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during last decade the integrable structures associated with conformal field theory ( cft ) and its @xmath0 extension were studied @xcite-@xcite . the main problem to which these papers were addressed is the simultaneous diagonalization of the corresponding infinite family of the integrals of motion ( i m ) . the method used for this purpose is the continuous version of the quantum inverse scattering method ( qism ) @xcite . the most successful results were obtained by means of the generalization of the baxter s @xmath1-operator relation @xcite . in this paper we will follow this way constructing the @xmath1-operator for analysis of the integrable structure arising in the superconformal field theory ( scft ) . this cft includes both usual left and right components of the energy - momentum tensor @xmath2 , @xmath3 and spin-@xmath4 local fields @xmath5 , @xmath6 where variable @xmath7 lies on a cylinder of circumference @xmath8 . the full symmetry algebra of the theory is @xmath9 ( @xmath10 means superconformal algebra).we will concentrate here on the left - chiral component- sca . the boundary conditions are : @xmath11 where `` + '' corresponds to ramond ( r ) sector and ``  '' to neveu - schwarz ( ns ) one . the @xmath10 gives possibility to construct two infinite abelian families of conserved charges @xcite,@xcite,@xcite where one is susy invariant and the other is not . here we will consider the integrable structure invariant under the supersymmetry transformation . in the classical limit the corresponding i m give the hamiltonians of the susy @xmath12=1 kdv hierarchy @xcite based on twisted affine lie superalgebra @xmath13 @xcite . the supertrace of the monodromy matrix of the associated l - operator in the 3-dimensional representation of @xmath14 ( see appendix 1 ) is a generating function for the i m . thus it s quantum generalization , the transfer matrix produces the i m of the scft . we will review some results from part i @xcite concerning the construction of the quantum counterpart of the monodromy matrix and it s basic properties in sec . 2 . using the super q - oscillator representations of the upper borel subalgebra of quantum affine superalgebra @xmath15 we define the @xmath16 operators . the transfer matrices in different evaluation representations can be expressed in such a way : @xmath17 where @xmath18 runs over integer and half - integer nonnegative numbers . @xmath16 operators satisfy quantum super - wronskian relation : @xmath19 one should note , that we use only @xmath20-dimensional `` @xmath21-induced '' representations ( sometimes called atypical ) of @xmath14 . it allows , however , to construct the fusion relations , see below . to construct the baxter - like relation we introduce additional `` quarter''-operators , constructed `` by hands '' from the @xmath1-operators : @xmath22 for odd integer @xmath23 . the baxter s relations are : @xmath24 @xmath25 the fusion relations have the following form very similar to the @xmath26 case @xcite : @xmath27 but these relations are only `` fusion - like '' because the `` quarter''-operators do not seem to correspond to any representation of @xmath14 . this will be the subject of sec . 3 . finally in sec . 4 we describe the truncation of these relations for different values of @xmath28 , being root of unity . in the part i @xcite we have explicitly constructed the quantum counterpart of the monodromy matrix of the linear problem for the l - operator of the susy n=1 kdv equation . this l - operator has the following explicit form : @xmath29 where @xmath30 , @xmath31 , @xmath32 are the chevalley generators of @xmath14 ( see appendix 1 ) , @xmath33 is a superderivative , the variable @xmath7 lies on a cylinder of circumference @xmath8 , @xmath34 is a grassmann variable , @xmath35 is a bosonic superfield . the poisson brackets for the field @xmath36 , obtained by means of the drinfeld - sokolov reduction are : @xmath37 and the following boundary conditions are imposed on the components of @xmath36 : @xmath38 , @xmath39 . making a gauge transformation of the auxiliary linear problem we obtain a new superfield @xmath40 , where @xmath41 and @xmath42 generate the superconformal algebra under the poisson brackets : @xmath43 the corresponding infinite series of the integrals of motion @xcite : @xmath44 allows us to construct integrable evolution equations . for example @xmath45 yields the susy n=1 kdv equation @xcite : @xmath46 which has the following form in components : @xmath47 these integrals of motion ( [ i m ] ) can be extracted from the monodromy matrix of the l - operator ( [ lop])@xcite : @xmath48 \big).\nonumber\end{aligned}\ ] ] it s quantum generalization can be represented in the quantum p - exponent form ( for the explanation of this notion see below ) : @xmath49 vertex operators @xmath50 have the following form @xmath51 , where : @xmath52=\frac{i}{2}\beta^2 , \quad   [ a_n , a_m]=\frac{\beta^2}{2}n\delta_{n+m,0},\qquad \{\xi_n,\xi_m\}=\beta^2\delta_{n+m,0}.\nonumber\\ & & : e^{\pm\phi(u)}:= \exp\big(\pm\sum_{n=1}^{\infty}\frac{a_{-n}}{n}e^{inu}\big ) \exp\big(\pm i(q+pu)\big)\exp\big(\mp\sum_{n=1}^{\infty}\frac{a_{n}}{n}e^{-inu } \big)\nonumber.\end{aligned}\ ] ] @xmath50 satisfy the following properties : @xmath53 these relations gave us possibility to show that the reduced universal r - matrix of the quantum affine superalgebra @xmath54 ( see appendix 1 ) with lower borel subalgebra represented by operators @xmath55 have the p - exponent multiplication property @xcite . thus we have called it the quantum p - exponent : @xmath56 the universal r - matrix represented by @xmath55 coincide with @xmath57 . note , that the monodromy matrix is : @xmath58 unlike the cases of quantum ( super)algebra @xmath59 , @xmath60 , @xmath61 , @xmath62 @xcite-@xcite , @xcite this object ca nt be represented in the usual p - ordered form due to the singularities arising from the products of fermion free fields . however as in all these cases @xmath63 which we will denote in the following as @xmath64 to simplify the notation satisfy the rtt - relation @xcite , @xcite : @xmath65 where @xmath18 , @xmath66 means that the corresponding object is considered in some representation of @xmath54 . thus the supertraces of monodromy matrix ( `` transfer matrices '' ) @xmath67 commute . it is very useful to consider the evaluation representations of @xmath54 , @xmath68 , where now symbol @xmath18 means integer and half - integer numbers ( see appendix 1 ) . denoting @xmath69 as @xmath70 we find that @xmath71 commute : @xmath72=0\end{aligned}\ ] ] the expansion of @xmath73 in @xmath74 ( the transfer matrix in the fundamental 3-dimensional representation ) is believed to give us as coefficients the local i m , the quantum counterparts of ( [ i m ] ) : @xmath75 where the densities @xmath76 are differential polynomials of @xmath2 and @xmath5 such that @xmath77 terms with derivatives . @xmath2 and @xmath5 have the following representation in terms of free fields ( [ freefields ] ) ( see @xcite ) : @xmath78 one of the basic features of the i m is that their commutators with certain vertex operators reduce to total derivatives @xcite : @xmath79=\partial_u(:o^{(k)}_{\pm}(u )   w_{\pm}(u):)=\partial_u\theta^{(k)}_{\pm}(u)\end{aligned}\ ] ] ( where @xmath80 is the polynomial of @xmath81 , @xmath82 and their derivatives ) . by means of this important property one can show ( see @xcite ) that such integrals commute with the transfer matrices : @xmath83=0.\end{aligned}\ ] ] in our case it is also important to show that the generator of the supersymmetry transformation @xmath84 commutes with @xmath85 . really , @xmath86=\pm i^{1/2}\partial_u : e^{\pm\phi(u)}:=\partial_u\xi^{\pm}(u ) . \end{aligned}\ ] ] thus , @xmath87 can be included in the family of i m and using again the results of @xcite it is easy to show that @xmath88=0 .   \end{aligned}\ ] ] let s consider evaluation representations @xmath68 of the quantum algebra @xmath54 which we have already mentioned in the introduction and in previous section . they are @xmath20 dimensional where @xmath18 runs over integer and half - integer numbers . we call these representations `` @xmath89-induced '' because each triple @xmath90 and @xmath91 form @xmath89 superalgebra @xcite , and @xmath20-dimensional representation space is a representation space for each @xmath89 subalgebra irreducible representation ( see appendix 1 ) . + we will also use in the following the corresponding reducible infinite - dimensional representations @xmath92 , which look like @xmath89 verma representations for the @xmath90 triple . to simplify the notation we denote @xmath93 as @xmath94 and @xmath95 as @xmath96 . + in order to construct the @xmath1-operator we introduce the auxiliary object , the super q - oscillator algebra @xcite with the following commutation relations : @xmath97=\pm\varepsilon_{\pm } , \qquad q^{\frac{1}{2}}\varepsilon_{+ } \varepsilon_{-}+q^{-\frac{1}{2}}\varepsilon_{-}\varepsilon_{+}=\frac{1}{q - q^{-1}}.\ ] ] using these relations it is possible to build realization for the upper borel subalgebra @xmath98 generated by @xmath99 : @xmath100 and we denote @xmath101 as @xmath102 . next , following @xcite we define two operators which do not depend on representation of oscillator algebra but depend on ( and completely determined by ) the commutation relations and cyclic property of the supertrace : @xmath103 where @xmath104 multiplying two @xmath105 operators we arrive to such a result ( see appendix 2 ) : @xmath106 where @xmath107 is the supertrace of the monodromy matrix in @xmath92 representation . defining then the @xmath1-operators : @xmath108 we find that the expression ( [ aa ] ) can be written in such a form : @xmath109 now let s recall that @xmath107 corresponds to infinite - dimensional representation @xmath110 . in order to obtain the transfer matrix corresponding to the finite - dimensional representation @xmath111 one should consider factor - representation + @xmath110/@xmath112 ( see appendix 1 ) . thus one obtains : @xmath113 sign + in this formula appears because of taking a supertrace . really , @xmath114 subrepresentation in @xmath110 representation space has a highest weight vector which is odd ( though our convention is that highest weight vector in @xmath115 representation is even ( see appendix 1 ) ) . that s why we have sign + instead of the usual . + rewriting ( [ ttt ] ) in terms of the @xmath16 operators we find : @xmath116 considering the case of 1-dimensional representation of @xmath54 ( @xmath117 ) we obtain a quantum super - wronskian ( qsw ) relation : @xmath118 in order to construct the baxter type relation @xcite we introduce the auxiliary `` quarter '' operators : @xmath119 for odd integer @xmath23 . let s consider the @xmath120 operator and multiply it on @xmath121 operators . using ( [ sw ] ) one can get the following relation of baxter type , which is similar to that one obtained in @xmath26 case @xcite : @xmath122 the main difference is that unlike the @xmath26 case we are not seeking for the eigenvalues of @xmath120 operator but we are interested in the operator corresponding to the 3-dimensional representation , @xmath123 , because we know that it is a generating function of the local i m @xcite . thus we are looking for the relation of baxter type which includes this operator . + in order to do this we need to proceed through some additional steps . first , let s unify the expressions ( [ tq ] ) , ( [ quarter ] ) : @xmath124 where @xmath125 , @xmath126 . using the qsw relation it is not hard to obtain the expressions for @xmath85 operators only in terms of @xmath127 or @xmath128 : @xmath129 for example for @xmath123 we obtain the `` triple relation '' : @xmath130 due to the unifying relation ( [ utq ] ) and using again ( [ sw ] ) we find the fusion relations : @xmath131 however , as we already mentioned in the introduction these relations are different from fusion relations obtained for ( super)algebras @xmath26 , @xmath132 , @xmath133 , @xmath134 ( see @xcite,@xcite,@xcite,@xcite ) in the sense that in all these cases operators present in the fusion relation correspond to the representations of the associated algebras . in our case we see that artificially constructed auxiliary quarter operators do not correspond to any representation of @xmath54 . + let s write the fusion relations for @xmath135 with @xmath74 multiplied on @xmath136 : @xmath137 multiplying both sides on @xmath121 we obtain second relation of baxter type : @xmath138 though the baxter @xmath1-operator method is very powerful there exists another approach which is useful for rational values of the central charge @xcite , it is the scft counterpart of baxter s commuting transfer matrix method . in this section we will show that when q is the root of unity the system of fusion relations becomes finite . so , let s put @xmath139 , @xmath140 , @xmath141 . then @xmath142 here @xmath125 , @xmath143 . note also that @xmath144 with the use of such a transform one obtains : @xmath145 recalling the formula ( [ tq ] ) for @xmath146 we arrive to the following expression : @xmath147 for example for @xmath117 we obtain : @xmath148 using the basic formula ( [ tq ] ) and property ( [ mon ] ) @xmath149 and for @xmath150 one arrives to the following relations : @xmath151 @xmath152 comparing ( [ n/2 ] ) and ( [ n/2 - 1/2 ] ) we obtain the following simple formula : @xmath153 in the case when @xmath154 , where @xmath155 , @xmath156 there exist additional number of truncations : @xmath157 the relation ( [ trunc ] ) allows us to to rewrite the fusion relation system in the thermodynamic bethe ansatz equations @xcite of @xmath158 type as in @xmath26 case @xcite : @xmath159 with the use of identification : @xmath160 where we have denoted @xmath161 the eigenvalue of @xmath85 . in this paper we studied algebraic relations arising from integrable structure of cft provided by the susy @xmath12=1 kdv hierarchy . as it was already mentioned in @xcite , @xcite , @xcite the construction of the @xmath1-operator as a `` transfer''-matrix corresponding to the infinite - dimensional q - oscillator representation could be also applied to the lattice models . the relations like baxter s and fusion ones will be also valid in the lattice case because they depend only on the decomposition properties of the representations . + the use of supersymmetry in the considered model is that the limit of @xmath162 when @xmath163 is very simple due to the expansion @xcite : @xmath164 where @xmath165 , @xmath166 for @xmath167 and @xmath168 are the integrals of motion of the susy @xmath12=1 kdv . thus @xmath169 . we hope that such a nice result remains valid in the quantum case . maybe this conjecture will give us possibility to simplify the study of the analytic properties of the eigenvalues of @xmath170 , @xmath1 operators and therefore the corresponding integral equations of destri - de vega type @xcite , @xcite . we will consider these questions in the third part of our manuscript @xcite . it will be interesting to consider the obtained tba equations ( [ tba ] ) in the context of the perturbed minimal models as it was done in @xcite . + in the following we also plan to study the quantization of @xmath171 susy kdv hierarchies , related with super - w conformal / topological integrable field theories . we are grateful to f.a . smirnov for useful discussions . the work was supported by the dynasty foundation ( a.m.z . ) and rfbr grant 03 - 01 - 00593 ( p.p.k . ) . & & \lambda^{-1}e_{\alpha_1}|s , l\rangle = c[s - l]^{+}[s+l+1/2 ]    & & \lambda e_{-\alpha_1}|s , l\rangle=    & & \lambda e_{-\alpha_1}|s , l\rangle=-    & & \lambda^{-1}e_{\alpha_0}|s , l\rangle=    & & \lambda^{-1}e_{\alpha_0}|s , l\rangle=    & & \lambda e_{-\alpha_0}|s , l\rangle = c[s - l][s+l+1/2]^{+ }    & & \lambda e_{-\alpha_0}|s , l\rangle =- c[s - l]^{+}[s+l+1/2 ]    where @xmath207=\frac{q^x - q^{-x}}{q - q^{-1}}$ ] , @xmath207^{+}=\frac{q^x+q^{-x}}{q+q^{-1}}$ ] , @xmath208^{+})^{-1}$ ] . + we also note ( this will be important in taking the supertrace ) that we choose the highest weight @xmath209 to be even . + the representations @xmath210 of @xmath54 , which play also a crucial role in the construction of the @xmath1-operators ( see sec . 3 ) are infinite - dimensional and their explicit form is similar to that of ( [ repr ] ) but with @xmath211 going from @xmath18 to @xmath212 . from a point of view of the @xmath89 subalgebra , generated by the triple @xmath90 these representation look like verma module for @xmath89 with the highest weight @xmath213 one can easily obtain that @xmath111 representations arise from the factor of two infinite - dimensional representations of @xmath214-type : @xmath110/@xmath114 . the product of two @xmath105-operators ( [ aa ] ) can be rewritten in the following form : @xmath215 where @xmath216 and @xmath217 . + by the fundamental property of the universal r - matrix : @xmath218 to rewrite the tensor product of two @xmath64-operators from the previous formula in such a way : @xmath219 where @xmath220 @xmath221 , @xmath222 and @xmath223 satisfy the following commutation relations : @xmath224=\mp\alpha_{\pm } , \quad[\mathcal{h},\beta_{\pm}]=\mp\beta_{\pm},\\ & & q^{1/2}\alpha_-\alpha_++q^{-1/2}\alpha_+\alpha_-=-\frac{\mu^2}{q - q^{-1 } } , \quad q^{1/2}\beta_+\beta_-+q^{-1/2}\beta_-\beta_+=\frac{\mu^{-2}}{q - q^{-1}}\nonumber\end{aligned}\ ] ] let s choose as @xmath225 ( [ chi ] ) the highest weight representations of the q - oscillator algebra ( @xmath226 ) with highest weight vectors @xmath227 such that : @xmath228 , where @xmath229 is the space spanned by the following vectors : @xmath230 takes such values when @xmath231 are linearly independent . it is easy to see how @xmath223 and @xmath232 act on these vectors : @xmath233 the action of @xmath234 is more complicated : @xmath235 where @xmath236[s+l+1/2]^{+},\quad   s - l\in\mathbb{z}\\ & & a(l , s)=c[s - l]^{+}[s+l+1/2 ] , \quad s - l\in\mathbb{z}+1/2\nonumber\end{aligned}\ ] ] and @xmath237 , the coefficients @xmath238 can be calculated but we do nt need them in the following . the elements @xmath239 and @xmath223 acting in the space spanned by @xmath231 for any constant @xmath240 coincide up to a constant shift in @xmath223 and shift in the `` @xmath241-direction '' of @xmath239 action coincides with action of the corresponding chevalley generators in the infinite dimensional @xmath210 evaluation representations ( [ repr ] ) . when we take the trace in ( [ aal ] ) there is no contribution of shift in `` @xmath241-direction '' and the constant shift in @xmath223 action can be easily calculated , that is : @xmath242 now recalling the definition of @xmath243 we find that in chosen highest weight representation @xmath244 therefore , @xmath245    20 v.v . bazhanov , s.l . lukyanov , a.b . zamolodchikov , comm . 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the algebraic structures related with integrable structure of superconformal field theory ( scft ) are introduced . 
 the scft counterparts of baxter s q - operator are constructed . 
 the fusion - like relations for the transfer - matrices in different representations and their truncations are obtained . 
 superconformal field theory , super - kdv , quantum superalgebras , q - operator 11.25.hf ; 11.30.pb ; 02.20.uw ; 02.20.tw