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based on a canonical quantization approach , the loop quantum gravity program gives a picture of the quantum geometry of space in terms of spin networks . dynamic enters the theory only through the hamiltonian constraint which is still poorly understood . spin foam models @xcite are alternative way to give dynamic to spin networks and are candidates for the construction of a quantum theory of gravity in a covariant framework . spin foam models are attempts to evaluate transition amplitude between spin network states . therefore it extends the structure of spin networks , which are one dimensional complexes , to two - complexes built from vertices , edges ( colored by intertwining operators ) and faces ( colored by certain type of group representations ) . a spin foam model would ultimately describe the quantum geometry of space - time and any slice of it would be a spin network describing the space structure at a given time . it is a major property of four dimensional pure gravity that it is a constrained topological field theory : it can be expressed as a @xmath0 theory with @xmath1 ( resp . @xmath2 ) group for the euclidean ( resp . lorentzian ) signature , where the @xmath3 field is constrained to be `` simple '' and more particularly to be of the type @xmath4    this property has given hope to construct spin foam models of quantum gravity by constraining spin foam models describing @xmath0 theory . indeed topological field theory of bf type can be exactly discretized with spin foam models . this is , in three dimensions , the result of the work of ponzano - regge @xcite and in 4 dimensions the work of ooguri @xcite . however without cosmological constant these models have infrared divergences . in the euclidean signature case these previous spin foam models can be regularized by the addition of a cosmological term and one obtains , in three dimensions , the turaev - viro model @xcite and in four dimensions the crane - yetter model @xcite . these spin foam models are built using representation theory of certain quantum groups . the barrett - crane model @xcite is a spin foam model which aim is to implement , in the ooguri model , the constraint of simplicity on @xmath3 . this can be done very easily in the euclidean case by restricting representations on the faces to be `` simple '' and by coloring the edges with a very peculiar intertwiner , known in the litterature as the barrett - crane intertwiner . soon after this proposal these authors gave a similar proposal @xcite in the lorentzian case . these models , which gave an impetus for a large amount of work on spin foam models of constrained @xmath0 theory , have several drawbacks :    \1 ) the weight of the vertices is well defined ( a 10j symbol ) but the weights on the edges and faces are not fixed and are still a matter of debate @xcite . this issue is very important for proving the finiteness of the amplitude after integrating over the area variables for a fixed 2-complex @xcite . \2 ) a meaning to the sum over 2-complexes has to be done in order to compute an amplitude , and up to now , there is no result on this important issue in the 4 dimensional case . note however the recent work of @xcite in the case of three dimensional riemmannian quantum gravity . \3 ) the simplicity constraint does not imply that the @xmath3 field is in the gravitational sector , and the relative weight of the gravitational sector compared to the other seems to be small @xcite . our work aims at defining and studying the barrett - crane model in the case of the quantum lorentz group @xmath5 for @xmath6 real . this is possibly interesting for different reasons . the first one is that one would like to obtain spin foam models for describing quantum gravity with a positive cosmological constant . this is important for the study of quantum gravity in de sitter space @xcite . the second one is that the use of quantum groups enhance convergence , particularly in the infrared . in @xcite three spin foam models are analyzed in the riemannian case . they differ only by the choice of the weight on the edges and on the faces . we will show that for any @xmath7- complex dual to a triangulation of the manifold , the integration over the coloring ( i.e area variables ) are finite for the quantum deformations of these three models . the third one is that , if the sum over 2-complexes can be defined , it is only in the presence of a cosmological constant . our work is divided in four sections . in section ii we recall elementary facts on spin foam models . in section iii we give a construction of the quantum hyperboloid and of the quantum lorentzian barrett - crane intertwiner . section iv is devoted to the study of quantum simple lorentzian spin networks . section v is an analysis of the cosmological deformation of the lorentzian barrett - crane model . we will prove the infrared finiteness property in this section . a large class of topological quantum field theories can be exactly discretized by spin foam models . for a review on this subject , see @xcite . this class of theories includes @xmath0 theory in any dimension and @xmath0 theory with cosmological constant in three and four dimensions . the barrett - crane model is a spin foam model which is advocated to be a discretization of the quantization of plebanski theory in four dimensions . the euclidean version has been defined in @xcite and the lorentzian one was studied in @xcite . in this last paper , the authors have also mentioned the idea to extend their model to the quantum deformation of the lorentz group as a possible regularization . after an overview of spin foam models ( lorentzian and euclidean ) in 3 and 4 dimensions , we recall the links between quantum spin foam models and @xmath0 theory with cosmological constant . we then explore the relations between quantum deformation of lorentzian barrett - crane model and quantum gravity in the presence of a cosmological constant .    in the sequel we will consider a riemannian ( resp . lorentzian ) manifold . we will denote @xmath8 the sign of the metric . we will denote @xmath9 and @xmath10 the isometry group of @xmath11    the einstein - hilbert action on a @xmath12 dimensional oriented riemannian ( resp.lorentzian ) manifold with cosmological constant @xmath13 is defined by @xmath14= \frac{1}{g } \int_{m } d^d x \ ;   \ ; ( { r}[g]-2 \lambda )   \sqrt{\vert   g\vert}.\end{aligned}\ ] ] let us introduce an orthonormal triad @xmath15 and a @xmath16 connection @xmath17 . the metric can be expressed in term of the cotriad @xmath18 as @xmath19 and the action ( [ actionofgravity ] ) is rewritten as the palatini action @xmath20 = \frac{1}{g } \int _ { m } d^d x \ ; \vert e \vert \ ; ( r_{\mu \nu}^{ij}[\omega]\ ;   e_i^{\mu } \ ; e_j^{\nu}-2\lambda)\end{aligned}\ ] ] where @xmath21 . this action can be conveniently written in term of forms , as :    @xmath22=\frac{1}{g } \int _ { m }    ( r^{ij}[\omega]\wedge { \star } ( e_i\wedge   e_j)-2\lambda { \star } 1)\ ] ]    where the @xmath23 is the hodge operator . from the orthonormality of @xmath24 , we have : @xmath25 where @xmath26 is the sign of @xmath27    the palatini action can be recast , in any dimension as : @xmath22=\frac{1}{(d-2)!g } \int _ { m }   s(e ) \epsilon_{i_1\cdots i_{d}}(r^{i_1i_2}\wedge e^{i_3}\wedge\cdots\wedge e^{i_d } - \frac{2\lambda}{d(d-1 ) } e^{i_1}\wedge\cdots\wedge e^{i_d}).\ ] ]      the very first spin foam model was introduced by ponzano and regge @xcite as an attempt to define a discrete version of three dimensional riemannian quantum gravity . the modern understanding of this construction follows the next lines of arguments . the action of @xmath0 theory in @xmath28 dimension for the @xmath10 group is given by : @xmath29 as a result in the case where @xmath30 this is exactly the action of palatini . the partition function of @xmath0 theory is @xmath31 \right).\end{aligned}\ ] ]    the action , being linear in the @xmath18 , we can integrate over the @xmath18 and we get : @xmath32(x)).\end{aligned}\ ] ] this partition function is therefore the `` volume '' of the space of flat connections on @xmath33 it can be exactly computed by the following discretization in the riemannian case . we choose any triangulation @xmath34 of the manifold : to each edge @xmath35 we associate an irreducible representation @xmath36 of @xmath37 . the ponzano - regge partition function is then given by @xmath38 where the sum runs over all irreducible representations of @xmath37 . we have assigned the weight @xmath39 to each edge @xmath35 and the weight @xmath40 to each tetrahedron @xmath41 colored with the six representations labelling the edges of @xmath41 . note that the @xmath42 we are using in this expression is invariant under the group of symmetry of the tetrahedron . the length of the edge @xmath35 in natural units is @xmath43 , where @xmath44 is the planck length . a simple explanation of this assertion comes from the classical limit of ponzano - regge sum : we let @xmath45 tends to zero while letting @xmath46 fixed to a classical length @xmath47 . thus the weight of a tetrahedron @xmath41 behaves as @xcite : @xmath48 } } \cos(\frac{s_r[t]}{\hbar}+\frac{\pi}{4})\end{aligned}\ ] ] with @xmath49 $ ] the volume of the tetrahedron and the regge action @xcite , given by @xmath50=\frac{1}{g}\sum_{\epsilon\in t}\theta_\epsilon \ ; l(\epsilon ) \ ; , \end{aligned}\ ] ] is a discretization , for @xmath51 , of ( [ actionofgravity ] ) . the partition function ( [ prsum ] ) can also be expressed as a sum over spin foams lying in the dual 2-skeleton of the given triangulation : in this way , the ponzano - regge model appears to be a spin foam model . note that the sum can be shown to be triangulation independent ( because of the biedenharn - elliot relation ) and thus it defines a topological invariant of the manifold @xmath52 . the ponzano - regge sum , as well as the partition function of @xmath0 theory , differs from the partition function of @xmath28 dimensional gravity . indeed the weight of a configuration of @xmath53 with @xmath54 differs from a weight of @xmath53 with @xmath55 in @xmath0 theory whereas it is the same in the action of gravity . this problem is too much understated in the literature but is well addressed in @xcite . the comparison between the two theories is still not very well understood . the ponzano - regge sum suffers from an infrared divergence , i.e when the variables @xmath36 become large . several possible regularization have been introduced , among them the turaev - viro one is the most natural and effective . it consists in replacing the classical lie algebra @xmath56 by the corresponding quantum group @xmath57 where @xmath6 is the root of unity @xmath58 . the edges are now labelled by irreducible representations of @xmath57 and the sum becomes : @xmath59 where @xmath60 is the quantum dimension of the representation @xmath36 , and @xmath61 , where @xmath62 is the quantum @xmath42 symbol associated to the tetrahedron @xmath41 , @xmath63 is the number of vertices of @xmath34 and @xmath64 . note that the number of irreducible representations of @xmath57 with non vanishing q - dimension is finite when @xmath6 is a root of unity , therefore the sum ( [ sumtv ] ) is a finite sum . the turaev - viro model is an exact discretization of @xmath0 quantum theory with a cosmological term , whose action is : @xmath65= \frac{1}{g } \int _ { m }   \epsilon_{ijk } \ ; ( r^{ij}[\omega]-\frac{\lambda}{3 } e^i\wedge e^j ) \ \ ; \wedge \ ; e^k.\end{aligned}\ ] ] when @xmath66 , this action is the same as the euclidean einstein - hilbert action with cosmological constant @xmath67 and the level @xmath68 is related to the cosmological constant by @xmath69 where @xmath70 is the cosmological length . note that the partition function @xmath71 is unchanged when one turns @xmath6 into @xmath72 because of the invariance of the quantum dimension and the quantum @xmath42-symbol in this transformation . this property corresponds in the @xmath0 theory to the invariance of the partition function @xmath73 when one tuns @xmath24 into @xmath74 and conjugate @xmath75 into @xmath76 . there are , up to our knowledge , different ways to understand that the turaev - viro sum is related to @xmath0 theory with cosmological term and that quantum groups necessarily appear in the quantization process . the first one , well exposed for example in @xcite uses the equivalence , due to witten @xcite , between @xmath0 theory with positive cosmological term and chern - simons theory with group @xmath77 in order to show that the turaev - viro sum is the partition function of @xmath0 theory with positive cosmological term one needs an additional result , shown by mathematicians , which is that the turaev - viro sum of a manifold @xmath52 is the modulus square of reshetikhin - turaev invariant of @xmath52 for the group @xmath78 the equality between reshetikhin - turaev invariant of @xmath52 and the partition function of chern - simons of m implies therefore the equality between the turaev - viro sum and the partition function of bf with cosmological constant as well as the relation between @xmath6 and @xmath79    an alternative way consists in studying the asymptotic behavior of the quantum @xmath42-symbol in the regime where @xmath80 this regime corresponds to the classical limit with a finite cosmological constant : physically , the length are bounded by @xmath81 because of the presence of a cosmological horizon . the asymptotic value of the quantum @xmath62 has been well understood in @xcite where , after reintroducing the natural units , the authors obtain : @xmath82 } } \cos(\frac{s_{r,\lambda}[t]}{\hbar}+\frac{\pi}{4}),\end{aligned}\ ] ] where @xmath49 $ ] is the volume of the tetrahedron and the regge action in presence of a cosmological constant is @xmath83=\frac{1}{g}\left(\sum_{\epsilon\in t}\theta_\epsilon \ ; l(\epsilon ) \ ; - \frac{1}{l_c^2 } v[t]\right)\;\;.\end{aligned}\ ] ]    these two previous arguments are rather indirect . a very clear understanding of the appearance of quantum groups in the quantization of @xmath0 theory with cosmological term is obtained using the combinatorial quantization of chern - simons theory as first understood in @xcite there are recent attempts to formally define 3-dimensional lorentzian spin foam models , with possible applications to 2 + 1 quantum gravity . an analogue of the ponzano - regge sum can be straightforwardly defined . one begins with the partition function of @xmath0 theory with @xmath84 group and applies the same method of integrating the @xmath53 field and replacing the delta function by its decomposition into characters of unitary representations of @xmath84 . one is therefore led to define a ponzano - regge model where the representations coloring the edges belong to the principal series and to the discrete series . the weight of the tetrahedron is given as the @xmath42 of these representations @xcite . this subject is still in its infancy , and important properties such as the study of particular regularization of this @xmath84 ponzano - regge sum and the proof of the asymptotic behaviour of the @xmath42 symbols are still unknown . it is expected that quantization of @xmath85 @xmath0 action with positive cosmological term is described by a turaev - viro model associated to @xmath86 for q real with @xmath87 due to the equality between the @xmath0 action with positive cosmological term and the chern - simons action associated to the lie algebra @xmath88 , there should exist a relation between the would be turaev - viro sum with @xmath86 and the would be reshetikhin - turaev invariants for the quantum algebra @xmath89 to be defined , these invariants , as well as the expectation value of observables , need regularization and up to now the litterature is empty on this subject . note however our work @xcite , where we quantize chern - simons theory with @xmath2 group using the program of combinatorial quantization of hamiltonian chern - simons theory . the action of @xmath0 theory in @xmath90 dimensions in the case of a simple compact lie group @xmath91 is @xmath92 = \int_m tr\left ( b \wedge f(a)\right)\end{aligned}\ ] ] where the field @xmath93 is a @xmath94-valued two - form , @xmath95 is the curvature associated to the @xmath94-connection @xmath96 and @xmath97 denotes the killing form on the lie algebra . this defines a topological theory which has been first studied in @xcite . in the case of @xmath98 , which is the case that we consider in this section , ooguri @xcite was the first to provide an exact discretization of @xmath0 theory in 4 dimensions through the construction of a spin foam model . let @xmath52 be a four dimensional oriented compact smooth manifold . let @xmath34 be any triangulation of this manifold and @xmath99 the set of @xmath100 simplices of @xmath34 . let @xmath101 be the set of irreducible representations of @xmath91 . if @xmath102 , we denote @xmath103 the vector space on which the representation @xmath104 acts . to each face @xmath105 of @xmath34 we associate an element @xmath106 of @xmath107 to each tetrahedron @xmath41 we define @xmath108 , to be the space of @xmath109intertwiners @xmath110 @xmath108 is endowed with the hermitian form @xmath111 in the @xmath37 case an orthonormal basis of @xmath108 is labelled by elements of @xmath101 as well . to each tetrahedron we associate an element @xmath112 of this orthonormal basis or equivalently an element @xmath113 the ooguri partition function is defined as : @xmath114 where the so - called @xmath115-symbol @xmath116 associated to each 4-simplex @xmath117 of the triangulation is constructed from the 10 representations labelling its 10 faces and the 5 representations labelling its 5 tetrahedra ( figure [ 15jsymbol1 ] ) .    -symbol . ]    for the same reason as for the ponzano - regge model , the sum ( [ osum ] ) is in general divergent . the crane - yetter model @xcite is a regularization of ( [ osum ] ) . it is the 4-dimensional analogue of the regularization of turaev - viro of the ponzano - regge sum and it is an exact discretization of @xmath0 theory with cosmological term whose action is : @xmath118    the regularization consists in replacing the classical lie group by the quantum group @xmath57 where @xmath6 is taken to be a root of unity . however , because it is impossible to have a projection of the 4-simplex on a plane without having at least two edges which intersect , it is important to be careful on the definition of the @xmath115 symbol and to correctly take account of the crossing .. this issue does not appear in the turaev - viro construction . the crane - yetter construction was generalized to the framework of 2-spherical categories in @xcite . we will refer to this work and use its notations because the construction is cristal - clear there . we take the quantum algebra @xmath57 for @xmath6 root of unit . we pick any total order on the set of vertices of the triangulation @xmath34 . this order induces , by lexicography , a total order on the set of faces , on the set of tetrahedra and on the set of 4-simplices . to each face @xmath119 we associate a representation of @xmath57 and we denote by @xmath120 its associated vector space . let @xmath117 be a @xmath90-simplex of @xmath34 , @xmath121 @xmath122 to each tetrahedron , appearing with positive sign in @xmath123 , for example @xmath124 , we associate the space @xmath125 of intertwiners @xmath126 @xmath127 to each tetrahedron appearing with negative sign in @xmath123 , for example @xmath128 , we associate the space @xmath129 of intertwiners @xmath130 @xmath131    we define the partition function of the 4-simplex @xmath132 defined by the graph shown in picture ( [ 15jsymbol2 ] ) .     where @xmath133 is a family of intertwiners . ] note that the definition of @xmath134 crucially depends on the ordering of the vertices . however , the sum @xmath135 does not depend on the chosen ordering nor on @xmath34 . it is therefore a topological invariant of the manifold . this invariant can be expressed solely in term of the signature and the euler characteristic of the manifold @xcite . furthermore , @xmath136 is the partition function of four dimensional @xmath0 theory with cosmological term whose action is : @xmath137 the cosmological constant @xmath13 is related to the quantum parameter @xmath6 by : @xmath138 where @xmath139 is the four dimensional planck length . this relation is derived in @xcite for example . pure gravity in @xmath90 dimensions is a constrained @xmath0 theory and the plebanski lagrangian is precisely designed to implement the contraint through the use of a lagrange multiplier @xcite . the einstein - hilbert action is written as : @xmath20 = \frac{1}{g } \int _ { m } d^4 x \ ; \vert e \vert \ ; ( r_{\mu \nu}^{ij}[\omega]\ ;   e_i^{\mu } \ ; e_j^{\nu}-2\lambda)\end{aligned}\ ] ] also written as :    @xmath20 = \frac{1}{g } \int _ { m } d^4 x   \epsilon_{ijkl } ( r_{ij}\wedge e^k\wedge e^l-\frac { 2\lambda}{4!}e^i\wedge e^j\wedge e^k \wedge e^l ) s(e)\end{aligned}\ ] ]    where @xmath140 is the curvature of the connection @xmath141 and @xmath142 is a basis of @xmath143    when @xmath144 this is exactly the form of a so(4 ) @xmath0 theory with cosmological constant , with @xmath145 and @xmath146 when @xmath3 is of this form it is said to belong to the gravitational sector . note , as usual , the sign @xmath147 whose possible effect in the quantum theory is still not understood . let us add some comments to clarify , for non experts , the condition of simplicity . if @xmath41 is an infinitesimal triangle , i.e @xmath148 with @xmath149 in the tangent space of a point @xmath150 in @xmath151 we can define @xmath152 the simplicity condition reads as : @xmath153 which simply means that que quadratic form @xmath154 in @xmath155 and @xmath156 vanishes . the condition of simplicity therefore is equivalent to : @xmath157}=0,\ ] ] which , when written explicitely in components , is exactly the three conditions : simplicity , intersection and normalisation of @xcite . note , of course , that this condition has nothing to do with the condition @xmath158 where the wedge means exterior product simultaneously on forms and on the elements of the lie algebra @xmath159 , which reads in components @xmath160}^{kl]}=0 , \;\;\text{i.e}\;\ ;   \epsilon^{\mu\nu\rho\sigma}\epsilon_{ijkl } b_{\mu\nu}^{ij } b_{\rho\sigma}^{kl}=0.\ ] ] when the @xmath3 field satisfies the condition that @xmath161 is nowhere equal to zero , @xmath3 is said to be non degenerate . it can be shown that the condition of simplicity on @xmath3 and the condition that @xmath3 is non degenerate imply that there exists a cotetrad @xmath18 such that @xmath162 the gravitational sector is the sector @xmath163    the plebanski lagrangian is designed to enforce the condition of simplicity on @xmath3 . the @xmath10 plebanski action is written as :    @xmath164= \frac{1}{g } \int_{m } d^4 x ( f_{ij}(\omega)\wedge b^{ij}- \frac{\lambda}{4}\epsilon_{ijkl } b^{ij}\wedge b^{kl } -\frac{1}{2}\phi_{ijkl } b^{ij}\wedge b^{kl}),\ ] ]    where @xmath165[kl]}$ ] is symmetric under the exchange of @xmath166 $ ] with @xmath167 $ ] and satisfies the constraint @xmath168 an analysis of this theory has been done in @xcite . let us pinpoint the results : integration on @xmath169 imposes the constraint that there exists a @xmath90-form @xmath170 such that @xmath171 in term of components this is equivalent to    @xmath172}= \tilde{\lambda}\epsilon_{\mu\nu\rho\sigma}\epsilon^{ijkl}.\label{simple1}\ ] ]    when @xmath3 is non degenerate this condition is also equivalent to    @xmath173}_{\rho\sigma}= \tilde{\lambda}\epsilon_{\mu\nu\rho\sigma}\epsilon^{ijkl}.\label{simple2}\ ] ]    any of these conditions are equivalent to the condition of simplicity .    as a result , integrating out the lagrange multiplier @xmath174 enforces @xmath3 to be simple . if one can design a method for restricting even more the @xmath3 field to the subsector @xmath175 one would therefore describe a quantization of the gravitational sector with a cosmological constant @xmath13 .    at the present time we are far from being able to implement and control this restriction . the barrett - crane model is a spinfoam model where the constraint of simplicity is imposed by a lattice analog of constraint ( [ simple2 ] ) : representations are balanced and the edges are colored by a simple intertwiner , the so - called barrett - crane intertwiner .    the basic idea is to find a way to implement the constraints ( [ simple2 ] ) directly in the ooguri model associated to the group @xmath10 . the euclidean barrett - crane model is based on simple geometric observations . it consists in describing each @xmath90-simplex of the triangulation @xmath34 in term of a set of @xmath176 simple bivectors @xmath177 associated to its faces @xmath105 . these bivectors satisfy a set of geometrical properties , given in @xcite . among them , the simplicity condition , i.e. @xmath178 , for all bivectors @xmath179 , is a discretization of the algebraic conditions ( [ simple2 ] ) . barrett and crane have postulated that these conditions should be imposed at the quantum level @xcite as follows .    in the euclidean case , they first write @xmath180 and label its irreducible representations by couple of non negative half - integers @xmath181 we can decompose this representation with respect to the diagonal subgroup @xmath37 and we denote @xmath182 and @xmath183    the barrett - crane euclidean spin foam model is defined through the following rules :    1 . faces are labelled with finite dimensional irreducible representations , edges are labelled by intertwiner operators . the representations coloring the faces of the triangulation are simple . simple representations are of the type @xmath184 , i.e. @xmath185 , which is equivalent to the fact that @xmath186 or equivalently that these representations have a non zero vector invariant under the action of the diagonal @xmath78 we denote by @xmath187 the set of simple representations . the area of a face @xmath105 colored by @xmath188 is then given by @xmath189 . 3 .   the intertwiners coloring the tetrahedron of the triangulation is unique and called euclidean barrett - crane intertwiner . this intertwiner is a 4-valent intertwiner between the simple representations coloring the faces of the tetrahedron and the trivial represenetation . its expansion into 3-valent intertwiners introduces only simple representations in the intermediate channel . up to a normalization , such an intertwiner is uniquely defined @xcite . the weight associated to each 4-simplex is constructed from the @xmath176 simple representations coloring its faces and from the 5 barrett - crane intertwiners labelling its boundary tetrahedra . we obtain the so - called @xmath190-symbol ( see figure [ 15jsymbol1 ] ) with @xmath191 fixed to the barrett - crane intertwiner . finally , the euclidean barrett - crane partition function is defined by : @xmath192 where @xmath193 is the value of the @xmath190 symbol of the colored simplex @xmath194    there is still some confusion about the value of the weights @xmath195 an empirical idea , pursued by @xcite , is to obtain a good balance between all the weights , in such a way that the expectation values of observables do not give trivial results ( zero value ) or do not explode to infinity . the work @xcite give a discussion of this important point and propose weights for @xmath196 on a basis of monte - carlo simulations . the goal of the work @xcite is to find in the discretized @xmath0 theory the configurations satisfying the constraint of being simple . the final conclusion is that the configurations of representations and intertwiners which are selected are precisely the one of the barrett - crane model . the method , unfortunately , can not fix the weights @xmath195    the lorentzian barrett - crane model @xcite is constructed along the same lines , considering instead of @xmath197 the lorentzian group @xmath2 or its universal covering @xmath198 . there are two types of models : the first one includes only spacelike faces and the second one includes timelike faces as well . these models have both been introduced in @xcite . a field theory on a group description has been given in @xcite . we will only discuss the first model . let us recall that principal irreducible representations of @xmath198 are labelled by a couple @xmath199 .    the lorentzian model of barrett - crane is defined using the same recipes as the euclidean one : the faces are colored by simple representations of type @xmath200 and the tetrahedra are colored by the lorentzian barrett - crane intertwiner .    the area of a face @xmath105 is related to the corresponding representation by the formula @xmath201    the lorentzian barrett - crane intertwiner is defined as in the euclidean case : it is unique up to a normalization and can be computed @xcite using the haar measure on the classical group ( this procedure needs a regularization because of the non - compactness of the group ) . the lorentzian barrett - crane partition function is given by ( [ bcsum ] ) where @xmath187 is now the set of simple representations of @xmath198 of type @xmath202 and the series is replaced by an integral over @xmath203 note that the existence of the @xmath190 symbol has been proven in @xcite . the lorentzian barrett - crane model is conjectured to be an exact discretization of the plebanski action associated to the lorentz group . the sum still suffers from an infrared divergence when @xmath204 goes to infinity . this is why it is very important to add a cut - off on the @xmath204 which can be implemented by adding a cosmological constant and going to the quantization of the lorentz group . this is the path that we will follow . note however that , up to now , another method has been applied . in @xcite , the authors have defined a spin foam model of lorentzian gravity by choosing appropriate weights @xmath205 and @xmath206 . this version of the lorentzian model has the marvelous property of being finite in the sense that for each triangulation , the integral over @xmath207 of the state sum is finite @xcite . therefore , there is no need to impose a cut off on the @xmath204 . however the work @xcite indicates that this model may well be trivial and that other choice of @xmath205 and @xmath206 are necessary and for them the finiteness seems to be more intricate let alone false .    the aim of our work is to define a quantum deformation of the lorentzian barrett - crane model . our goal is to introduce a natural cut - off , the cosmological constant , and show that the spin foam model , for a large class of choice of @xmath208 , has the property of being finite for each triangulation    we will be concentrating on the case where the quantum parameter is real and related to the cosmological constant by : @xmath209 where the cosmological length @xmath81 is defined by @xmath210 though it is consistent with all the results of the litterature , it is still conjectural that lorentzian plebanski theory with positive cosmological constant can be described by this type of spin foam model with this regime of @xmath211 the computation of the asymptotic of the 4-simplex could be a useful tool to support this conjecture . the quantum lorentzian spin foam model is then the barrett - crane model associated to the quantum group @xmath212 where @xmath6 is taken to be real . the representation theory of this quantum group is well exposed in @xcite where a plancherel theorem was proved and the intertwiners between principal unitary representations have been computed in @xcite . the main results will be recalled in the next section . note that principal unitary representations of @xmath212 are labelled by a couple @xmath213 where @xmath68 is an integer and @xmath214 is a real bounded by @xmath215 . these representations are the quantum analogue of principal representations of @xmath216 . the quantum lorentzian model is thus obtained considering only the representations @xmath200 where @xmath214 is now bounded . the area @xmath217 of a face colored by @xmath214 is given by ( [ aireface ] ) , and satisfies therefore the inequalities : @xmath218 in the regime where @xmath219    note that these inequalities are intuitively the expected ones : we can not define area less than the planck area and the maximal area of a space like surface is given by the area of the cosmological horizon .    before studying the model we recall mathematical results on the quantum lorentz group and other related objects . in this section , we introduce the fundamental tools needed to define a cosmological deformation of the lorentzian barrett - crane model . we acquaint the reader with the structure of the quantum lorentz algebra @xmath212 and its representations . we then describe the structure of the three dimensional quantum hyperboloid @xmath220 which appears to be a pile of quantum fuzzy spheres . we then define the cosmological deformation of the barrett - crane intertwiner as an integral on @xmath220 . the construction of this intertwiner mimics the classical one @xcite . the classical hyperboloid @xmath221 is the submanifold of the four dimensional minskowski space @xmath222 defined as @xmath223 where the metric is given by @xmath224 . @xmath221 admits two connected components @xmath225 , diffeomorphic to @xmath226 , which are the two sheets @xmath227 and @xmath228    the orthochronous proper lorentz group @xmath229 acts transitively on @xmath230 and therefore @xmath230 is the orbit of any element @xmath231 as a result @xmath230 is the homogeneous space @xmath232 where @xmath233 is the isotropy group ( the small group ) of @xmath234 this group is @xmath235 this analysis extends directly to the universal covering group @xmath236 of @xmath229 , and we get : @xmath237 the natural measure on minkowski space induces a measure @xmath238 on the hyperboloid which is invariant under the action of the lorentz group . the same analysis leads to @xmath239    the homogeneous space @xmath240 has also two other descriptions . the first one is associated to the polar decomposition . let @xmath241 the isomorphism between the four dimensional minkowski space @xmath222 and the space @xmath242 of @xmath243 hermitian matrices . we denote by @xmath244 the subset of positive - definite hermitian matrices of determinant equal to @xmath245 this map identifies the hyperboloid and @xmath244 : @xmath246 the polar decomposition of the group @xmath198 says that : @xmath247 as a result the polar decomposition gives another understanding of the identification @xmath248 the identification is obtained using the map @xmath249    another decomposition of @xmath198 , the iwasawa decomposition will provide another description of the hyperboloid . the iwasawa decomposition says : @xmath250 where @xmath96 is the group of diagonal positive matrices of determinant 1 and @xmath251 is the nilpotent group of lower trigonal matrices with diagonal elements equal to 1 . the iwasawa decomposition implies that the hyperboloid can also be identified as : @xmath252    expressing the polar decomposition in term of the iwasawa one is straightforward and we get : @xmath253 where @xmath254 is a point on @xmath255    the action of the lorentz group on @xmath256 , is mapped to the natural right action of @xmath236 on the homogeneous space @xmath257 .    using @xmath258 as a level function , we obtain that @xmath230 is a cone over @xmath259 foliated by the spheres of equation @xmath260 note that the radius of the sphere containing the point @xmath261 is equal to @xmath262 these spheres are invariant under the right action of @xmath37 on @xmath263    the measure on the hyperboloid is identified to the measure on @xmath257 invariant under the right action of @xmath236 . as a result if we identify @xmath257 with @xmath264 , the haar measure on @xmath236 factorizes as @xmath265 where @xmath266 is the right invariant integral on @xmath264 . the measure on the hyperboloid is therefore identified with @xmath266 . we will now describe a construction of the quantum algebra @xmath267 and of its algebra of function @xmath268 this is obtained through the use of the quantum duality principle . after having completed this task , we introduce the notion of quantum hyperboloid .    the quantum lorentz algebra @xmath212 and the algebra @xmath269 of functions on this quantum group is recalled briefly in the appendix a.1 . the quantum lorentz algebra is the quantum double of @xmath57 . this construction is in this case the quantum analog of the iwazawa decomposition of the lorentz algebra i.e. @xmath270 where @xmath271 is the real algebra of @xmath243 lower triangular complex matrices with real diagonal of zero trace . a complete survey of this construction is exposed in @xcite . we let @xmath272 and define @xmath273 , to be the hopf algebra generated by the elements @xmath274 , @xmath275 , and the relations :    @xmath276    this algebra can be endowed with a structure of ribbon quasi - triangular hopf algebra . we will only recall the structure of coproduct which reads :    @xmath277 we will denote by @xmath278 or @xmath279 the @xmath278-matrix and by @xmath280 . finite dimensional unitary irreducible representations @xmath281 of @xmath57 are labelled by a spin @xmath282 and let us define @xmath283 the vector space associated to the representation @xmath281 . this representation is of dimension @xmath284 and we denote by @xmath285 the orthonormal basis such that the action of the generators on this basis is given by the following expressions : @xmath286 where we have denoted as usual @xmath287_q=\frac{q^z - q^{-z}}{q - q^{-1}}.$ ] the quantum dimension of this representation is defined by @xmath288_q$ ] where @xmath289 we denote by @xmath290 the dual basis .    through the use of the coproduct , representations can be tensorized . for any unitary representations @xmath291 , we define the clebsch - gordan maps @xmath292 ( resp . @xmath293 ) as intertwiners from @xmath294 to @xmath295 ( resp . intertwiners from @xmath295 to @xmath296 . with a particular choice of normalisation ( see @xcite ) , these elements are unique and their matrix elements are the 3j coefficients of @xmath297    we will denote by @xmath298 the matrix elements of @xmath299 , i.e the linear forms on @xmath300 @xmath301 the algebra of quantum deformations of polynomial functions on @xmath37 is the hopf - algebra generated as a vector space by @xmath298 and denoted @xmath302    by a direct application of the definition , the algebra structure is given by : @xmath303 where we have denoted , as usual , @xmath304 , @xmath305 and @xmath306 . the existence of the @xmath278 matrix implies the braiding relations : @xmath307 because the spin @xmath308 when tensored sufficiently many times contains any prescribed spin , @xmath309 can equivalently be viewed as the hopf algebra generated by the matrix elements of @xmath310 with the defining relation ( [ polsu2 ] ) and the additional relation @xmath311 explicit relations are recalled in the appendix of @xcite for example . a very important notion in the theory of quantum groups is the principle of quantum duality . we will only give the general idea of this principle , leaving aside delicate issues related to the topology of the spaces . let @xmath312 a lie algebra . if @xmath313 is a quantization , of hopf algebra type , of the enveloping algebra of @xmath312 , it endows a structure of lie algebra on the dual of the lie algebra @xmath314 . it is obtained as follows : if @xmath315 , we denote @xmath316 we can define @xmath317 it can be shown that @xmath318 as a result the transpose map @xmath319 maps @xmath320 , is antilinear and satisfies jacobi identity because @xmath321 is coassociative . the principle of `` quantum duality '' expresses the fact that : if @xmath312 is a lie algebra , such that @xmath313 is a quantization of @xmath322 , then @xmath314 admits a quantization , denoted , @xmath323 , and we have @xmath324 as hopf algebras . a typical example is @xmath325 in this case a direct computation , using the structure of @xmath57 shows that @xmath326 therefore we can define , through the use of the quantum duality principle : @xmath327    note that such definition can only hold at the non - commutative level i.e when @xmath6 is non equal to @xmath245 indeed when @xmath328 , @xmath57 is cocommutative , therefore the dual @xmath329 is a commutative algebra , which prevents to be isomorphic to @xmath330 this principle can also be used at the level of function spaces . let @xmath91 be the lie group of lie algebra @xmath312 , let @xmath331 be the lie group of lie algebra @xmath332 we can define @xmath333 . the quantum duality principle implies that we have @xmath334 as hopf algebras .    as a result , in our case , @xmath98 , @xmath335 the quantum duality principle tells us that we can define @xmath336    we will now describe the space @xmath337 , which is a star - hopf algebra . we will also show that it is very natural to define the quantum hyperboloid @xmath338 as being , as in the classical case , the hopf algebra @xmath339 . we have already described the structure of hopf algebra of @xmath302    we can now define spaces of functions of different classes on @xmath264 . the first one is the quantization of the compact supported functions on @xmath340 it is defined as : @xmath341 with the following ( `` multiplier'')-hopf algebra structure : @xmath342 which is the dual of the hopf algebra @xmath302 note that by definition an element @xmath343 in @xmath344 is such that only a finite number of @xmath345 are non zero .    the coproduct maps @xmath344 into the larger space @xmath346 which is the reason why we have to work in the scheme of multiplier hopf algebras . the quantum algebra of functions ( generally unbounded ) on @xmath264 is denoted @xmath347 and defined as @xmath348 it can be endowed with a structure of hopf algebra with the same formulas for the product , the coproduct and the star as ( [ multfunc ] , [ coprodfunc ] , [ starfunc ] ) . it also possess a unit element defined by @xmath349 where @xmath350 is the identity matrix of @xmath351 . the very definition of @xmath352 comes from the existence of a haar measure . @xmath344 admits a right invariant integral @xmath353 defined as follows : @xmath354 which satisfies @xmath355 for any @xmath356    we define @xmath357 , the algebra of compact supported functions on the quantum hyperboloid , as being , as in the classical case , the algebra @xmath358 .    therefore , as an algebra , it has the structure @xmath359    this description is the deformation of the foliation of @xmath360 by quantum fuzzy spheres . quantum fuzzy spheres have been introduced and studied in @xcite . the construction of the quantum hyperboloid that we are proposing has a scaled limit @xmath361 studied recently by @xcite . we will see more on this topic in the sequel . we shall now group together the quantization of @xmath56 and @xmath362 in order to obtain @xmath363 . this is done through the construction of the quantum double exposed in the appendix a.1 . let us now introduce the quantum algebra of functions on @xmath236 , good references on this subject being @xcite . we define the space of compact supported functions on @xmath236 by @xmath364 as algebras . the expression of the coproduct is written in @xcite . there exists a haar measure @xmath365 on @xmath366 which expression is @xmath367 , where @xmath368 denotes the normalized haar measure on @xmath369 . the functional @xmath365 is a right and left integral for @xmath370    the matrix elements of the 2-dimensional left spinorial representation of @xmath212 can be seen as quantum deformations of polynomial functions on @xmath371 we denote by @xmath91 this @xmath243 matrix of non commuting elements . in @xcite the iwasawa decomposition of this matrix has been done , and we have @xmath372 where @xmath373 is a @xmath243 matrix satisfying the defining relations of @xmath374 and @xmath375 is a lower triangular matrix of the form :    @xmath376    with the relations :    @xmath377    note that these relations , which can be seen as the defining relations of @xmath337 are also the defining relations of @xmath57 ( this is the quantum duality principle ) through the identification :    @xmath378    therefore the element @xmath379 is represented in @xmath344 as being the elements @xmath380 with the analog representation for @xmath381 and @xmath382    it is explained in @xcite why the classical limit of the integral @xmath383 on @xmath344 is the continuous integral @xmath384    of particular importance for the understanding of this classical limit is the element @xmath385 note that the classical limit of this element is exactly @xmath386 where @xmath387 is the radius of the sphere . in the quantum case , @xmath388 is in the center of the algebra and therefore is represented by a scalar @xmath389 on each subspace of matrices @xmath351 . its value is @xmath390 the value of the radius of the fuzzy sphere @xmath351 is therefore defined by @xmath391 and explicitely given by @xmath392[i+1].$ ]    the quantum analog of the relation ( [ x0c ] ) is @xmath393 . since @xmath388 is a casimir with positive eigenvalues , the time component @xmath394 is a central element , with discrete spectrum @xmath395 on each components of @xmath351 .    the classical limit , @xmath396 is recovered as follows . a quantum fuzzy sphere indexed by @xmath282 recovers its classical limit when @xmath397 tends to a classical value @xmath398 in that case , this classical sphere is located at time @xmath399 the value @xmath400 is interpreted as the distance on the hyperboloid between any point on this classical sphere and the point located at @xmath401      let us now recall basic notions on the representation theory of the quantum lorentz group . irreducible unitary representations of @xmath212 have been classified . they include the principal series on which the plancherel measure is concentrated . the representations of the principal series are labelled by a couple @xmath402 with @xmath403 with @xmath404-\frac{\pi}{\kappa } , \frac{\pi}{\kappa}[$ ] . we denote @xmath405 the principal representation associated to @xmath406 acting on the vector space @xmath407 as explained in @xcite , @xmath405 is a harisch - chandra representation which decomposes in representations of @xmath57 as : @xmath408 a basis of @xmath409 is given by @xmath410 where , for @xmath282 fixed , @xmath411 is the basis of the @xmath57 module @xmath283 ( [ su2rep ] ) . @xmath409 contains a @xmath57-invariant vector if and only if @xmath412 . these representations are called simple representations . if @xmath406 is simple , we will denote @xmath413 the @xmath57 invariant vector of unit length @xmath414 as a result the matrix element @xmath415 is a linear form on @xmath416 . we will also use the notation @xmath417 . @xmath418 is an element of @xmath419 . because this function is invariant on the left by @xmath57 , the iwasawa decomposition implies that @xmath420 as a result @xmath421 defines a function on the quantum hyperboloid . moreover this function is invariant under the right action of @xmath297 as a consequence it is diagonal : @xmath422 with @xmath423    we can compute exactly its expression . this is done using the work @xcite and the necessary background is recalled in appendix a.1 : we have @xmath424 we finally obtain @xmath425}{[i\rho][2j+1]}1_j.\ ] ] note that @xmath421 satisfies @xmath426    it is an easy check to see that in the classical limit , i.e @xmath427 , @xmath428 the coefficient @xmath429 which is exactly the classical zonal function used in @xcite . the barrett - crane model is defined through the use of a particular intertwiner , which is nowadays called barrett - crane intertwiner ( bc intertwiner ) . its quantum deformation is easy to define and this is the purpose of what is coming next . it will be very important in the sequel to generalize to the quantum deformation the following type of integrals . if @xmath430 is a principal representation of @xmath431 we also denote @xmath430 the associated unitary representation of the group @xmath371 if @xmath432 is a smooth compact supported function on @xmath236 , the following bounded operator can be defined : @xmath433 these type of integrals have a direct generalization to the quantum group case although there is no more notion of point nor of the meaning of @xmath434 let @xmath430 be a principal representation of @xmath416 , we denote @xmath435 the basis of @xmath436 as defined in the appendix a.1 . @xmath437 the dual basis of @xmath438 is a basis of @xmath439 as a result we can define @xmath440 its exact expression in a basis being given in the appendix a.1 . we have shown in @xcite ( eq . 93 ) that if @xmath441 is an element of @xmath442 the following operator is well defined and is moreover of finite rank and corank ( i.e it is a finite dimensional matrix ) : @xmath443 this operator is the quantum analog of @xmath444 more generally if @xmath445 is a family of principal representations of @xmath236 , integrals of the type @xmath446 often appear in the construction of interwiners . they will be generalized as follows : @xmath447    we will have to extend these operators in the case where @xmath432 is the constant function @xmath448 . in particular we will show that when @xmath449 , the operator @xmath450 is well defined . note that this operator , in the case @xmath451 has already been studied in detail in @xcite . the basic definition of the q - deformation of the lorentzian bc intertwiner is similar to the classical one . the q - bc lorentzian intertwiner is a n - valent intertwiner between simple representations of the quantum lorentz group such that any expansion of it into 3-valent intertwiners introduces only simple representations in the intermediate channel . q - bc lorentzian intertwiner . let @xmath452 a sequence of simple representations of the quantum lorentz group . we will denote @xmath453=\bigotimes\limits_{i=1}^{n } \stackrel{\alpha_i}{\mathbb v}$ ] and @xmath454= \bigotimes\limits_{i=1}^{n}\omega(\alpha_i).$ ] the quantum lorentzian intertwiner @xmath455:\mathbb{v}[\alpha ] \rightarrow { \mathbb c}$ ] is defined by : @xmath456 \ ; =   < \omega[\alpha]\vert ( \bigotimes\limits_{i=1}^n\stackrel{\alpha_i}{\pi})(1).\end{aligned}\ ] ] it is represented by the left graph of caption ( [ intertwiner ] ) .        from its formal definition , @xmath455 $ ] , if it exists , is an interwiner . we are therefore led to study the convergence property of the series defining @xmath455.$ ] we note first that , for @xmath457 $ ] , @xmath455(z)=h(f[\alpha](z))$ ] where @xmath458(z)$ ] is an element of @xmath347 because the vector @xmath454 $ ] is invariant by the right action of @xmath297 as a consequence , the q - bc lorentzian intertwiner is defined by an integral on the quantum hyperboloid whose convergence will be analyzed in the sequel .    from the expression of the matrix element @xmath405 ( see the appendix a.1 ) , it is straigthforward to see that : @xmath459 as a result we obtain :    let @xmath460 a family of simple representations , let @xmath461 a family of spins and @xmath462 the magnetic numbers . we denote by @xmath463 = \bigotimes_{i=1}^n   \stackrel{a_i}{e}_{a_i}\!\!(\alpha_i)$ ] and we have : @xmath464\;\vert \ ; \stackrel{a}{e}_{a}[\alpha ] > \ ; = \ ; \sum_m [ d_m ] \ ; c(m , a , a ) \ ; \prod_{i=1}^n \lambda_{0a_i}^{mm}(\alpha_i),\end{aligned}\ ] ]    where @xmath465 the series ( [ sum ] ) converges when @xmath449 and in this case the q - bc lorentzian intertwiner ( [ qli ] ) is well defined . + the clebsch - gordan coefficients are bounded by one , as a result we get the bound : @xmath466(2m+1)^n.\ ] ] from the asymptotics computed in @xcite , we have the following bound when @xmath52 goes to infinity : @xmath467 as a consequence , @xmath468 \ ; c(m , a , a ) \ ; \prod_{p=1}^n \lambda_{0a_p}^{mm}(\alpha_p )    = { \cal o}(m^{2n } q^{2m(n-2)})$ ] which implies that the series ( [ sum ] ) converges absolutely when @xmath469 @xmath470    we will now prove that the four valent q - bc intertwiner satisfies the property that its expansion in three valent intertwiners only includes simple representations in the intermediate channel :    [ decompositionoftheqbcintertwiner ] the four valent q - bc lorentzian intertwiner @xmath455 $ ] is decomposed in three valent @xmath212 intertwiners as follows : @xmath471 \ ; = \ ; \int d\beta p(\beta)\ ; f(\alpha,\beta ) \ ; \psi_{\beta,\beta}^{\;\;0 } ( \psi_{\alpha_1,\alpha_2}^{\;\;\beta } \otimes \psi_{\alpha_3,\alpha_4}^{\;\;\beta } ) \end{aligned}\ ] ] where the measure @xmath472 is the plancherel measure and the kernel @xmath473 has its support on the set of simple representations . its explicit expression is given in the following proof . + the proof of the decomposition ( [ decomposition ] ) is a direct application of the relations ( [ orthogonalityrelation],[schurlemma ] ) given in the appendix . indeed , one first applies the completeness relation ( [ orthogonalityrelation ] ) to the q - bc intertwiner : @xmath474 \ ; = \ ; < \omega[\alpha ] \vert   \int d\beta   d\gamma \ ; m(\alpha_1,\alpha_2,\beta ) m(\alpha_3,\alpha_4,\gamma )   \phi^{\alpha_1   \alpha_2}_{\;\ ; \beta } \otimes \phi^{\alpha_3   \alpha_4}_{\;\ ; \gamma }   ( \stackrel{\beta}{\pi } \otimes \stackrel{\gamma}{\pi})(1 )   \psi^{\;\ ; \beta}_{\alpha_1 \alpha_2 } \otimes \psi^{\;\ ; \gamma}_{\alpha_3 \alpha_4}\;\;.\end{aligned}\ ] ] the use of the orthogonality relation ( [ schurlemma ] ) implies : @xmath474 \ ; = \ ; < \omega[\alpha ] \vert \int d\beta \ ; \frac{1}{p(\beta)}m(\alpha_1,\alpha_2,\beta ) m(\alpha_3,\alpha_4,\beta )   \phi^{\alpha_1   \alpha_2}_{\;\ ; \beta } \otimes \phi^{\alpha_3   \alpha_4}_{\;\ ; \beta } \phi^{\beta,\beta}_{\;\;0 }   \ ; \psi_{\beta,\beta}^{\;\;0 } \ ; \psi^{\;\ ; \beta}_{\alpha_1 \alpha_2 } \otimes \psi^{\;\ ; \beta}_{\alpha_3 \alpha_4}\;\;.\end{aligned}\ ] ]    moreover the clebsh - gordan coefficient @xmath475 with @xmath476 is different from zero if and only if @xmath155 is a non zero @xmath57-invariant vector . such a vector exists if @xmath477 is a simple representation and by definition the normalization of @xmath478 is chosen such that @xmath479 as a consequence , the q - bc intertwiner can be decomposed as ( [ decomposition ] ) where @xmath480 @xmath470    note that we have proven the property of decomposition of the q - bc lorentzian intertwiner in a particular channel . it is straightforward to extend this proof to the two other channels , as in the classical case @xcite . a very important property satisfied by the q - bc intertwiner is the invariance under braiding , which can be formulated as follows . let @xmath481 be an n - uplet of principal representations and let @xmath482 be an element of the braid group on @xmath483 strands . to @xmath482 is associated a unique permutation on @xmath483 points , denoted @xmath484 we associate , as usual , to @xmath482 a linear operator @xmath485 \rightarrow \mathbb { v}[\sigma(b)(\alpha)]$ ] by representing the braid by a product of @xmath486 matrices of @xmath416 . note that this is well defined because the @xmath486 matrix of @xmath487 evaluated in the tensor product of principal representation is well defined , which is a non trivial fact ( see @xcite ) . the property of invariance under braiding states that : @xmath488{\mathbb r}(b)=\iota[\alpha].\ ] ]    this means that in the picture ( [ braiding ] ) representing the barrett - crane intertwiner , the lines are lines and not ribbons and the black dot is a completely symmetric pivot . it is sufficient to show this property for an elementary braid , i.e of length one . let @xmath489 be the transposition of the first two variables . @xmath490 p_{12}r_{12 } & =   & < \omega[\tau(\alpha ) ]   \vert ( \bigotimes_{i=\tau(1)}^{\tau(n ) } \stackrel{\alpha_i}{\pi } ) ( \delta^{(n)}(x^a ) ) p_{12}{\mathbb r}_{12 } h(x_a)\\ \label{line1 } & = & < p_{12}\omega[\tau(\alpha)]\vert ( \bigotimes_{i=1}^n \stackrel{\alpha_i}{\pi})(\tau ( \delta^{(n)})(x^a)){\mathbb r}_{12 }   h(x_a ) \label{line2 } \\ & = &   < { \mathbb r}_{12}\omega[\alpha ] \vert ( \bigotimes_{i=1}^n \stackrel{\alpha_i}{\pi } ) ( \delta_{}^{(n)}(x^a ) )   h(x_a ) \label{line3 } \\ & = & \iota[\alpha]\;\;.\end{aligned}\ ] ] we have successively used the definition of the q - bc lorentzian intertwiner , the quasitriangular relation @xmath491 and the invariance @xmath492 shown in the appendix ( [ ronomega1 ] ) . @xmath470    remark : this property can also be shown using the decomposition in three valent interwiners . the fact that the action of the @xmath486 matrix on three valent intertwiner amounts to multiply it by the ratio of ribbon elements and the fact that the ribbon element is trivial in a simple representation is sufficient to conclude . finally , we can generalized the q - bc lorentzian intertwiner as follows :    the generalized q - bc lorentzian intertwiner    let @xmath452 and @xmath493 two sequences of simple representations of the quantum lorentz group . following ( [ qli ] ) , we define the generalized q - bc intertwiner @xmath494 \ ; : \ ; \mathbb{v}[\alpha ] \rightarrow    \mathbb{v}[\beta]$ ] by : @xmath495 \ ; = \sum_{a , b }   ( \bigotimes_{i=1}^p \stackrel{\beta_i}{\pi})(\delta^{(p)}(s(x^{b } ) ) ) \vert \omega[\beta ] >   < \omega[\alpha ] \vert   ( \bigotimes_{j=1}^n\stackrel{\alpha_j}{\pi})(\delta^{(n)}(x^{a}))\ ; h ( x_{b}x_{a})\;\;.\end{aligned}\ ] ]    it is represented by the right graph of caption ( [ intertwiner ] ) and satisfies the following properties :    1 . @xmath496 $ ] is an intertwiner and is well defined in the following sense : the series @xmath497 \vert \stackrel{a}{e}_a(\alpha)>$ ] converges . any expansion of @xmath494 $ ] in three valent intertwiners introduces only simple representations in intermediate channels . 3 .   @xmath494 $ ] is invariant under braiding . @xmath498{\mathbb r}(b)=\iota[\beta;\alpha]= { \mathbb r}(b')^{-1}\iota[\sigma(b')(\beta);\alpha].\ ] ]     + this proposition is proved along the same lines as the previous propositions . @xmath470    the graph of the right handside of caption as one vertex @xmath155 and @xmath483 edges entering , colored by @xmath499 $ ] , and @xmath500 colored edges going out , colored by @xmath501 $ ] . we will denote this graph by @xmath502 $ ] . it will be useful to define the vertex function @xmath503,\mathbb{v}[\beta])$ ] by : @xmath504 > < \omega[\alpha ] \vert   ( \bigotimes_{j=1}^n\stackrel{\alpha_j}{\pi})(\delta^{(n)}(x^{a}))\ ; h ( x_{b}x_{a})\end{aligned}\ ] ] as a consequence , the quantum lorentzian intertwiner is recovered from the vertex function as @xmath494 \ ; = \ ; h(f_v)$ ] . in this section we will define what is meant by evaluation of a closed simple spin network for the quantum lorentz group . rather than studying the property of convergence in general we will concentrate on the evaluation of particular simple closed graph which are the basic ingredients for the construction of a spin foam amplitude . in particular we will give a formula for the weight associated to a 4-simplex , the so - called quantum @xmath190-symbol .    of course , as in the classical case , problems of divergences due to the non - compactness of the quantum lorentz group appear as soon as one tries to evaluate naively a closed spin network . this difficulty is handled by using a regularization similar to the classical case . moreover , quantum groups introduce problems of non trivial braidings in the definition of spin networks which have to be handled as well . evaluation of simple closed spin network has been studied in @xcite in the case of @xmath1 . in the case of @xmath505 it has been studied in @xcite . later on a study of simple closed spin network in the case where the group is the lorentz group has been done in @xcite . the property of integrability of simple spin network has been unraveled in @xcite . evaluation of simple spin networks , in the case of the quantum lorentz group , contains therefore the two difficulties :    -the network is a graph embedded in three space and its projection on a plane contains possible crossings    - the integrability , in the sense of baez and barett , has to be studied baring in mind that in the quantum case property of convergences are usually enhanced . the barrett - crane intertwiner being formulated as an integral on the hyperboloid , the evaluation of a simple spin network is given by a multiple integral on a product of hyperboloid . let @xmath506 where @xmath150 and @xmath507 are two points in the hyperboloid @xmath508 , @xmath509 the distance between them and @xmath200 the simple representation coloring the edge linking @xmath150 to @xmath507 . a simple spin network is an unoriented graph @xmath510 where the edges are colored by simple representations . we denote @xmath511 the set of vertices of @xmath510 and @xmath512 the set of edges of @xmath510 . if @xmath35 is an edge it defines a pair of vertices @xmath513    the evaluation of a simple spin network @xmath510 is given by the integral : @xmath514    however because of the non - compactness of the lorentz group , such an integral is divergent . to regularize it , one chooses any vertex @xmath515 , and define @xmath516 as being the integral @xmath517    it can be shown @xcite that this integral , if it converges , does not depend on @xmath518 and does not depend on the chosen @xmath519    quantum lorentzian spin networks are defined in an analogous way . from an algebraic point of view our construction is the same as yetter s one @xcite because the quantum lorentz algebra @xmath5 is twist equivalent , as a hopf algebra , to @xmath520 the only , but essential , difference is contained in the star structure and in the category of representation which is considered . a simple lorentzian q - spin network is a graph @xmath510 embedded in @xmath521 colored by simple representations @xmath499 $ ] of @xmath5 . the operation of evaluation of this network is defined in the sequel and when it is finite is an invariant under isotopy . we first define the evaluation formally , along the lines of yetter . we first project the graph @xmath510 on a plane keeping track of the under - over crossings . we then apply the method of reshetikhin - turaev @xcite to compute the evaluation , where the coupons are q - bc intertwiner and the ribbons are colored by simple representations . note that the structure of ribbons is invisible because the ribbon element on simple representations is equal to @xmath245    we will consider only graphs which are at least trivalent for each vertex , this implies that the q - bc intertwiner is well defined for each vertex . unfortunately the definition of reshetikhin - turaev invariant uses traces over the vector spaces associated to the representations , therefore as such it is ill - defined . the method of regularization of this evaluation is similar to the classical one . we associate to each vertex @xmath155 the vertex function @xmath522 we tensor them and pair the spaces ``  la reshetikhin - turaev '' while keeping track of the braiding by introducing @xmath486 matrices . the obtained result is an element @xmath523 where @xmath483 is the the number of vertices . the formal evaluation of @xmath510 would be the integral @xmath524 , while the regularization consists in picking an arbitrary vertex and integrating @xmath525 over the quantum hyperboloid associated to the remaining vertices . the invariance of the integral under @xmath5 implies first that @xmath526 and secondly that this @xmath527 is independant of the choice of the removed vertex . when @xmath527 is finite @xmath510 is said to be integrable . we have used the following convention for evaluating simple q - spin networks . we select a projection of the graph in such a way that all the vertices are aligned and we enumerate them from the left to the right . we first define the quantum analog of @xmath528 we will define it as an element of @xmath529 which is @xmath530 we will use the convenient abuse of notation of viewing this as a non commutative function of two variables @xmath531    using the sweedler notation , @xmath532 we obtain that @xmath533    this definition is a direct consequence of the definition of the q - bc interwiner and of the relation @xmath534    using the definition of the coproduct ( [ coprodfunc ] ) , we obtain :    @xmath535    this function satisfies the following convolution property : @xmath536 where @xmath537 is the plancherel measure and @xmath538 is the integral over the quantum hyperboloid . this property is a direct consequence of ( [ schurlemma ] ) .    first , let us consider the simple network @xmath539 $ ] with two vertices linked by a set of @xmath500 edges colored by @xmath499=(\alpha_1,\cdots,\alpha_p)$ ] .    . ]    it is straightforward to see that the associated graph function , represented by the graph ( [ gamma2 ] ) , is given by : @xmath540 the evaluation of this spin - network , defines the so - called @xmath541 functions , and are given by : @xmath542)=\theta^q_p(\rho_1, ... ,\rho_p)= h(\prod_{j=1}^p \stackrel{\alpha_i}{k } ) = \sum_m [ d_m]_q^2 \prod_{i=1}^p \lambda^{mm}_{\;\;0}(\alpha_i)=\\   & & = \sum_{n=1}^{+\infty}\frac{1}{[n]_q^{p-2}}\prod_{j=1}^p   \frac{\sin(2n\nu_j)}{\sin(2\nu_j)}=g^q_p(\nu_1, ... ,\nu_p),\label{gqp}\end{aligned}\ ] ] where we have denoted @xmath543    we will need properties of these series and will concentrate on a precise study of these series when @xmath544 and @xmath545 these series have appeared in other contexts .    when @xmath328 , @xmath546 is , up to a constant , the value found by witten @xcite of the volume of the moduli space @xmath547 of flat @xmath37 connections on the sphere with @xmath500 punctures associated to conjugacy classes @xmath548 , with @xmath549 where @xmath550    if we fix the variables @xmath551 and study the behaviour @xmath552 goes to @xmath553 , which amounts to recover the ordinary lorentzian barrett - crane model , these series are equivalent , up to a normalization of @xmath552 , to the integrals @xmath554 considered by barrett and crane .    when @xmath544 , @xmath555 has appeared naturally in the work @xcite , and it is precisely the definition of the coefficient @xmath556    the functions @xmath546 have also been introduced by one of us and a.szenes in @xcite , as the value of cyclic traces on non commutative deformations of @xmath547 . when @xmath557 , it is trivial to show that these series are uniformally convergent for @xmath558 , whereas it is a distribution in the case where @xmath559 and is equal to @xmath560    as a result , when @xmath558 and @xmath557 the functions @xmath561 are continuous functions ( even smooth functions ) of @xmath562    the functions @xmath563 and @xmath564 can be expressed in term of jacobi @xmath565 functions as shown in the appendix a.2 .    in particular for @xmath566 we obtain that the evaluation of this simple spin network is equal to : @xmath567    in particular we can recover the classical behaviour , when @xmath427 , computed in @xcite . from the modular properties of @xmath568 we have for any real variable @xmath150 : @xmath569 as a result we get @xmath570 where @xmath105 is a function of @xmath552 . the evaluation of @xmath571 is crucial for the computation of the weight of the spin foam on edges . we can compute it exactly ( see the appendix a.2 ) and we have the @xmath572analogue of the formula of barrett - crane : @xmath573    when @xmath328 , @xmath546 , being volume of a symplectic manifold , is non negative . however it can be equal to zero , which arizes when the manifold is empty . let us define the function @xmath574 , which is the characteristic function of the set @xmath321 defined by @xmath575 and for @xmath576 , denote by @xmath577 the unique real number satisfying @xmath578    when @xmath544 we have @xmath579    the behaviour of @xmath580 is different , because it is always strictly positive . indeed , in the appendix we have shown that @xmath581 with @xmath582    we will need a similar result for @xmath583 we will use the following formula , which is trivial from the definition of @xmath584 as series : @xmath585    ( note that when q=1 , this equation follows from the fact that the moduli space of the sphere with 4 punctures is obtained by gluing the moduli spaces of two three punctured spheres . ) . as a result , using the lower bound for @xmath586 , we obtain @xmath587 this has the far reaching consequence that the function @xmath571 is bounded from below by @xmath588 this behaviour is very different from the case where @xmath589 . in the riemannian case , @xmath590 are positive half - integers , and @xmath591 can vanish if the selection rules on @xmath592 are not satisfied . in the lorentzian barrett - crane model , @xmath590 are any positive number , and the infimum of @xmath591 is @xmath553 , though it is never reached .      we will now give an expression , in term of multiple series , of the evaluation of a colored 4-simplex ( the 10j symbol ) . important properties will show up on this expression :    -it is finite    -it is a continuous function of @xmath593    -it does not depend on the choice of crossing . let us consider the graph embedded in @xmath594 , whose vertices and edges are those of the @xmath90 simplex colored by a sequence @xmath595 of simple representations . when one projects this graph on a hyperplane there is a choice of crossing which has to be made . we will denote @xmath596 the associated simple spin network . the following computation is done with the choice of upper crossing which is depicted in the figure ( [ 10jsymbol ] ) .     spin network . ]    with the convention of the picture , it is straightforward to compute the graph function : @xmath597 where we have abusively denoted @xmath598    using the properties @xmath599 and @xmath600 this function can be written as : @xmath601 we evaluate this spin network by applying @xmath602 we obtain : @xmath603    using the invariance on the left and on the right of @xmath365 and the braiding relation , we obtain @xmath604 where @xmath605    we eliminate the @xmath486 matrix using the property ( [ ronomega2 ] ) , and the expression reduces to : @xmath606    it remains to evaluate the four integrals . we denote by @xmath607 the basis of @xmath608 which satisfies @xmath609    as a result , in this basis , ( [ coprodfunc ] ) is expressed as : @xmath610    and the action of the antipode @xmath611 is given by :    @xmath612    where @xmath613 is the matrix defined as :    @xmath614^{1/2}e^{-i\pi a}\clebpsi{a}{a}{0}{m}{n}{0}.\ ] ]    it is then straighforward to perform the integrations of ( [ evaluationof4simplex ] ) , and we get :    @xmath615_q[d_{m_5}]_q[d_{m_9}]_q[d_{m_{10}}]_q x(\{m_i\})\ ] ]    where the function @xmath616 is the evaluation of the @xmath57 spin network drawn in picture ( [ graph ] ) .    . ]    we now evaluate this spin network . each box is replaced by the corresponding @xmath617 intertwiner multiplied by a number . for example the box on the upper left is replaced by @xmath618 where @xmath619_q^{1/2}}{[d_{m_4}]_q^{1/2 } } \sixj{m_9}{m_8}{m_4}{m_3}{m_{10}}{m_2 } .\ ] ] as a result we obtain : @xmath620_q[d_{m_3}]_q^{1/2}[d_{m_6}]_q^{1/2}[d_{m_7}]_q^{1/2}[d_{m_8}]_q^{1/2 } } { [ d_{m_4}]_q^{1/2}[d_{m_5}]_q[d_{m_9}]_q^{1/2}}\sixj{m_3}{m_7}{m_6}{m_1}{m_2}{m_8 } \sixj{m_4}{m_7}{m_5}{m_6}{m_{10}}{m_3}\nonumber\\ & & \sixj{m_{10}}{m_6}{m_5}{m_1}{m_9}{m_{2 } } \sixj{m_4}{m_8}{m_9}{m_1}{m_5}{m_7 }   \sixj{m_9}{m_8}{m_4}{m_3}{m_{10}}{m_2}.\end{aligned}\ ] ]    using racah - wigner symmetry , we finally obtain the nice formula : @xmath621_q^{1/2 } \lambda^{m_i m_i}_{\;\;00 } ( \alpha_i ) \right ) \sixj{m_3}{m_7}{m_6}{m_1}{m_2}{m_8 } \sixj{m_4}{m_7}{m_5}{m_6}{m_{10}}{m_3 } \nonumber \\ & & \sixj{m_2}{m_6}{m_1}{m_5}{m_9}{m_{10 } } \sixj{m_4}{m_8}{m_9}{m_1}{m_5}{m_7 } \sixj{m_9}{m_8}{m_4}{m_3}{m_{10}}{m_2}\label{4simplexevaluated}\;\;. \end{aligned}\ ] ]    this expression is a multiple series which converges absolutely . indeed , each @xmath622-symbol is bounded by one and when @xmath52 goes to infinity , @xmath623_q^{1/2 } \lambda^{m m}_{\;\;00 } ( \alpha ) = { \cal o}(mq^m)$ ] . as a result the graph @xmath624 is integrable . because the convergence is uniform in @xmath551 , @xmath625 is a continuous function of the variables @xmath551 . note a very important property satisfied by @xmath625 : it is a function of @xmath6 symmetric in the exchange @xmath626 this is a direct consequence of the invariance of the @xmath42 symbols under the exchange @xmath627    the evaluation of the graph @xmath628 proceeds along the same lines , and its evaluation gives the same answer ( [ 4simplexevaluated ] ) . this is compatible with the invariance @xmath627 with all the tools introduced in the previous section , we can now built cosmological lorentzian spin foam models . let @xmath34 be a triangulation of the lorentzian manifold @xmath52 , we define by @xmath99 the set of @xmath100simplices of the triangulation . the partition function of the spin foam model is defined as usual : @xmath629}\prod_{f\in \delta_2 } d\rho_f \prod_{f\in \delta_2 } a_2 ( \rho_f )   \prod_{t\in \delta_3 } a_3(\rho_1^t, ... ,\rho_4^t )   \prod_{s\in \delta_4}a_4(\rho_1^s, ... ,\rho_{10}^s ) , \ ] ] where @xmath630 is the value of the @xmath190 symbol of the @xmath631 simplex @xmath117 . note that the value of @xmath632 and @xmath633 is still a matter of debate . see @xcite for a clear survey of the situation . in this article three choices of @xmath205 and @xmath206 have been analyzed . we give their straighforward q - deformation in the sequel :    1)@xmath634    2)@xmath635    3)@xmath636    where @xmath637 is the plancherel measure . it is therefore straighforward to show that the integral @xmath638 is finite for the three previous cases . the integration is on a compact set , the function @xmath639 is continuous , and because of the lower bound on @xmath640 , the function @xmath206 is also continuous for the three cases . as a result @xmath638 is a convergent integral for these three cases and more generally in the case where @xmath208 are continuous functions of the @xmath214 s . in our work we have defined cosmological deformation of lorentzian spin foam models . the only analysis that we have made of this model is its finiteness property for a fixed triangulation . presently there are lots of work on the study of lorentzian spin foam models and its relations to quantum gravity . central questions are the `` derivation '' of these models from the lagrangian field theory , the restriction to the gravitational sector , the analysis of the asymptotic properties of the weight of a simplex , the relation to the hamiltonian constraint , the possible non - perturbative definitions of the sum over the 2-complexes , the importance of the degenerate metrics , the nature of the spectrum of observables ( continuous or discrete ? ) . they are all related to the following three questions : how fondamental is the barret - crane model ? what are the observables and their spectrum ? how do we do a semi - classical study of spin - foam models ? the same questions can be formulated in the context of cosmological deformation of spin foam models as well . even more pressing questions are present : how do we precisely connect the plebanski lagrangian field theory with a cosmological constant to cosmological deformation of the barrett - crane model ? how do we compute @xmath205 and @xmath206 ? how do we do a semi - classical expansion around de sitter space ? what precisely are the states of quantum lorentzian gravity with a positive cosmological constant ? how do we couple these models to matter and extract information about physical quantities ?    the fate of the spinfoam model approach depends crucially on the possibility of answering to these questions . there are of course extensions of our work which can be readily done . the first one is the inclusion of faces of timelike type . the technology of quantum groups that we have developped in the past should give an easy answer to this extension .    the spinfoam model that we have defined should also appear as the perturbative expansion of a field theory on a non commutative analogue of @xmath641 . another well studied question nowadays , in the classical case , is the asymptotic properties of the @xmath190 symbol . a similar question can be asked in the cosmological deformation case . its formulation is as follows : we fix @xmath642 , define @xmath551 by @xmath643 , and we would like to understand the behaviour of @xmath644 in the limit where @xmath45 goes to 0 . it is straightforward to see that @xmath645_q^{1/2}\frac{\sin((2m_j+1)\omega_j ) } { \sin(\omega_j)}r_q(m_1, ... ,m_{10})\ ] ]    where @xmath646 is the product of the five @xmath647 symbols and @xmath648    the asymptotic properties of the simplex is therefore the classical expansion of this series around @xmath328 . the form of this series is very reminiscent of the series that we have studied in @xcite , for which we have developped tools ( residue formulas ) for computing the asymptotic of such series . we hope to apply analog of these methods to analyze the behaviour of the @xmath90-simplex . we thank l.crane and c.rovelli for stimulating and enthousiastic discussions on spin foam models . we also thank e.buffenoir for interesting discussions . the lie algebra @xmath88 is also the real lie algebra of @xmath649 therefore we will use indifferently the notation @xmath5 or @xmath212 for its quantization . we define the quantum lorentz group @xmath5 as being the quantum double of @xmath57 . this construction is a standard and very important construction in the theory of quantum groups . it is exposed in detail in @xcite . the quantum double of @xmath57 is the hopf algebra @xmath650 . @xmath651 denotes the hopf algebra @xmath309 with permuted coproduct . the notation @xmath652 reminds the reader that inside @xmath653 , @xmath654 and @xmath655 do not commute but satisfy braided identities . in @xcite we have classified and given simple expressions for the unitary irreducible representations of @xmath89 we will only need the representation of the principal series which we denote @xmath405 where @xmath656 matrix elements of @xmath405 are linear forms on @xmath416 and therefore are elements of @xmath657    using the decomposition @xmath658 we have expressed in @xcite the matrix elements of @xmath405 in terms of a basis of @xmath659 we will use the following convenient basis which elements are @xmath660 . we will denote this basis by @xmath435 . the dual @xmath661 is @xmath416 ( more precisely contains @xmath662 as a hopf sub - algebra ) . one of its basis is the dual basis @xmath437 which is the basis @xmath663 with the duality bracket given by : @xmath664    we have given explicit formulas for the representation @xmath405 in this basis . namely : @xmath665 where @xmath666 are coefficients defined in terms of analytic continuation of @xmath42 symbols of @xmath57 and whose properties are studied in depth in @xcite . an explicit expression is given in equation ( 89 ) of @xcite . as a result we get the expression : @xmath667        the @xmath486 matrix of @xmath672 is given by the construction of the quantum double , its evaluation on principal representations has been computed in @xcite : @xmath673 in particular this expression implies that : @xmath674      in @xcite we have made a study of intertwining operators for the tensor product of principal representations . we have shown that if @xmath676 label principal representations , the space of intertwiners @xmath677 is one dimensional when @xmath678 and zero otherwise . we have chosen a particular normalized intertwiner , denoted @xmath679 , which satisfies the following completeness relation : @xmath680 with @xmath681 a normalization factor given by an infinite product expression and the integral @xmath682 means , as usual , an integral over @xmath683 and a sum over the integer @xmath684 .      note that when @xmath676 are simple the expression for @xmath52 reduces to a simple ratio of theta functions , as explained in @xcite and recalled in the appendix a.2 of the present work . as explained in the sequel , this function @xmath52 is exactly the evaluation of the so - called `` theta '' graph @xmath688                      to express @xmath707 in term of theta functions , one can use the following trick which will be useful later on . we first expand @xmath708 , and we interchange the sum in @xmath707 by summing first over @xmath709 we therefore obtain that @xmath710          this enables us to show that @xmath555 is strictly positive and that we have the lower bound : @xmath717 to prove this lower bound we use the infinite product formula for @xmath568 and @xmath718 . as a result we obtain the inequalities , when @xmath150 is real , : @xmath719 as a result we obtain that @xmath720 where @xmath721
we study the quantum deformation of the barrett - crane lorentzian spin foam model which is conjectured to be the discretization of lorentzian plebanski model with positive cosmological constant and includes therefore as a particular sector quantum gravity in de - sitter space . 
 this spin foam model is constructed using harmonic analysis on the quantum lorentz group . 
 the evaluation of simple spin networks are shown to be non commutative integrals over the quantum hyperboloid defined as a pile of fuzzy spheres . 
 we show that the introduction of the cosmological constant , removes all the infrared divergences : for any fixed triangulation , the integration over the area variables is finite for a large class of normalization of the amplitude of the edges and of the faces .