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lattice gauge field theory brings the representation theory of an underlying manifold and its quantum invariants into the same setting . consider the case of a cylinder over a compact , oriented surface with boundary . a lattice model of the surface determines an algebra of gauge invariant fields ( i.e. observables ) . in the classical case , based on a connected , simply connected lie group @xmath0 , observables are the characters of the fundamental group of the lattice represented in @xmath0 . wilson loops can be understood as traces of conjugacy classes in the fundamental group of the lattice . for the theory based on a drinfeld - jimbo deformation of a simple lie algebra @xmath1 , the observables are a deformation quantization of the @xmath0-characters of the surface with respect to the standard poisson structure @xcite . in the case of @xmath2 this , together with the classical isomorphism @xcite , allowed us to prove that the algebra of observables is the kauffman bracket skein algebra of the surface , completed as an algebra over formal power series . in this paper we return to an analytic setting in which the deformation parameter is any complex number other than a root of unity . the analogous theorem relating observables and the kauffman bracket skein algebra is again true . the proof is based on the combinatorial equivalence between the temperley - lieb algebra and the quantized invariant theory of @xmath3 ; it does not explicitly use the relationship with surface characters . the paper is organized as follows . section [ kauffman ] recalls definitions and associated formulas of the kauffman bracket skein algebra . section [ reps ] summarizes , for the generic parameter , the construction of a quasi - triangular matrix model of quantum @xmath3 . section [ lgft ] outlines basic definitions and constructions of lattice gauge field theory . section [ temperley ] describes the correspondence between skeins and intertwiners in the verlinde algebra for quantum @xmath3 . finally , section [ observable ] contains a proof of the main theorem . let @xmath4 be a complex number that is neither @xmath5 nor a root of unity . suppose that @xmath6 is a compact , oriented surface with boundary and @xmath7 is a closed interval . denote by @xmath8 the set of isotopy classes of framed links in @xmath9 , including the empty link . let @xmath10 be the vector space with basis @xmath8 . define @xmath11 to be the subspace of @xmath10 spanned by all expressions of the form @xmath12 and @xmath13 , where the framed links in each expression are identical outside the balls pictured in the diagrams . the kauffman bracket skein algebra @xmath14 is the quotient @xmath15 . multiplication is given by laying one link over the other . more precisely , if @xmath16 and @xmath17 are in @xmath8 , isotop them so that @xmath16 lies in @xmath18 , and @xmath17 in @xmath19 $ ] . then @xmath20 is the union of these two links in @xmath21 $ ] . extend linearly to a product on @xmath10 . since @xmath11 is an ideal , the product descends , making the skein module into a skein algebra . since the algebra structure depends on the specific product structure of @xmath9 , rather than its topological type , we use the notation @xmath22 . we use the standard convention of modeling a skein in @xmath22 on a framed , admissibly colored , trivalent graph . an admissible coloring is an assignment of a nonnegative integer to each edge so that the colors at each vertex form admissible triples . a triple @xmath23 is admissible if @xmath24 , @xmath25 , @xmath26 and @xmath27 is even . the corresponding skein in @xmath22 is obtained by replacing each edge labeled with the letter @xmath28 by the @xmath28-th jones  wenzl idempotent ( see @xcite , or @xcite ) , and replacing trivalent vertices with kauffman triads ( see ( * ? ? ? * fig .  14.7 ) ) . if @xmath29 is a trivalent spine of @xmath6 then the set of skeins carried by admissible colorings of @xmath29 forms a basis @xmath30 for @xmath22 . if @xmath6 is an annulus , @xmath30 consists of skeins obtained by labeling the core with a jones  wenzl idempotent . one may think of the core as a `` trivalent '' spine with one vertex , whose admissible labels are @xmath31 . the space @xmath22 also has a basis @xmath32 consisting of all links with simple diagrams on @xmath6 , i.e.  with no crossings and no trivial components . the details of the following can be found in @xcite . let @xmath33 be the unital hopf algebra on @xmath34 , @xmath35 , @xmath36 , @xmath37 , with relations : @xmath38 let @xmath39 denote the irreducible @xmath40-dimensional representation of @xmath33 . fixing an ordered basis for @xmath39 we define linear functionals @xmath41 to be the coefficient in the @xmath42-th row and @xmath43-th column in the representation @xmath39 . the @xmath44 form a basis for the stable subalgebra @xmath45 of the hopf algebra dual @xmath46 . ( here @xmath47 . ) define @xmath48 and give it the product topology . a typical element of @xmath49 is a sequence of arbitrarily chosen matrices in which the @xmath42-th term is an @xmath50 matrix . let @xmath51 be the homomorphism corresponding to the representation @xmath39 . the homomorphism @xmath52 given by @xmath53 , is injective and its image is dense in @xmath49 ( see @xcite ) . the algebra @xmath49 is the completion of @xmath33 by equivalence classes of cauchy sequences in the weak topology from @xmath45 . it has the structure of a topological ribbon hopf algebra . the projection of @xmath49 onto its @xmath40-st factor is an irreducible representation of @xmath49 , also denoted by @xmath39 . composing @xmath44 with this projection yields a function on @xmath49 , also called @xmath44 . thus @xmath45 is understood to lie in @xmath54 . in this section we recall basic definitions and constructions of lattice gauge field theory . details can be found in @xcite . let @xmath55 be an oriented , ciliated graph ; i.e.  the edges are oriented and the vertices carry a linear ordering of the adjacent edges . one can think of this as instructions for building a strip and disk model of an oriented surface @xmath6 having @xmath55 as a strong deformation retract . vertices correspond to the disks , edges to strips , and the ciliation determines how to glue the strips to the disks . the surface @xmath6 is called the envelope of @xmath55 . define a space of connections : @xmath56 and define an algebra of fields : @xmath57 = \bigotimes_{\text{edges $ e$ } }   ( { { _ qsl_2}})_e.\ ] ]    note that fields are functions on connections in the obvious way . the connections form a coalgebra with comultiplication as defined in @xcite . multiplication of fields is the convolution product dual to comultiplication of connections . there is an action of the gauge algebra , @xmath58 on the space of connections , and adjointly on fields . the invariant part of the gauge fields under this action is called the observables , @xmath59 . the multiplication of fields restricts to make @xmath59 into an algebra , which is a deformation of the @xmath60-characters of @xmath61 . let @xmath62 be a representation of the hopf algebra @xmath49 , that is @xmath62 is a finite dimensional left module over @xmath49 . the dual vector space to @xmath62 carries two distinct @xmath49-module structures . when @xmath49 acts on the left , the dual module is denoted by @xmath63 . the action is : @xmath64 for any @xmath65 , @xmath66 and @xmath67 in the dual vector space to @xmath62 . when @xmath49 acts on the right , denote the dual by @xmath68 with @xmath69    there is an alternate description of observables in terms of `` colorings '' of the lattice . assigning a representation @xmath70 to each positively oriented edge @xmath71 of the lattice @xmath55 determines a map @xmath72 this yields , at each vertex , a tensor product of representations coming from the edges adjoining that vertex taken in the order given by the ciliation . use the representation @xmath70 for the edges @xmath71 starting at a vertex and the dual @xmath73 for the edges terminating there . the resulting representation at a vertex @xmath74 is denoted by @xmath75 . finally , choose @xmath76 for each vertex @xmath74 . the element @xmath77 is evaluated on a connection @xmath78 by mapping @xmath78 to @xmath79 and then re - parsing to an element of @xmath80 . by ( * corollary 1 ) every observable is a linear combination of observables of this form . now assume that @xmath55 is a trivalent lattice colored by irreducible representations . the coloring is admissible if , at each vertex , the integers corresponding to the colorings of the incident edges form an admissible triple .    [ basis ] suppose that @xmath55 is an admissibly colored trivalent lattice . for each @xmath75 there exists a non - zero dual element invariant under the right action of @xmath49 . the tensor product of these invariants over all vertices defines an observable . the set of such observables , one for each admissible coloring , is a basis for @xmath81 . let @xmath82 be an admissible coloring . note that admissibility implies a @xmath83-dimensional invariant subspace in each @xmath84 . hence there is a non - zero observable @xmath85 . since @xmath86 is nonzero , there exists @xmath87 so that @xmath88 . let @xmath89 be the `` parsing '' map . let @xmath90 be the map @xmath91 where the inclusion is given by forming a sequence that has all zero matrices except for the @xmath40-st entry corresponding to @xmath39 . define @xmath92 to be the connection @xmath93 . clearly @xmath94 . since @xmath95 unless @xmath96 , @xmath92 is annihilated by all observables constructed from colorings different than @xmath97 .    from this we conclude that any set of observables constructed from distinct colorings is independent . by @xcite they span . there is a map @xmath98 that assigns to each framed link a wilson operator . classically , a wilson operator is the trace of the holonomy of a connection along some fixed loop . the construction in the quantum setting is described in @xcite . in a theory based on the drinfeld - jimbo @xmath2 and its hopf algebra dual , the observables are isomorphic to the kauffman bracket skein algebra of the lattice envelope . since @xmath2 is a complete topological algebra over @xmath99 $ ] , it was necessary to complete the skein algebra as well . the isomorphism was inferred from the classical isomorphism , the agreement of poisson brackets , and the @xmath100-adic completion . in section [ observable ] we will show directly that the map @xmath101 is an isomorphism . in this section we recall the correspondence between skeins and intertwiners in the category of representations @xmath39 of @xmath49 . consider a rectangle @xmath102 with @xmath103 distinguished points ; @xmath104 of them on @xmath105 and @xmath104 on @xmath106 . take the space of blackboard framed tangles with @xmath104 arc components ending at the distinguished points . its quotient by the kauffman bracket skein relations is denoted by @xmath107 . this quotient has an algebra structure given by placing the bottom of a rectangle on the top of another in such a way that the distinguished points meet . [ rect ] the algebra @xmath107 is isomorphic to @xmath108 , the space of @xmath49-linear maps of @xmath109 to itself .    in the basis @xmath110 of @xmath111 , the isomorphism is given by a tangle functor which makes the following assignments . a local maximum is sent to the morphism @xmath112 defined by @xmath113 @xmath114 a local minimum is associated to the morphism @xmath115 given by @xmath116    [ joneswenzl ] the isomorphism takes the @xmath104-th jones - wenzl idempotent to the intertwiner that projects @xmath109 onto its highest weight invariant subspace . let @xmath117 be the closed upper half plane . for any @xmath104 , choose @xmath103 distinguished points on the @xmath118-axis , @xmath119 , and form a space of blackboard - framed tangles with @xmath104 arc components ending at the distinguished points . the quotient of this space by the kauffman bracket skein relations is denoted @xmath120 . @xmath121    the isomorphism is given by the same tangle functor as for fact [ rect ] . an admissible triple @xmath122 determines a skein in @xmath123 consisting of a kauffman triad with all three legs attached to the @xmath118-axis . fact [ joneswenzl ] gives a canonical inclusion of @xmath124 into @xmath125 . [ triad ] an admissible triple @xmath122  equivalently , a kauffman triad  corresponds to a nonzero vector in the @xmath83-dimensional space @xmath126 . our goal is to prove a theorem analogous to ( * ? ? ? * theorem 10 ) , but replacing power series by complex numbers .    [ iso ] let @xmath55 be a lattice and let @xmath6 be its envelope . assume that @xmath127 is not a root of unity . the algebra of observables of lattice gauge field theory on @xmath55 based on @xmath128 is isomorphic to @xmath129 . the following description of @xmath101 is implicit in @xcite . since @xmath59 is a homeomorphism invariant of @xmath6 , we can assume that the lattice @xmath55 comes from giving orientation and ciliation to a trivalent spine @xmath131 of @xmath6 . let @xmath132 be an element of the basis @xmath32 of @xmath22 ( i.e. , a link with a simple diagram ) . choose an orientation of @xmath132 . to compute the image of @xmath132 under @xmath101 , first perform the composition : @xmath133 here @xmath134 is the number of strands of @xmath132 running along the edge @xmath71 of @xmath55 , and @xmath135 means comultiplying @xmath136 times in the factor corresponding to @xmath71 . third , treat each ciliated vertex as a half plane with the cilium at infinity . up to isotopy , the link @xmath132 now appears as a collection of oriented caps in each half plane . the cap pictured on the left of figure [ caps ] is associated with the map @xmath140 where @xmath141 , and @xmath142 is given by @xmath143 for the cap on the right . notice that @xmath147 sends the switch map of @xcite to the map @xmath148 and sends multiplications to contractions . hence our description of @xmath144 for @xmath149 coincides with the wilson operator . it follows from @xcite that @xmath101 does not depend on the choice of orientation of @xmath132 . extend it linearly to all of @xmath22 . it is a homomorphism of algebras @xmath22 and @xmath59 . in order to prove that @xmath101 is an isomorphism we factor it into two maps which are isomorphisms on the level of vector spaces . the first map , expressed in the basis @xmath32 , is given by a diagonal matrix with @xmath83 s and @xmath150 s on the diagonal . the second map , @xmath151 does not require a choice of orientation of a link @xmath132 . to compute the image of @xmath149 under @xmath152 first perform the composition ( [ phi ] ) . second , apply the map @xmath153 to each copy of @xmath154 . third , treat each vertex as a half plane and associate the map @xmath155 from equation ( [ max ] ) to the ( unoriented ) caps . by fact [ triad ] , the map @xmath152 takes an element of the basis @xmath30 ( i.e. , a skein obtained by an admissible coloring of @xmath131 ) to an observable coming from coloring the edges of @xmath55 with corresponding irreducible representations of @xmath49 . thus , by proposition [ basis ] , the map @xmath152 takes the basis @xmath30 of @xmath22 to a basis of @xmath59 .
we prove that the kauffman bracket skein algebra of a cylinder over a surface with boundary , defined over complex numbers , is isomorphic to the observables of an appropriate lattice gauge field theory .