Here x, y and z are Cartesian coordinates, n xyz is number density (molecule cm -3 ) corresponding to the cell or elemental volume at x, y and z, T(x,y,z) is temperature (K) of the cell, is integrated absorption cross section, g D is Gaussian line shape with thermal Doppler effect (temperature based line broadening)

and convective Doppler effect (transverse velocity based line broadening), is wave-number (cm -1 ), i and j are lower and upper states of the spectroscopic transition, is length scale of a cell along line of sight (cm), L is length of cavity (cm) and w 0 is waist of laser beam (cm). The integrated absorption cross section is estimated here as

where, Q is partition function, E is energy level (cm -1 ) and k B is Boltzmann constant (0.69503 cm -1 K -1 ).

Required integrated absorption cross section at T ref (296 K) is extracted from the 2012 HITRAN database [START_REF] Rothman | [END_REF]. Equilibrium partition function [18] is considered here since the flow simulation has assumption of equilibrium between translational, rotational and vibrational modes of temperature. Gaussian line profile, from Eq. 2, is calculated as, is most probable speed (m/s), R is Universal gas constant (J/mol K) and M is molecular weight (kg/mol). Prediction of line shape using this methodology is advantageous since it gives absorption line shape for any laser length at a given axial location. Thus it helps in understanding contribution of each region of jet in building the integrated absorption line.

Results and Discussion

Mach number, transverse velocity, temperature and density variations in the hypersonic free-jet, obtained from CFD simulations, are shown in Fig. 1. Jet probing stations, considered during experiments, and experimentally recorded P(1) lines at those stations, are shown in the transverse velocity contours.

Figure 1 helps to understand expansion of the gas and thus evolved jet structure encountered by the laser during measurement. In the expansion process, gas attains sonic state at the exit of nozzle from reservoir conditions and then expands to very low pressure in the isentropic core. Mach disk limits expansion of gas in the axial direction but present simulation is terminated before the Mach disk so as to analyze the flow field in the neighborhood of nozzle exit. Compression waves limit expansion of the gas in transverse direction. Subsequently, there is jet boundary or shear layer in the transverse direction which is portion of the jet decelerated by static residual gas in the chamber. However chamber fluid also gets entrained while decelerating the jet. Thus, it is evident here that, laser at any axial location interacts with four regions of Sufficient increment in translational temperature would be evidenced while transiting from region C to D.

Representative small volume from regions A to D, for probing station X3, are highlighted here using numbers 1 to 4 respectively. This representation is thought to be useful to get the CFD based line shape at these individual stations and onwards to visualize the process of integration of all such local lines in getting the complete integrated line shape. Here, region 1 is a cylinder of diameter 1 mm (same as laser beam) and length 10 mm along the laser beam with its geometric center on the jet axis. both the simulation techniques. This Gaussian shape can be expected in the axial region of the jet since these numerical techniques do not account for clustering as per their formulations. Further, MOC based simulation does not predict double peak structure of P(1) line in the second phase of integration for complete isentropic core or silent zone of the flow (II). Unlike MOC, shallow dip can be noticed in the CFD based synthetic P(1) line for this axial and off-axial parts of the flow (II) which are predominantly inviscid. Major reasons for this computational artifact are the governing equations for 2D axi-symmetric fluid flows which hold singularity at the axis and the associated computational cost required to overcome the same. Such artifact can also be noticed in the findings of Mate et al. [19] where density variation close to the axis is not in agreement with the experiments and expected MOC based variation. In case of CFD simulations, double peak structure of P(1) line becomes prominent in the next integration phase where non-isentropic region 3 is also included (III). Since computations with MOC are not valid in this region, here (III) corresponding MOC based P(1) line is same as of II and represents contribution of entire isentropic core only. Moreover, P(1) line is seen to be deeper and wider after accounting for regions like 4 in the process of complete integration along the line of sight (IV). Therefore, at the end of integration in the line of sight (IV), CFD based P(1) line is compared with the experimentally obtained P(1) line which as well exhibits same double peak structure. Here also (IV), MOC based P(1) line, which has a plateau at the peak, is same as II and refers to isentropic core. Thus it is evident from this process of integration (I-IV) that the isentropic core is not responsible for the double peaked structure in P(1) line corresponding location X3. Besides this, it is the non-isentropic region of the jet which is responsible for such shape of P(1) line. For better understanding, variation of flow properties along the line of sight is plotted in Fig. 3 and P(1) lines of individual regions (1-4), as per their transverse locations, are plotted in Fig. 4. These two figures help in establishing correlation between flow properties in the probing direction with the integrated line shape. Here, intensity of each Gaussian in Fig. 4 is by the virtue of local temperature and density given in Fig. 3. However, displacement of each Gaussian on the frequency axis is attributed to the component of velocity along the line of sight. There is reduction in peak intensity of the local Gaussian of isentropic core with increase in distance from the jet axis. Along with this, these Gaussians can be seen to be linearly displaced on the frequency axis due to linear variation of transverse velocity with increase in distance from the axis. This fact has its basis on conical expansion of the jet in the silent zone or diverging streamline pattern in the isentropic core (Fig. 1). Due to such a flow structure, no central dip in the P (1) line can be expected from MOC based computations (Fig. which can be easily explained by warming of the jet in these regions (Fig. 3). It also leads to enhanced thermal broadening of local Gaussians. Nevertheless, it is apparent in Fig. 4 that there is loss of linearity in displacement of these Gaussians on frequency axis with increase in distance from the axis. It is due to reduction in the rate of increase of transverse velocity in the presence of compression waves or due to decrement in transverse velocity in the jet boundary. Such transverse velocity variation is equivalent to deflection of streamline towards the jet axis (Fig. 1). Thus, local Gaussians from non-isentropic region exhibit resistance in their outward shift on frequency axis. Therefore, when this region gets accounted for obtaining integrated P(1) line, it necessarily adds the thermally broadened Gaussians to the P(1) line obtained from lone isentropic core of the jet. This mechanism justifies the double peak structure noticed in experimental and synthetic lines in the absence of condensation. Synthetic P(1) line of CFD shows shift in peak at wavenumber of 0.0106 cm -1 for location X3 (Fig. 2 (b), IV). Experimental P(1) line of the same location also shows shift of peak at 0.0080 cm -1 and thus supports the demonstration in getting double peaked P(1) line. Thus Figs. 2 to 4 clearly illustrate the reasoning behind double peaked P(1) line for station X3. Similar process can be executed for any other axial station. However, it has been seen that the isentropic core (A and B) is not responsible for double peaked line structure and it is evident from the Fig. 1 that the size of the non-isentropic regions (C and D) increase with increase in streamwise distance.

Therefore it is expected to have shallow peak near the nozzle exit owing to thin jet boundary. This observation justifies the small dip detected in experimental and synthetic P(1) lines at X1 location (Fig. 1).

It also supports the deeper P(1) line noticed at X2 and X3 probing stations (Fig. 1). Same jet structure based fluid dynamic reasoning is valid for other absorption lines as shown in Fig. 5. Here the central dip in P(2), P(4) and P(6) lines can be justified in the similar manner. However, P(9) line shows single peak structure due to the fact that its major contribution is from high temperature fluid from jet boundary which has negligible transverse velocity.

Conclusion

Experiments are successfully carried out for probing hypersonic jet of carbon monoxide seeded in argon using cw-CRDS. Viscous simulations are then performed using ANSYS-15 to understand the structure of the jet. Method of characteristics based fits are also employed to simulate the silent zone of the jet. Results from these simulations are then used to demonstrate the construction of absorption lines obtained in the line of sight measurements. In this process, it has been noticed that the isentropic core or silent zone is not responsible for double peaked absorption lines. However, transverse velocity variation in the non-isentropic region is found to add thermally broadened Gaussians for building the double peaked absorption lines. Moreover, height of peak or size of the dip is found dependent on size of this region.

Thus present studies are found useful in understanding the absorption line shape and correlating it with structure of the jet. In light of this conclusion, double peaked or anomalous line shapes [20] of monomers reported in the literature are expected to have relevance with the jet profile where explicit evidence of condensation is absent. However, in case of clustering at axis as well, loss of monomers and warmer nonisentropic part of the jet should be accounted for double peaked absorption lines.

  of sight (CFD) and for limited sight (MOC) at any laser probing station. At a given probing station, laser beam of 1 mm diameter with Gaussian intensity distribution and zero divergence is introduced in such 3D domains. Since the chosen mesh for computations has length scale of the order 0.02 mm, it ensures participation of sufficient number of cells (identified by x, y and z co-ordinates) in the laser beam for predicting the absorption line shape. Then, the flow properties viz. component of velocity in the line of sight, density and temperature are used to get the synthetic line from CFD and MOC. Equation 2 is used for calculation of absorption coefficient integrated along the line of sight.

Some of the terms of this equation

  or component of velocity in the line of sight (m/s), c is Speed of light (m/s),

  the jet viz. near axis flow region with negligible transverse velocity (A), off axial flow with finite transverse velocity (B), region of compression waves (C) and the jet boundary (D). Colder regions A and B are necessarily isentropic while warmer regions C and D are non-isentropic in nature. Only transverse velocity variation is prominent in the transition from regions A to B while there is marginal change in any other flow property. Boundary between regions B and C is characterized by decrement of Mach number.

1 )

 1 Regions 2-4 are also circular cylinders of same diameter but of length 5mm on either side of jet axis. They have their geometric centers at a distance of 10, 20 and 30 mm respectively from jet axis. Probing station X3 is chosen for demonstration since it is comprised of sufficiently thick regions (A to D) due to which the representative volumes (1 to 4) are of equal and adequate sizes and are well separated. But these specifications of representative small volumes (1 to 4) should be dependent on axial probing station since thickness of regions A to D vary in the flow direction. For example, dimension of jet, along the laser, is very small at 5 mm from nozzle exit (X1). Here jet would be comprised of comparatively larger sized regions A and B with negligible thicknesses of regions C and D. Due to change in sizes of regions A to D in axial direction, experimentally obtained P(1) lines at different probing stations (Fig. are expected tohave differential contributions from these regions. Apart from this, since present experiments are conducted for high stagnation temperature conditions, it has led to strong thermal non-equilibrium[12] in the jet. Therefore, clustering of carbon monoxide is disregarded due to freezing of vibrational temperature in the expansion process. These facts strengthen the possibility of correlation between observed double peaked absorption lines with the jet structure.

Figure 2

 2 Figure2shows the contribution of each region in building P(1) line for location X3. Process of integration is illustrated in Fig.2(a) while Fig. 2(b) gives the outcome of this process of integration

4 ,

 4 2 (b), II). However flattening of the peak of such MOC based P(1) line is dependent on the local diameter of isentropic core (divergence of the flow or convective Doppler effect) and thermal line broadening (Thermal Doppler effect). Further, non-isentropic region of the jet needs to be considered to explain such a line shape. As per Fig. for non-isentropic regions, peak intensity of the local P(1) line continues to reduce with increase in transverse coordinate,

Fig. 1 CFD

 1 Fig.1 CFD Simulation based variation of flow properties in the jet (a) Mach number, (b) Transverse velocity (m/s) (c) Temperature (K) and (d) Density (kg/m 3 ). Laser probing stations considered in the experiments and corresponding P(1) lines are shown in the transverse velocity field (b).

13

 13

Fig. 2 .

 2 Fig.2. (a) Process of integration along the line of sight. (b) 3 0 vibrational band at X3 (1-4) and corresponding process of obtaining integrated P(1) line from simulations along with its comparison with experiment (I-IV).

Fig. 3

 3 Fig.3 Flow properties obtained from CFD along the line of sight at X3.

Fig. 4

 4 Fig.4 Illustration of building of the integrated P(1) line at X3 using the local Gaussian distributions.

Fig. 5

 5 Fig.5 Comparison of simulated and experimental lines at X3.

Acknowledgements

Authors would like to acknowledge the financial support from CEFIPRA (Project No. 4905-1 "Kinetics and spectroscopy in extreme environments: Applications to astrophysics and astrochemistry"). RG acknowledges the "Programme National de la Planétologie" (INSU CNRS) for its financial support. RG and SK would like to thank GDR Spec Mo 3152 for the financial support.


Samuel Belliard
email: samuel.belliard@cea.fr

NON-ABELIAN SYMMETRIES OF THE HALF-INFINITE XXZ SPIN CHAIN

Keywords: 81R50, 81R10, 81U15 Open XXZ spin chain, Coideal subalgebras, Non-Abelian symmetry, Quantum integrable models

The non-Abelian symmetries of the half-infinite XXZ spin chain for all possible types of integrable boundary conditions are classified. For each type of boundary conditions, an analog of the Drinfeld-Jimbo presentation is given for the corresponding symmetry algebra. In particular, two new algebras arise that are, respectively, generated by the symmetry operators of the model with triangular and special Uq(gl 2 )-invariant integrable boundary conditions.

Introduction

Identifying non-Abelian infinite dimensional symmetries of quantum integrable systems provides a very efficient starting point for transferring the Hamiltonian and eigenstates formulation into the langage of operator algebra, representation theory and related special functions, on which the non-perturbative analysis is thus based. Besides the restricted class of models with conformal symmetry for which the Virasoro algebra and its representation theory play a central role, in the thermodynamic limit of lattice systems it is known that quantum groups, related current algebras and infinite dimensional q-vertex operators representations provide an appropriate mathematical framework. As an important example, in the thermodynamic limit N → ∞ of the finite XXZ spin chain with N sites and periodic boundary conditions, the U q ( sl 2 ) algebra emerges as a hidden non-Abelian symmetry of the Hamiltonian [START_REF] Foda | Corner transfer matrices and quantum affine algebras[END_REF][START_REF] Jimbo | Quantum group symmetry and lattice correlation functions[END_REF][START_REF] Jimbo | Correlation functions of the XXZ model for ∆ < -1[END_REF]. Based on the representation theory of the U q ( sl 2 ) quantum affine algebra at level one and its current algebra, an explicit characterization of the Hamiltonian's spectrum, corresponding eigenstates, as well as multiple integral representations of correlation functions and form factors of local operators has been given [START_REF] Jimbo | Correlation functions of the XXZ model for ∆ < -1[END_REF]. This approach has been later on extended to lattice systems with periodic boundary conditions associated with higher spins [START_REF] Idzumi | Quantum affine symmetry in vertex models[END_REF] or higher rank affine Lie algebras [START_REF] Koyama | Staggered Polarization of Vertex Models with Uq( sl(n))-Symmetry[END_REF], as well as for certain class of boundary conditions, see for instance [START_REF] Jimbo | XXZ chain with a boundary[END_REF][START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF][START_REF] Baseilhac | Correlation functions of the half-infinite XXZ spin chain with a triangular boundary[END_REF][START_REF] Baseilhac | Form factors of the half-infinite XXZ spin chain with a triangular boundary[END_REF][START_REF] Kojima | Diagonalization of transfer matrix of supersymmetry Uq(sl(m + 1|n + 1)) chain with a boundary[END_REF].

For lattice models with general integrable boundary conditions, identifying the non-Abelian symmetries in the thermodynamic limits has remained essentially unexplored although coideal subalgebras of quantum affine Lie algebras are natural candidates. For instance, in the thermodynamic limit N → ∞ of the finite open XXZ spin chain it was expected that non-Abelian infinite dimensional symmetries emerge, that are associated with certain coideal subalgebras of U q ( sl 2 ). Recall that the Hamiltonian of the half-infinite XXZ spin chain is formally defined as (see also [START_REF] Jimbo | XXZ chain with a boundary[END_REF][START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF][START_REF] Baseilhac | Correlation functions of the half-infinite XXZ spin chain with a triangular boundary[END_REF][START_REF] Baseilhac | Form factors of the half-infinite XXZ spin chain with a triangular boundary[END_REF]):

H 1 2 XXZ = - 1 2 ∞ k=1 σ k+1 1 σ k 1 + σ k+1 2 σ k 2 + ∆σ k+1 3 σ k 3 + h B (1.1) with boundary interaction h B = - (q -q -1 ) 4 ( + --) ( + + -) σ 1 3 - 1 ( + + -) k + σ 1 + + k -σ 1 -.
Here σ 1,2,3 and σ ± = (σ 1 ± iσ 2 )/2 -see Appendix A -are usual Pauli matrices, ∆ = (q + q -1 )/2 denotes the anisotropy parameter and ± , k ± are scalar parameters associated with the right boundary field. Formally, the Hamiltonian acts on an infinite tensor product of 2-dimensional vector spaces. Note that the ordering of the tensor components in (1.1) is such that:

V = • • • ⊗ C 2 ⊗ C 2 ⊗ C 2 . (1.2)
Given the Hamiltonian of the half-infinite XXZ spin chain (1.1), until recently the hidden non-Abelian symmetry for q generic and any type of boundary conditions1 k ± , ± has remained unknown. However, at the end of the article [START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF], it was pointed out that for the generic case (k ± = 0 and ± = 0) and the diagonal case (k ± = 0 and ± = 0) the respective hidden non-Abelian symmetries are associated with two different coideal subalgebras of U q ( sl 2 ): the q-Onsager algebra [T03, B04] and the augmented q-Onsager algebra [BC12, BB13] (see also [START_REF] Ito | The augmented tridiagonal algebra[END_REF]).

In this letter, we present a unified picture that complete the preliminary observations of [START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF]. Namely, for each type of boundary conditions, the non-Abelian symmetry algebra of the half-infinite XXZ spin chain is identified and characterized through generators and relations. In particular, two new coideal subalgebras are obtained.

The text is organized as follows. In Section 2, four different types of coideal subalgebras of U q ( sl 2 ) are defined though generators and relations in a Drinfeld-Jimbo's type of presentation. The coaction and counit maps are given in each case. Then, it is shown that these algebras are symmetry algebras of the Hamiltonian (1.1) for generic ( ± = 0, k ± = 0), triangular ( ± = 0, k + = 0, k -= 0), diagonal ( ± = 0, k ± = 0) and special2 boundary conditions ( + = 1, -= 0, k ± = 0), respectively. Note that since the case of upper triangular (k -= 0, k + = 0) and lower triangular (k + = 0, k -= 0) boundary conditions are related through conjugation of the Hamiltonian (1.1) by the spin-reversal operator ν = ∞ j=1 σ x j , it is sufficient for our purpose to restrict our attention to the case of lower boundary conditions. In Section 3, we propose an alternative and simpler derivation of the symmetry operators which is based on the remarkable connection between the infinite dimensional algebra A q introduced in [BK05] and the q-Onsager algebra. Concluding remarks follow in Section 4.

Notation . The q-commutator X, Y q = qXY -q -1 Y X is introduced, where q is the deformation parameter assumed not to be a root of unity.

Four different types of q-Onsager symmetry algebras

In the first part of this Section, four different types of coideal subalgebras with central extension are introduced through basic generators and relations. The coaction and counit maps are given. Note that two of these algebras have already appeared in the literature: the q-Onsager algebra [T03, B04] and the augmented q-Onsager algebra [IT09, BC12, BB13]. The two others, the so-called triangular q-Onsager algebra and U q (gl 2 ) invariant q-Onsager algebra3 are new. In a second part, depending on the choice of boundary conditions, by analogy with [J92, eq. (3.2)] it is shown that the Hamiltonian (1.1) commutes with all generators of one of these four algebras.

2.1. The q-Onsager algebra. The q-Onsager algebra O q ( sl 2 ) is an example of tridiagonal algebra [T03]. Including a central extension, it is generated by two elements W 0 , W 1 , a central element Γ and unit. The defining relations are:

W 0 , W 0 , W 0 , W 1 q q -1 = ρ W 0 , W 1 , (2.1) W 1 , W 1 , W 1 , W 0 q q -1 = ρ W 1 , W 0 , W 0 , Γ = W 1 , Γ = 0,
where ρ is a scalar. Without loss of generality, let us define

ρ = (q + q -1 ) 2 k + k - (2.2)
where k ± are nonzero scalars. By analogy with the situation for Hopf algebras, one endows the q-Onsager algebra with the coaction map δ : O q ( sl 2 ) → U q ( sl 2 ) ⊗ O q ( sl 2 ) defined by:

δ(W 0 ) = (k + e 1 + k -q -1 f 1 q h1 ) ⊗ 1 + q h1 ⊗ W 0 , (2.3) δ(W 1 ) = (k -e 0 + k + q -1 f 0 q h0 ) ⊗ 1 + q h0 ⊗ W 1 , δ(Γ) = q c ⊗ Γ and counit E : O q ( sl 2 ) → C: E(W 0 ) = + , E(W 1 ) = -, E(Γ) = E(1) = 1. (2.4)
This induces an homomorphism ψ = (id × E) • δ : O q ( sl 2 ) → U q ( sl 2 ) from the q-Onsager algebra to a subalgebra of U q ( sl 2 ):

ψ(W 0 ) = k + e 1 + k -q -1 f 1 q h1 + + q h1 , (2.5) ψ(W 1 ) = k -e 0 + k + q -1 f 0 q h0 + -q h0 , ψ(Γ) = q c .
2.2. The triangular q-Onsager algebra. The triangular q-Onsager algebra O t q ( sl 2 ) is generated by three elements T 0 , T 1 , P1 , a central element Γ and unit. The defining relations are:

T 0 , T 0 , P1 q -1 = ρ t T 0 , T 1 , T 1 , T 1 , P1 q = ρ t T 0 , T 1 , (2.6) T 1 , T 0 q -1 = ρt Γ, T 0 , Γ = T 1 , Γ = P1 , Γ = 0.
Let us define:

ρ t = k -(q + q -1 ) 2 and ρt = -+ -(q -q -1 ), (2.7)
where k -, ± are nonzero scalars. We endow this algebra with the coaction map δ t : O t q ( sl 2 ) → U q ( sl 2 ) ⊗ O t q ( sl 2 ) defined by:

δ t (T 0 ) = k -q -1 f 1 q h1 ⊗ 1 + q h1 ⊗ T 0 , (2.8) δ t (T 1 ) = k -e 0 ⊗ 1 + q h0 ⊗ T 1 , δ t ( P1 ) = k -[e 1 , e 0 ] q + q c [f 1 , f 0 ] q ⊗ 1 + q c ⊗ P1 + (q 2 -q -2 ) q -1 e 1 q h0 ⊗ T 1 + q c f 0 q c ⊗ T 0 , δ t (Γ) = q c ⊗ Γ and counit E t : O t q ( sl 2 ) → C: E t (T 0 ) = + , E t (T 1 ) = -, E t ( P1 ) = p, E t (Γ) = E t (1) = 1. (2.9)
This induces an homomorphism ψ t = (id × E t ) • δ t from the triangular q-Onsager algebra to a subalgebra of U q ( sl 2 ):

ψ t (T 0 ) = k -q -1 f 1 q h1 + + q h1 , (2.10) ψ t (T 1 ) = k -e 0 + -q h0 , ψ t ( P1 ) = (q 2 -q -2 ) -q -1 e 1 q h0 + + f 0 q h1+h0 + k -[e 1 , e 0 ] q + q c [f 1 , f 0 ] q + pq c , ψ t (Γ) = q c .
2.3. The augmented q-Onsager algebra. The augmented q-Onsager algebra O d q ( sl 2 ) has been introduced in [BB13], as a generalization4 of the augmented tridiagonal algebra introduced in [START_REF] Ito | The augmented tridiagonal algebra[END_REF]. It is generated by four elements K 0 , K 1 , Z 1 , Z1 , a central element Γ and unit. The defining relations are:

K 0 K 1 = K 1 K 0 = -+ Γ , (2.11) K 0 Z 1 = q -2 Z 1 K 0 , K 0 Z1 = q 2 Z1 K 0 , K 1 Z 1 = q 2 Z 1 K 1 , K 1 Z1 = q -2 Z1 K 1 , Z 1 , Z 1 , Z 1 , Z1 q q -1 = ρ d Z 1 ( K 1 K 1 -K 0 K 0 )Z 1 , Z1 , Z1 , Z1 , Z 1 q q -1 = ρ d Z1 (K 0 K 0 -K 1 K 1 ) Z1 , Z 1 , Γ = Z1 , Γ = K 0 , Γ = K 1 , Γ = 0
with the identification:

ρ d = (q 3 -q -3 )(q 2 -q -2 ) 3 q -q -1 . (2.12)
We endow this algebra with the coaction map δ d : O d q ( sl 2 ) → U q ( sl 2 ) ⊗ O d q ( sl 2 ) defined by:

δ d (K 0 ) = q h1 ⊗ K 0 , δ d (K 1 ) = q h0 ⊗ K 1 , (2.13) δ d (Z 1 ) = q c ⊗ Z 1 + (q 2 -q -2 ) q -1 e 0 q h1 ⊗ K 0 + f 1 q c ⊗ K 1 , δ d ( Z1 ) = q c ⊗ Z1 + (q 2 -q -2 ) f 0 q c ⊗ K 0 + q -1 e 1 q h0 ⊗ K 1 , δ d (Γ) = q c ⊗ Γ and counit E d : O d q ( sl 2 ) → C: E d (K 0 ) = + , E d (K 1 ) = -, E d (Z 1 ) = E d ( Z1 ) = 0, E d (Γ) = -+ , E d (1) = 1. (2.14)
This induces an homomorphism ψ d from the augmented q-Onsager algebra to a subalgebra of U q ( sl 2 ):

ψ d (K 0 ) = + q h1 , ψ d (K 1 ) = -q h0 , (2.15) ψ d (Z 1 ) = (q 2 -q -2 ) + q -1 e 0 q h1 + -f 1 q h1+h0 , ψ d ( Z1 ) = (q 2 -q -2 ) -q -1 e 1 q h0 + + f 0 q h1+h0 , ψ d (Γ) = -+ q c .
2.4. The U q (gl 2 ) invariant q-Onsager algebra: The U q (gl 2 ) invariant q-Onsager algebra O i q ( sl 2 ) is generated by six elements e, f, q h , X, Y, Ỹ, a central element Γ and unit. The defining relations are:

[e, f] = q h -q -h q -q -1 , [e, q h ] q = [q h , f] q = 0, (2.16) [q h , X] = 0, [X, e] = Y, [f, X] = Ỹ, [q h , Ỹ] q = 0, [e, Y] q = 0, [q h , Y] q -1 = 0, [f, Ỹ] q -1 = 0, [e, Ỹ] = (q + q -1 )Xq h , [Y, f] = (q + q -1 )Xq h , [X, [Y, Ỹ]] = (q 2 -
q -2 ) 3 (q Ỹfee -q -1 Yeff + (q + q -1 ) 2 Xefq h )q h + q 2 (q 2 -q -2 )(q + q -1 ) 2 (q 2 -q -2 -1) Ỹe + (q 2 -q -2 + 1)Yf -(q + q -1 )Xq h q 2h + (q 2 -q -2 )(q + q -1 ) 2 q -2 Ỹe -q 2 Yf + q 2 (q + q -1 )Xq h , Y, Γ = Ỹ, Γ = X, Γ = e, Γ = f, Γ = q h , Γ = 0.

We endow this algebra with the coaction map δ i : O i q ( sl 2 ) → U q ( sl 2 ) ⊗ O i q ( sl 2 ) defined by:

δ i (e) = e 0 ⊗ 1 + q h0 ⊗ e, δ i (f) = f 0 ⊗ q -h + 1 ⊗ f, δ i (q h ) = q h0 ⊗ q h , (2.17) δ i (X) = ([e 1 , e 0 ] q -[f 1 , f 0 ] q -1 q c ) ⊗ 1 + q c ⊗ X + (q 2 -q -2 )(q h0+1 e 1 ⊗ e -q c-1 f 1 ⊗ fq h ), δ i (Y) = ( e 1 ,
e 0 , e 0 q -q c+1 (q + q -1 )f 1 q h0 ) ⊗ 1 + q h0+c ⊗ Y (2.18) +(q 2 -q -2 ) q h0 [e 1 , e 0 ] q ⊗ e + (q 2 -1)q 2h0 e 1 ⊗ e 2 + q c+h0 f 1 ⊗ q 2h -1

q 2 -1 , δ i ( Ỹ) = q c f 1 , f 0 , f 0 q -1 -q -1 (q + q -1 )e 1 q h0 ⊗ q -h + q c ⊗ Ỹ -(q 2 -q -2 ) q c [f 0 , f 1 ] q ⊗ f (2.19) +q 1+h0 e 1 ⊗ q h -q -h q -q -1 + (q -q -1 )q c f 1 ⊗ f 2 q -h , δ i (Γ) = q c ⊗ Γ and counit E i : O i q ( sl 2 ) → C: E i (e) = E i (f ) = E i (X) = E i (Y) = E i ( Ỹ) = 0, E i (q h ) = E i (Γ) = E i (1) = 1. (2.20)
This induces an homomorphism ψ i = (id⊗E i )•δ i from the U q (gl 2 ) invariant q-Onsager algebra to a certain subalgebra of U q ( sl 2 ):

ψ i (e) = e 0 , ψ i (f) = f 0 , ψ i (q h ) = q h0 , (2.21) ψ i (X) = [e 1 , e 0 ] q -[f 1 , f 0 ] q -1 q c , ψ i (Y) = e 1 ,
e 0 , e 0 q -q c+1 (q + q -1 )f 1 q h0 , ψ i ( Ỹ) = q c f 1 , f 0 , f 0 q -1 -q -1 (q + q -1 )e 1 q h0 , ψ i (Γ) = q c . 2.5. Non-Abelian symmetry algebras of the Hamiltonians. In the thermodynamic limit of the XXZ spin chain Hamiltonian, it is well-known that the Hamiltonian commutes with the generators of the quantum affine algebra U q ( sl 2 ) acting on an infinite tensor product representation [START_REF] Foda | Corner transfer matrices and quantum affine algebras[END_REF][START_REF] Jimbo | Quantum group symmetry and lattice correlation functions[END_REF][START_REF] Jimbo | Correlation functions of the XXZ model for ∆ < -1[END_REF]. In this subsection, we basically use similar arguments in order to characterize the hidden non-Abelian symmetries of the open XXZ spin chain for four different types of boundary conditions. As an example, let us consider the Hamiltonian (1.1). On the semi-infinite tensor product vector space (1.2), the generators (2.5) of the q-Onsager algebra act as (see also [START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF]):

W (∞) 0 = ∞ j=1 • • • ⊗ q σ3 ⊗ q σ3 ⊗ (k + σ + + k -σ -) sitej ⊗II ⊗ • • • ⊗ II + + • • • ⊗ q σ3 ⊗ q σ3 , (2.22) W (∞) 1 = ∞ j=1 • • • ⊗ q -σ3 ⊗ q -σ3 ⊗ (k + σ + + k -σ -) sitej ⊗II ⊗ • • • ⊗ II + -• • • ⊗ q -σ3 ⊗ q -σ3 ,
where the coaction map (2.3) is iterated repeatedly and (2.5) is used. The action of the operators (2.22) on (1.1) is easy to compute. On one hand, introduce the local density

h i = σ i+1 1 σ i 1 + σ i+1 2 σ i 2 + ∆σ i+1 3 σ i 3 .
By straightforward calculations, one first observes:

h i , W (∞) 0 = • • • ⊗ q σ3 ⊗ q σ3 ⊗ Γ i+1,i ⊗ II ⊗ • • • ⊗ II (2.23) where Γ i+1,i = (2k + σ + -2k -σ -) ⊗ (∆ -q σ3 )σ 3 + (∆q σ3 -1)σ 3 ⊗ (2k + σ + -2k -σ -).
Summing up the local densities, after some simplifications one ends up with:

∞ i=1 h i , W (∞) 0 = - 1 2 (q -q -1 ) • • • ⊗ q σ3 ⊗ q σ3 ⊗ (k + σ + -k -σ -).
On the other hand, the contribution from the boundary term in (1.1) is non-vanishing. It gives:

h B , W (∞) 0 = 1 2 (q -q -1 ) • • • ⊗ q σ3 ⊗ q σ3 ⊗ (k + σ + -k -σ -).
Combining the above expressions together and repeating the analysis for W (∞) 1

, we conclude that the q-Onsager algebra is a symmetry algebra of the Hamiltonian with generic boundary conditions:

H 1 2 XXZ , W (∞) 0 = 0 and H 1 2 XXZ , W (∞) 1 = 0. (2.24)
Although the calculations become quickly more involved, the same technique is then extended to the Hamiltonian (1.1) with triangular ( ± = 0, k + = 0, k -= 0), diagonal ( ± = 0, k ± = 0) and special boundary conditions ( + = 1, -= 0, k ± = 0), respectively. With respect to the boundary conditions chosen, it is found that the corresponding Hamiltonian is commuting with all generators of the triangular q-Onsager, augmented q-Onsager and U q (gl 2 ) invariant q-Onsager algebras acting on (1.2), respectively. In the next Section, we present an alternative and much simpler derivation of the symmetry operators generating the four different types of q-Onsager algebras.

An alternative derivation of the symmetry operators

In the q-Onsager approach of the finite open XXZ spin chain [START_REF] Baseilhac | A deformed analogue of Onsager's symmetry in the XXZ open spin chain[END_REF] with left and right generic non-diagonal boundary conditions, it is known that all local or non-local conserved quantities are generated from elements in a certain quotient 5 of the infinite dimensional algebra A q [BK07] (see also [START_REF] Baseilhac | A new (in)finite dimensional algebra for quantum integrable models[END_REF]). In the literature, the generators of A q are usually denoted {W -k , W k+1 , G k+1 , Gk+1 |k ∈ Z + }, and W 0 , W 1 are called the fundamental generators. Importantly, W 0 , W 1 satisfy the defining relations of the q-Onsager algebra (2.1) [B04, B05].

By analogy with Onsager's [Ons44], Dolan-Grady's [DG82] and Davies' [D90] works on the Onsager algebra, generalizing the results of [BK05, Subsection 2.4.2] it was shown in [BB10, Example 2] through a brute force calculation that the first few 'descendants' generators W -1 , W 2 , G 1 , G1 , G 2 , G2 can be written as polynomials in the two fundamental generators W 0 , W 1 only. Including a central element γ, for instance one finds:

G 1 = W 1 , W 0 q -(q -q -1 ) + -γ , (3.1) W -1 = 1 ρ (q 2 + q -2 )W 0 W 1 W 0 -W 2 0 W 1 -W 1 W 2 0 + W 1 - (q -q -1 ) 2 ρ + -γW 0 , G 2 = 1 ρ(q 2 + q -2 ) (q -3 + q -1 )W 2 0 W 1 2 -(q 3 + q)W 1 2 W 2 0 + (q -3 -q 3 )(W 0 W 1 2 W 0 + W 1 W 2 0 W 1 ) -(q -5 + q -3 + 2q -1 )W 0 W 1 W 0 W 1 + (q 5 + q 3 + 2q)W 1 W 0 W 1 W 0 +ρ(q -q -1 )(W 2 0 + W 1 2 ) -(q 2 + q -2 )(q -q -1 ) 2 + -γ(qW 1 W 0 -q -1 W 0 W 1 ) -( 2 + + 2 -)γ 2 q -q -1 q 2 + q -2 -2 + 2 -γ 2 (q -q -1 ) 3 ρ(q 2 + q -2 ) -ρq 2 (γ 2 -1) (q + q -1 ) 3 (q 2 + q -2 )
.

The expressions for the elements G1 , W 2 , G2 are obtained from G 1 , W -1 , G 2 by exchanging W 0 ↔ W 1 in the above formulae.

5 Because the generators of Aq act on a finite dimensional vector space V (N ) of dimension 2 N , they satisfy additionnal relations that are q-deformed analogs of Davies' relations. Compare the first reference of [BK07, eqs. (17),( 18)] to the first reference of [D90, eq. (2.6a),(2.6b)] for details.

Actually, under certain assumptions it is possible to show that all descendants generators of the algebra A q admit a unique explicit polynomial formulae in terms of the two fundamental generators W 0 , W 1 [START_REF] Baseilhac | A note on the Oq( sl 2 ) algebra[END_REF]. Roughly speaking, they take the form:

W -k = F -k (W 0 , W 1 ), G k+1 = F k+1 (W 0 , W 1 ), (3.2) W k+1 = F -k (W 1 , W 0 ), Gk+1 = F k+1 (W 1 , W 0 ) for all k ∈ N,
where {F k (X, Y ), k ∈ Z} is a family of two-variable polynomials in X, Y . Note that the polynomial formulae that are obtained in [START_REF] Baseilhac | A note on the Oq( sl 2 ) algebra[END_REF] can be independently derived using the connection between the algebra A q and the reflection equation algebra [START_REF] Baseilhac | A new current algebra and the reflection equation[END_REF]. Details will be reported elsewhere.

We now show how each realization of the four different types of q-Onsager algebras in terms of U q ( sl 2 ) Chevalley generators (2.5), (2.10), (2.15), (2.21) can be recovered in a straightforward manner starting from the polynomial formulae (3.1) for the first few descendant generators. Define:

W 0 = ψ(W 0 ), W 1 = ψ(W 1 ), γ = ψ(Γ), (3.3)
with (2.5). Observe that the polynomials (3.2) can be systematically expanded in terms of the Chevalley generators and parameters k ± , ± . The explicit expressions for the first few examples (3.1), sufficient for our purpose, are reported in Appendix B. Then, we specialize case by case the boundary parameters into these expressions. According to the specialization chosen, the corresponding 'reduced' descendant generators do produce the symmetry operators of each of the q-Onsager algebras exhibited in the previous Section. Explicitly, we obtain:

• Triangular boundary conditions k -= 0, k + = 0, ± = 0. W 0 | k+→0 = ψ t (T 0 ), W 1 | k+→0 = ψ t (T 1 ), G1 k + k+→0 = ψ t ( P1 ).
• Diagonal generic boundary conditions k ± = 0, ± = 0.

W 0 | k±→0 = ψ d (K 0 ), W 1 | k±→0 = ψ d (K 1 ), G 1 k -k±→0 = ψ d (Z 1 ), G1 k + k±→0 = ψ d ( Z1 ).
• Special diagonal boundary conditions k ± = 0, -= 0, + = 1.

W 1 | k±→0, + →1, -→0 = 0, W 0 | k±→0, + →1, -→0 = ψ i Γq -h , G 1 k -k±→0, + →1, -→0 = ψ i (q 2 -q -2 )Γq -1 eq -h , G1 k + k±→0, + →1, -→0 = ψ i (q 2 -q -2 )Γf , W 2 | k±→0, + →1, -→0 = ψ i Γ q + q -1 q 1+h + q -1-h + (q -q -1 ) 2 fe , W -1 | k±→0, + →1, -→0 = ψ i q -q -1 (q + q -1 ) 2 Xq -h Γ , G 2 k -k±→0, + →1, -→0
= ψ i -q -2 (q -q -1 ) (q + q -1 ) ΓYq -h + q -1 (q -q -1 ) 2 (q + q -1 ) ΓeXq h ,

G2

k + k±→0, + →1, -→0 = ψ i -q (q -q -1 ) (q + q -1 ) Γ Ỹ + (q -q -1 ) 2 (q + q -1 ) fX .

Finally, as an alternative check of the analysis of the previous Section, we now show that according to the choice of boundary conditions, the corresponding above set of operators acting on the vector space (1.2) commute with the associated Hamiltonian. For generic boundary conditions k ± , ± , from (2.24) recall that the two operators W 0 , W 1 acting on (1.2) commute with the Hamiltonian. As a corollary of (2.24) and (3.2), for generic boundary conditions it implies that any descendant generator is commuting with (1.1):

H 1 2 XXZ , F k (W (∞) 0 , W (∞) 1 ) = 0, H 1 2 XXZ , F k (W (∞) 1 , W (∞) 0 ) = 0 for all k ∈ Z. (3.4)
The sum c = h 0 + h 1 is the central element of the algebra. The Hopf algebra structure is ensured by the existence of a comultiplication ∆ : U q ( sl 2 ) → U q ( sl 2 ) ⊗ U q ( sl 2 ), antipode S : U q ( sl 2 ) → U q ( sl 2 ) and a counit E :

U q ( sl 2 ) → C with ∆(e i ) = e i ⊗ 1 + q hi ⊗ e i , ∆(f i ) = f i ⊗ q -hi + 1 ⊗ f i , ∆(h i ) = h i ⊗ 1 + 1 ⊗ h i , (4.2) S(e i ) = -q -hi e i , S(f i ) = -f i q hi , S(h i ) = -h i S(1) = 1 and E(e i ) = E(f i ) = E(h i ) = 0 , E(1) = 1 .
Note that the opposite coproduct ∆ can be similarly defined with ∆ ≡ σ •∆ where the permutation map σ(x⊗y) = y ⊗x for all x, y ∈ U q ( sl 2 ) is used.

The (evaluation in the principal gradation) endomorphism π ζ : U q ( sl 2 ) → End(V ζ ) is chosen such that (V ≡ C 2 )

π ζ [e 1 ] = ζσ + , π ζ [e 0 ] = ζσ -, π ζ [f 1 ] = ζ -1 σ -, π ζ [f 0 ] = ζ -1 σ + , π ζ [q h1 ] = q σ3 , π ζ [q h0 ] = q -σ3 , (4.3)
in terms of the Pauli matrices σ ± , σ 3 : σ + = 0 1 0 0 , σ -= 0 0 1 0 , σ 3 = 1 0 0 -1 .

APPENDIX B: Descendant generators in the Drinfeld-Jimbo presentation of U q ( sl 2 )

G 1 k - = (q 2 -q -2 ) -q c f 1 + + q -1 e 0 q h1 + k -f 1 q h1 e 0 (4.4) + k + qe 1 e 0 -q -1 e 0 e 1 + (qf 0 f 1 -q -1 f 1 f 0 )q c , W -1 = q + q -1 q 1+h1 + q -1-h1 + (q -q -1 ) 2 f 1 e 1 q c (4.5) + + (q -q -1 ) (q + q -1 ) 2 qe 1 e 0 -q -1 e 0 e 1 + (qf 0 f 1 -q -1 f 1 f 0 )q c q h1 + k + (q + q -1 ) 2 (q -q -1 )(qf 0 f 1 -q -1 f 1 f 0 )q c e 1 + q -2 (q + q -1 )f 0 q h0 -(e 1 ) 2 e 0 + (q 2 + q -2 )e 1 e 0 e 1 -e 0 (e 1 ) 2

+

k - (q + q -1 ) 2 q -1 (q + q -1 )e 0 + q -1 (q -q -1 )f 1 q h1 (qe 1 e 0 -q -1 e 0 e 1 ) -q c-1 (f 1 ) 2 f 0 -(q 2 + q -2 )f 1 f 0 f 1 + f 0 (f 1 ) 2 , G 2 k - = - q -q -1 q + q -1 (q + q -1 )q c e 0 q h1 + q 2 -q -2 q + q -1 q c f 1 (qe 0 e 1 -q -1 e 1 e 0 ) (4.6) -q 2c (f 1 ) 2 f 0 -(q 2 + q -2 )f 1 f 0 f 1 + f 0 (f 1 ) 2 + + q -q -1 q + q -1 (q + q -1 )q 2c+1 f 1 + q 2 -q -2 q + q -1 q c-1 (qf 0 f 1 -q -1 f 1 f 0 )e 0 q h1 -q -1 (e 0 ) 2 e 1 -(q 2 + q -2 )e 0 e 1 e 0 + e 1 (e 0 ) 2 + O(k + ) + O(k -).

Note that for simplicity, the terms of order k + and k -in G2 k-are not explicitly written, as they do not contribute in the limit k ± = 0. Also, the descendant generators G1 k+ , W 2 and G2 k+ can be derived in terms of the generators of the Drinfeld-Jimbo presentation using the map x 0 → x 1 , x 1 → x 0 , with x ∈ {h, e, f }, k ± → k ∓ and ± → ∓ on the expressions (4.4), (4.5) and (4.6) respectively.

In (1.1), we implicitly assume that + + -= 0.

The corresponding Hamiltonian can be understood as the thermodynamic limit of the Uq(sl 2 ) invariant spin chain studied in[START_REF] Pasquier | Common structures between finite systems and conformal field theories through quantum groups[END_REF].

However, note that the generators of the Uq(gl 2 ) invariant q-Onsager algebra have already appeared in[START_REF] Regelskis | Reflection algebras for SL(2) and GL[END_REF][START_REF] Kolb | Quantum symmetric Kac-Moody pairs[END_REF].

In[START_REF] Ito | The augmented tridiagonal algebra[END_REF], the special case Γ = 1 is considered. In[START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF], the central element is not explicitly introduced and the first relation is replaced by [K 0 , K 1 ] = 0. Note that for the level one q-vertex operators representations constructed in[START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF], one has Γ = q.

Acknowledgements: P.B. thanks T. Kojima for discussions. S.B. thanks V. Regelskis for discussions and also LMPT for hospitality, where part of this work has been done. S.B. is supported by a public grant as part of the "investissement d'avenir" project, reference ANR-11-LABX-0056-LMH, LabEx LMH. P.B. is supported by C.N.R.S.

Specializing the boundary parameters accordingly, it implies that the four different types of q-Onsager algebras are symmetry algebras of the Hamiltonian.

Let us remark that the defining relations of the current algebra associated with the q-Onsager algebra are given in [START_REF] Baseilhac | A new current algebra and the reflection equation[END_REF]. For the triangular, augmented and U q (gl 2 ) invariant q-Onsager algebras, the defining relations of the corresponding current algebras can be derived, respectively, by taking appropriate limits of the relations in A q , given by [BS09, Definition 3.1] and [BS09, Proposition 3.1].

Concluding remarks

It is instructive to consider the limit q → 1 of the four different types of q-Onsager algebras described in Section 2. In particular, in this limit the q-Onsager, the augmented q-Onsager and the U q (gl 2 ) invariant q-Onsager algebras specialize to three different invariant fixed-point subalgebras of U ( sl 2 ). For the q-Onsager algebra with q → 1, the corresponding automorphism is the Chevalley involution of U ( sl 2 ), given by θ(e i ) = f i , θ(h i ) = -h i . For the augmented q-Onsager algebra with q → 1, one considers the composition of the Chevalley involution with the outer automorphism of U ( sl 2 ). It reads:

For the U q (gl 2 ) invariant q-Onsager algebra at q → 1, one considers the composition of the Chevalley involution with the Lusztig's automorphism of

From that point of view, three of the algebras introduced in Section 2 can be understood as a q-deformation of these invariant fixed-point subalgebras of U ( sl 2 ) [START_REF] Kolb | Quantum symmetric Kac-Moody pairs[END_REF]. For the q-Onsager and augmented q-Onsager algebras, see [START_REF] Belliard | Coideal algebras from twisted Manin triples[END_REF] for details. Note that it is a simple exercice to apply the technique of [START_REF] Belliard | Coideal algebras from twisted Manin triples[END_REF] to the U q (gl 2 ) invariant q-Onsager algebra. Besides these three algebras, let us point out that the triangular q-Onsager algebra gives an example of coideal subalgebra that does not correspond to an invariant fixed-point subalgebra of U q ( sl 2 ) even at q → 1.

From the point of view of physics, in the thermodynamic limit of the open XXZ spin chain with Hamiltonian (1.1), the solution of the model (spectrum, eigenstates, multiple integral representations of correlation functions and form factors) can be derived using infinite dimensional (q-vertex operators) representations of the symmetry algebras here considered. Clearly, the q-vertex operators for each symmetry algebra follow from the representation theory of U q ( sl 2 ). Note that the solution for special, diagonal and triangular boundary conditions is given in [JKKKMW95, [START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF][START_REF] Baseilhac | Correlation functions of the half-infinite XXZ spin chain with a triangular boundary[END_REF][START_REF] Baseilhac | Form factors of the half-infinite XXZ spin chain with a triangular boundary[END_REF]. For the Hamiltonian with generic boundary conditions, although q-vertex operators for the q-Onsager algebra are known [START_REF] Baseilhac | The half-infinite XXZ chain in Onsager's approach[END_REF], the solution remains an open problem.

Finally, an interesting problem would be to extend the analysis here presented to integrable models with higher rank symmetries of q-Onsager's type (see for instance [START_REF] Baseilhac | Generalized q-Onsager algebras and boundary affine Toda field theories[END_REF]). The higher rank infinite dimensional algebra extending A q , yet unknown, could be a starting point for the identification of the symmetry algebras. APPENDIX A: Drinfeld-Jimbo presentation of U q ( sl 2 ) Define the extended Cartan matrix {a ij } (a ii = 2, a ij = -2 for i = j). The quantum affine algebra U q ( sl 2 ) is generated by the elements {h j , e j , f j }, j ∈ {0, 1} which satisfy the defining relations [h i , h j ] = 0 , [h i , e j ] = a ij e j , [h i , f j ] = -a ij f j , [e i , f j ] = δ ij q hi -q -hi q -q -1 together with the q-Serre relations [e i , [e i , [e i , e j ] q ] q -1 ] = 0 , and [f i , [f i , [f i , f j ] q ] q -1 ] = 0 . (4.1)