Yang-Baxter R-operators for osp superalgebras
aa r X i v : . [ m a t h - ph ] S e p Yang-Baxter R -operatorsfor osp superalgebras A.P. Isaev a,b,d , D. Karakhanyan a,c , R. Kirschner e a Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia b Physics Faculty, M.V. Lomonosov State University, Moscow c Yerevan Physics Institute, 2 Alikhanyan br., 0036 Yerevan, Armenia d St.Petersburg Department of Steklov Mathematical Institute of RAS,Fontanka 27, 191023 St. Petersburg, Russia e Institut f¨ur Theoretische Physik, Universit¨at Leipzig,PF 100 920, D-04009 Leipzig, Germany
Abstract
We study Yang-Baxter equations with orthosymplectic supersymmetry. We extenda new approach of the construction of the spinor and metaplectic ˆ R -operators withorthogonal and symplectic symmetries to the supersymmetric case of orthosymplecticsymmetry. In this approach the orthosymplectic ˆ R -operator is given by the ratio oftwo operator valued Euler Gamma-functions. We illustrate this approach by calcu-lating such ˆ R operators in explicit form for special cases of the osp ( n | m ) algebra, inparticular for a few low-rank cases. We also propose a novel, simpler and more elegant,derivation of the Shankar-Witten type formula for the osp invariant ˆ R -operator anddemonstrate the equivalence of the previous approach to the new one in the generalcase of the ˆ R -operator invariant under the action of the osp ( n | m ) algebra. The similarities between the orthosymplectic supergroups
OSp ( N | M ) (here M = 2 m isan even number) and their orthogonal SO ( N ) and symplectic Sp ( M ) bosonic subgroupscan be traced back to the existence of invariant metrics in the (super)spaces V ( N | M ) , V N and V M of their defining representations. These similarities lead to the consideration ofthe supergroup OSp and its superalgebra osp in full analogy with the unified treatment(see e.g. [1]) of the groups SO , Sp and their Lie algebras. Moreover these similaritiesare inherited in the study of solutions of the Yang-Baxter equations that possess suchsymmetries.In the present paper, we continue our study [2] of the solutions of the Yang-Baxterequations symmetric with respect to ortho-symplectic groups. We start with the gradedRLL-relations with the R -matrix in the defining representation R ∈ End( V ( N | M ) ⊗ V ( N | M ) )and find the L -operator, L ( u ) ∈ End( V ( N | M ) ) ⊗ A , where A is a super-oscillator algebrainvariant under the action of the OSp ( N | M ) group. Then this L operator allows one e-mail:[email protected] e-mail: [email protected] e-mail:[email protected] RLL relations) to define a richer and more complicated family ofsolutions of the Yang-Baxter equations, namely the ˆ R -operators, which take values in inthe tensor product A ⊗ A and are expressed as an expansion over the invariants in
A ⊗ A .The orthogonal and symplectic groups are embedded in the ortho-symplectic super-group
OSp , and the ˆ R -operators invariant under the so ( N ) and sp ( M ) algebras can be obtainedfrom the OSp -invariant ˆ R -operator as special cases. In the orthogonal case the algebra A is the N -dimensional Clifford algebra and the operator ˆ R is called the spinor R -matrix. Inthe symplectic case the algebra A is the oscillator algebra and ˆ R is called the metaplectic R -operator.The standard approach to the problem of finding the spinor ( so -invariant) ˆ R -operatorwas developed in [3], [4] and is based on the expansion of the ˆ R -operator over the invariants I k realized in the spaces A ⊗ A . Here the factors A are the Clifford algebras with thegenerators ( c a ) αβ , where α , β and a are respectively spinor and vector indices. Then theinvariants I k are given by the contraction of the antisymmetrized products of c ( a . . . c a k )1 ∈A ⊗ I and c ( b . . . c b k )2 ∈ I ⊗ A with the invariant metrics ε a i b i . In that approach we obtainthe spinor ˆ R -operator as a sum over invariants I k with the coefficients r k which obeyrecurrence relations. Analogous formulae of the Shankar-Witten (SW) type for the ˆ R -operators were deduced for the symplectic case in [1] and then were generalized for theortho-symplectic case in [2]. Note that we cannot consider these expressions for the ˆ R -operators as quite satisfactory, since they do not provide closed formulas for the consideredˆ R -operators. For example, in the symplectic and ortho-symplectic cases, the sum over I k is infinite.On the other hand, it is known that an analogous ˆ R -operator invariant under the sℓ (2)algebra can be represented (see [5], [6]) in a compact form of the ratio of two operator-valued Euler Gamma-functions. Surprisingly, as it was shown in a recent paper [7], the so and sp invariant ˆ R -operators (for special Clifford and oscillator representations of so and sp ) are also represented in the Faddeev-Tarasov-Takhtajan (FTT) form of the ratioof two operator-valued Euler Gamma-functions.In the present paper, we generalize the results of [7] to the supersymmetric case andshow that the osp invariant ˆ R -operator can also be represented in the FTT form. Thisis the main result of our paper. The natural conjecture is that the osp -invariant SWtype ˆ R -operator given as a sum over invariants I k is equal to the osp -invariant FTT typeˆ R -operator given by the ratio of two Gamma-functions. This conjecture is based on thefact that both ˆ R -operators are solutions of the same system of finite-difference equationswhich arise from the RLL relations.A complete proof of this conjecture is still missing. In the present paper we proposeanother simpler and more elegant derivation of the SW type formula for the osp invariantˆ R -operator. This new derivation supports the conjecture of the equivalence of the SWand FTT expressions for the ˆ R -operators. Indeed, in the previous derivation, the roleof invariant, ”colorless”, elements in A ⊗ A is played by the operators I k . In the newderivation, we prove that the invariants I k are polynomials of one invariant I ∼ z onlyand rewrite the RLL relation itself into a ”colorless” form from the very beginning interms of a system of finite-difference equations in the variable z .We relate this new system of equations to both the SW and the FTT expressions forthe ˆ R -operator. On one hand, the FTT type ˆ R -operator is its solution. On the otherhand we show that the expansion of the SW type ˆ R -operator over I k satisfies this systemof finite-difference equations as well. 2he paper is organized as follows. In Section 2, we recall some basic facts of thelinear algebra on the superspace V ( N | M ) with N bosonic and M fermionic coordinatesand briefly formulate the theory of supergroups OSp ( N | M ) and their Lie superalgebras osp ( N | M ). In this section we fix our notation and conventions. In Section 3, we definethe osp -invariant solution of the Yang-Baxter equation as an image of a special elementof the Brauer algebra in the tensor representation in super-spaces V ⊗ r ( N | M ) . Section 4 isdevoted to the formulation of the graded RLL relations. In this Section, we find a special L -operator that solves the RLL relations in the case of the osp algebra and introduce(see also [2]) the notion of the linear evaluation of the Yangian Y ( osp ). In Section 5we define the super-oscillator algebra A and describe the super-oscillator representationfor the linear evaluation of the Yangian Y ( osp ). In particular, we define the set of Osp invariant operators I k in A ⊗ A and their generating function.In terms of these invariant operators we construct in Section 6 the osp invariant ˆ R -operators in the super-oscillator representation. We find two forms for such ˆ R -operator.One of these forms represents the ˆ R -operator as a ratio of Euler Gamma-functions. Forthe sℓ (2) case this type solution was first obtained in [5] (see also [6]) and we call thesesolutions the FTT type ˆ R -operators. Another form of the osp invariant ˆ R -operator inthe super- oscillator representation generalizes the SW solution [3] of the spinor-spinor so -invariant ˆ R -operator. This solution (see eqs. (6.6) and (6.8)) for the osp -invariantˆ R -matrix in the super- oscillator representation was first obtained in our paper [2] byusing the methods developed in [4], [8] and [1]. In [2] we have generalized formulas for the so -type R -matrices (in the Clifford algebra representation) obtained in [3], [8] (see also[9],[10],[11],[12],[1]). In [2] we have also generalized the formulae for sp -type R -matrices (inthe oscillator, or metaplectic, representation of the Lie algebra sp ), which were deducedin [1]. It has been shown in [2] that all these so - and sp -invariant R -matrices are obtainedfrom (6.6), (6.8) by restriction to the corresponding bosonic Lie subalgebras of osp .In Section 7 the result for the FTT type R operator is studied in detail in particularcases of osp ( N | M ). The arguments of the Gamma-functions involve the invariant oper-ator z ∼ I which decomposes into a bosonic and a fermionic part. The finite spectraldecomposition of the fermionic part is considered and used to decompose the R operatorwith respect to the correspoding projection operators.In Section 8 we present the new and more direct derivation of the solutions (6.6) and(6.8). Two Appendices are devoted to the proofs of the statements made in the main bodyof the paper. Consider (see, e.g., [13], [2]) a superspace V ( N | M ) with graded coordinates z a ( a = 1 , . . . , N + M ). The grading grad( z a ) of the coordinate z a will be denoted as [ a ] = 0 , z a is even then [ a ] = 0 (mod2), and if the coordinate z a is odd then[ a ] = 1 (mod2). It means that the coordinates z a and w b of two supervectors z, w ∈ V ( N | M ) commute as follows z a w b = ( − [ a ][ b ] w b z a . (2.1)Let the superspace V ( N | M ) be endowed with a bilinear form( z · w ) ≡ ε ab z a w b = z a w a = z b w a ¯ ε ab , ( z · w ) = ǫ ( w · z ) , (2.2)3hich is symmetric for ǫ = +1 and skewsymmetric for ǫ = −
1. In eq. (2.2) we define w a ≡ ε ab w b , where, in accordance with the last relation in (2.2), the super-metric ε ab andinverse super-metric ¯ ε ab have the properties ε ab ¯ ε bd = ¯ ε db ε ba = δ da , ε ab = ǫ ( − [ a ][ b ] ε ba ⇔ ¯ ε ab = ǫ ( − [ a ][ b ] ¯ ε ba . (2.3)We stress that the super-metric ε ab is an even matrix in the sense that ε ab = 0 iff [ a ]+ [ b ] =0 (mod2): ε ab = ( − [ a ]+[ b ] ε ab . (2.4)In other words the supermatrix ε ab is block-diagonal and its non-diagonal blocks vanish.Using (2.4), the properties (2.3) can be written as ε ab = ǫ ( − [ a ] ε ba = ǫ ( − [ b ] ε ba , ¯ ε ab = ǫ ( − [ a ] ¯ ε ba = ǫ ( − [ b ] ¯ ε ba . (2.5)Further, we use the following agreement on raising and lowering indices for super-tensorcomponents z ...c d...a = ε ab z ...c b d... , z a...c d... = ¯ ε ab z ...cbd... . (2.6)According to this rule, we have ε ab = ¯ ε ac ¯ ε bd ε cd = ¯ ε ba and the metric tensor with the upperindices ε ab does not coincide with the inverse matrix ¯ ε ab . Further, we use only the inversematrix ¯ ε ab and never the metric tensor ε ab .Consider a linear transformation in V ( N | M ) z a → z ′ a = U ab z b , (2.7)which preserves the grading of the coordinates grad( z ′ a ) = grad( z a ). For the elements U ab of the supermatrix U from (2.7) we have grad( U ab ) = [ a ] + [ b ]. The ortho-symplectic group OSp is defined as the set of supermatrices U which preserve the bilinear form (2.2) underthe transformations (2.7)( − [ c ]([ b ]+[ d ]) ε ab U ac U bd = ε cd ⇒ ( − [ c ]([ b ]+[ d ]) U ac U bd ¯ ε cd = ¯ ε ab . (2.8)Now we write the relations (2.8) in the coordinate-free form as ε h U ( − ) U ( − ) = ε h ⇔ U ( − ) U ( − ) ¯ ε i = ¯ ε i , (2.9)where the concise matrix notation is used¯ ε i ∈ V ( N | M ) ⊗ V ( N | M ) , U = U ⊗ I , U = I ⊗ U , (( − ) ) a a b b = ( − [ a ][ a ] δ a b δ a b , ( − ) ∈ End( V ( N | M ) ⊗ V ( N | M ) ) . (2.10)Here ⊗ denotes the graded tensor product:( I ⊗ B )( A ⊗ I ) = ( − [ A ] [ B ] ( A ⊗ B ) , ( A ⊗ I )( I ⊗ B ) = ( A ⊗ B ) , and [ A ] := grad( A ), [ B ] := grad( B ). We remark that in our paper we use the conventionin which gradation is carried by the coordinates, while there is another convention inwhich gradation is carried by the basis vectors (see e.g. [14]). The relation of these twoformulations is explained in [2].Consider the elements U ∈ OSp which are close to unity I : U = I + A + . . . . Heredots denote the terms which are much smaller than A . In this case, the defining relations42.8) give conditions for the supermatrices A which are interpreted as elements of the Liesuperalgebra osp of the supergroup OSp :( − [ c ]([ b ]+[ d ]) ε ab ( δ ac A bd + A ac δ bd ) = (cid:16) ( − [ c ]+[ c ][ d ] ε cb A bd + ε ad A ac (cid:17) = 0 , (2.11)or equivalently A cd = − ǫ ( − [ c ][ d ]+[ c ]+[ d ] A dc . (2.12)The coordinate free form of relation (2.11) is ε h ( A + ( − ) A ( − ) ) = 0 ⇔ ( A + ( − ) A ( − ) )¯ ε i = 0 . (2.13)One can directly deduce these relations from equalities (2.9).The set of super-matrices A , which satisfy (2.11), (2.13), forms a vector space over C which is denoted as osp . One can check that for two super-matrices A, B ∈ osp thecommutator [ A, B ] = AB − BA , (2.14)also obeys (2.11), (2.13) and thus belongs to osp . It means that osp is an algebra. Anymatrix A which satisfies (2.11), (2.13) can be represented as A ac = E ac − ( − [ c ]+[ c ][ d ] ε cb E bd ¯ ε da (2.15)where || E ac || is an arbitrary matrix. Let { e fg } be the matrix units, i.e., matrices with thecomponents ( e fg ) b d = δ fd δ bg . If we substitute E = e fg = ¯ ε fg ′ ε gf ′ e f ′ g ′ in (2.15), then weobtain the basis elements { e G fg } in the space osp of matrices (2.13):( e G fg ) ac ≡ ( e fg ) ac − ( − [ c ]+[ c ][ d ] ε cb ( e fg ) b d ¯ ε da = ¯ ε fa ε gc − ǫ ( − [ c ][ a ] δ fc δ ag . (2.16)Now any super-matrix A ∈ osp which satisfies (2.11), (2.13) can be expanded over thebasis (2.16) A ac = a gf ( e G fg ) ac , (2.17)where a gf are the components of the super-matrix. Since the elements ( e G fg ) ac are even,i.e., ( e G fg ) ac = 0 iff [ f ] + [ g ] + [ a ] + [ c ] = 0 (mod2), then from the condition grad( A ac ) =[ a ] + [ c ] we obtain that grad( a gf ) = [ g ] + [ f ]. It means that the usual commutator (2.14)appears as a super-commutator for the basis elements e G fg :[ A, B ] ac = [ a gf ( e G fg ) , b nk ( e G kn )] ac = a gf b nk (cid:0) [ e G fg , e G kn ] ± (cid:1) ac , where in the component form the super-commutator is (cid:0) [ e G a b , e G a b ] ± (cid:1) a c ≡ ( e G a b ) a b ( e G a b ) b c − ( − ([ a ]+[ b ])([ a ]+[ b ]) ( e G a b ) a b ( e G a b ) b c . (2.18)We notice that the elements of the matrices e G ab are numbers. However, the super-commutator (2.18) is written for e G ab as for the graded elements with deg( e G ab ) = [ a ] + [ b ].Now we substitute the explicit representation (2.16) in the right-hand side of (2.18)and deduce the defining relations for the basis elements of the superalgebra osp :( − [ b ][ a ] · [ e G a b , e G a b ] ± = − ( − [ a ][ a ] ¯ ε a a e G b b + ǫ δ a b e G a b ++( − [ a ][ a ] ε b b e G a a − ǫ ( − [ a ]([ b ]+[ a ]) δ a b e G a b , (2.19)5here we have omitted the matrix indices. Below we use the standard component-freeform of notation, where we substitute ( e G a i b i ) a k b k → e G ik (here i and k are numbers 1 , , V ( N | M ) in V ⊗ N | M ) ). In this notation, taking into account (2.18), therelation (2.19) is written as[( − ) e G ( − ) , e G ] = [ ǫ P − K , e G ] , (2.20)where we introduce two matrices K , P ∈
End( V ⊗ N | M ) ): K a a b b = ¯ ε a a ε b b , P a a b b = ( − [ a ][ a ] δ a b δ a b . (2.21)The matrix P is called superpermutation since it permutes super-spaces, e.g., using thismatrix one can write (2.1) as P abcd w c z d = z a w b . Note that the generators (2.16) of the Liesuper-algebra osp can be expressed in terms of P and K as e G = K − ǫ P , (2.22)and after substituting (2.22) into (2.20) can be written (2.20) in the form[( − ) e G ( − ) , e G ] + [ e G , e G ] = 0 . (2.23)One can explicitly check the relation (2.23) by making use of the identities for the operators P and K presented in Appendix A .Note that conditions (2.13) for the osp generators A ac = ( e G fg ) ac , given in (2.16) and(2.22), can be written as K ( e G + ( − ) e G ( − ) ) = 0 , ( e G + ( − ) e G ( − ) ) K = 0 . (2.24)One can verify that these conditions are equivalent to K (( − ) e G ( − ) + e G ) = 0 , (( − ) e G ( − ) + e G ) K = 0 . (2.25)Using (2.25) and the commutation relations of super-permutation P and generators e G (seeappendix A ) P ( − ) e G ( − ) = e G P , ( − ) e G ( − ) P = P e G , (2.26)we write (2.20) as[( − ) e G ( − ) , e G ] = [ ǫ P − K , ( − ) e G ( − ) ] . (2.27)It means that the defining relations (2.19) can be written in many equivalent forms. Atthe end of this section we note that the matrix (2.22) is the split Casimir operator for theLie superalgebra osp in the defining representation. Consider the three
OSp invariant operators in V ⊗ N | M ) : the identity operator , the super-permutation operator P and metric operator K . According to definition (2.21), the super-permutation P is a product of the usual permutation P and the sign factor ( − ) , P = ( − ) P , or in components P a a b b = ( − [ a ][ a ] δ a b δ a b , (3.1)6hile the operator K is defined as K = ¯ ε i ε h , or in components K a a b b = ¯ ε a a ε b b . (3.2)Their OSp invariance means that (see (2.9)) U ( − ) U ( − ) K = K U ( − ) U ( − ) ,U ( − ) U ( − ) P = P U ( − ) U ( − ) . (3.3)In particular, it follows from these relations that the comultiplication for the supermatrices U ∈ Osp ( N | M ) has the graded form ∆( U ) = U ( − ) U ( − ) . In fact this comultipli-cation follows from the transformation (2.7) applied to the second rank tensor z a · z a .Using the operators P , K one can construct a set of operators { s i , e i | i = 1 , . . . , n − } in V ⊗ n ( N | M ) : s i = ǫ P i,i +1 ≡ ǫI ⊗ ( i − ⊗ P ⊗ I ⊗ ( n − i − , e i = K i,i +1 ≡ I ⊗ ( i − ⊗ K ⊗ I ⊗ ( n − i − , (3.4)which define the matrix representation T of the Brauer algebra B n ( ω ) [16], [17] with theparameter ω = ε cd ¯ ε cd = ǫ ( N − M ) . (3.5)Recall that here N and M are the numbers of even and odd coordinates, respectively.Indeed, one can check directly (see Appendix A ) that the operators (3.4) satisfy thedefining relations for the generators of the Brauer algebra B n ( ω ) s i = 1 , e i = ωe i , s i e i = e i s i = e i , i = 1 , ..., n − ,s i s j = s j s i , e i e j = e j e i , s i e j = e j s i , | i − j | > , (3.6) s i s i +1 s i = s i +1 s i s i +1 , e i e i +1 e i = e i , e i +1 e i e i +1 = e i +1 ,s i e i +1 e i = s i +1 e i , e i +1 e i s i +1 = e i +1 s i , i = 1 , ..., n − . (3.7)This presentation of the Brauer algebra can be obtained in the special limit q → T (3.4) of the generators s i , e i ∈ B n ( ω )acts in the space V ⊗ n ( N | M ) .Let us consider the following linear combination of the unit element ∈ B n ( ω ) andthe generators s i , e i ∈ B n ( ω )ˆ ρ i ( u ) = u ( u + β ) s i − ( u + β ) + u e i ∈ B n ( ω ) , (3.8)where u is a spectral parameter and β = 1 − ω . (3.9) Proposition 1. (see [19],[2]). The element (3.8) satisfies the Yang-Baxter equation ˆ ρ i ( u )ˆ ρ i +1 ( u + v )ˆ ρ i ( v ) = ˆ ρ i +1 ( v )ˆ ρ i ( u + v )ˆ ρ i +1 ( u ) , (3.10) and the unitarity condition ˆ ρ i ( u )ˆ ρ i ( − u ) = ( u − u − β ) . T (3.4) of the element (3.8) isˆ R ( u ) ≡ ǫ T (ˆ ρ ( u )) = u ( u + β ) P − ǫ ( u + β ) + ǫu K . (3.11)Here we suppress index i for simplicity. It follows from (3.10) that ˆ R ( u ) satisfies the braidversion of the Yang–Baxter equationˆ R ( u − v ) ˆ R ( u ) ˆ R ( v ) = ˆ R ( v ) ˆ R ( u ) ˆ R ( u − v ) . (3.12)Thus, in the supersymmetric case the braid version (3.12) of the Yang–Baxter equation isthe same as in the non supersymmetric case. Further we use the following R -matrix R ( u ) = P ˆ R ( u ) = ( u − ω u − ǫ P ) + u K = u ( u + β ) − ǫ ( u + β ) P + u K , (3.13)which is the image of the elements [19]: ρ i ( u ) = u ( u + β ) − ( u + β ) s i + u e i ∈ B n ( ω ) . Proposition 2.
The standard R -matrix R ( u ) = P ˆ R ( u ) , which was defined in (3.13),satisfies the graded version of the Yang–Baxter equation [22] R ( u − v )( − ) R ( u )( − ) R ( v ) = R ( v )( − ) R ( u )( − ) R ( u − v ) . (3.14) Proof.
Substituting ˆ R ij ( u ) = P ij R ij ( u ) = ( − ) ij P ij R ij ( u ) into (3.12) and moving all usualpermutations P ij to the left we write (3.12) in the form R ( u − v )( − ) R ( u )( − ) R ( v ) = R ( v )( − ) R ( u )( − ) R ( u − v ) . (3.15)The matrix R ∈ End( V ⊗ N | M ) ) is an even matrix since the following condition holds R i i j j = 0 iff [ i ] + [ i ] + [ j ] + [ j ] = 0 (mod2) . (3.16)This follows from the same property for the matrices , P , K which compose the operator R ( u ). Therefore, for arbitrary k we have R ij ( − ) ik ( − ) jk = ( − ) ik ( − ) jk R ij . (3.17)where the operator ( − ) ik is defined in (2.10) ( i and k are numbers of only two super-spaces V ( N | M ) in the product V ⊗ n ( N | M ) where the operator ( − ) ik acts nontrivially). Usingthe property (3.17), one can convert (3.15) into the form R ( u − v )( − ) R ( u )( − ) R ( v ) = R ( v )( − ) R ( u )( − ) R ( u − v ) , (3.18)and after the change of the spectral parameters we obtain the graded version of Yang–Baxter equation (3.14). Remark 1.
We stress that the sign operators ( − ) in (3.14) can be substituted by theoperators ( − ) by means of manipulations similar to (3.17). Moreover, if R ij ( u ) solvesthe Yang-Baxter equation (3.14), then the twisted R -matrix ( − ) ij R ij ( u )( − ) ij is also asolution of (3.14). Remark 2.
Eqs. (3.11), (3.13) give unified forms for solutions of the Yang-Baxterequations (3.12), (3.14) which are invariant under the action of all Lie (super)groups SO , Sp and OSp . Recall that for the SO case the R -matrix (3.13) was found in [24] and forthe Sp case it was indicated in [25]. For the OSp case such R -matrices were considered inmany papers (see, e.g., [23], [14], [26]). 8 Graded RLL-relation and the linear evaluation of Yangian Y ( osp ) We start with the following graded form of the RLL-relation (see, e.g., [26] and referencestherein) R ( u − v ) L ( u )( − ) L ( v )( − ) = ( − ) L ( v )( − ) L ( u ) R ( u − v ) , (4.1)where the R-matrix is given in (3.13). This graded form of the RLL relations is alsomotivated by the invariance conditions (3.3). It is known (see, e.g., [2], [14] and referencestherein) that eqs. (4.1) with the R-matrix (3.13) are defining relations for the super-Yangian Y ( osp ). In [2] we proved the following statement. Proposition 3.
The L -operator L ab ( u ) = ( u + α ) δ ab + G ab , (4.2) where α is an arbitrary constant, solves the RLL-relation (4.1) iff G ab is a traceless matrixof generators of the Lie superalgebra osp , i.e., it satisfies equations (cf. (2.24)) K n G + ( − ) G ( − ) o = 0 = n G + ( − ) G ( − ) o K , (4.3) defining relations for osp -algebra (cf. (2.20)) G ( − ) G ( − ) − ( − ) G ( − ) G = [ K − ǫ P , G ] , (4.4) and in addition obeys the quadratic characteristic identity G + β G − ǫω str (cid:0) G (cid:1) = 0 , (4.5) where as usual β = 1 − ω/ . The L -operator (4.2), where the elements G ab satisfy the conditions (4.3), (4.6) and (4.5),is called the linear evaluation of the Yangian Y ( osp ). Remark 3.
The relations (4.4) are written after the exchange 1 ↔ − ) G ( − ) G − G ( − ) G ( − ) = [ ǫ P − e K , G ] , (4.6)where e K = K = ( − ) K ( − ) , or e K a a b b = ¯ ε a a ε b b . Now we are able to compare thedefining relations (4.4), (4.6) with (2.20), (2.27), where the elements G ab are representedas matrices e G ab acting in the super-space V ( N |M ) , namely, the commutation relations(4.6) turn into the commutation relations (2.20) after the change of the definition of thesupermetric ε ab → ε ba = ǫ ( − [ a ] ε ab (see also the discussion in Remark 5 below). Remark 4.
The conditions (4.3) for the generators of osp read in component form (cf.(2.12)): G ab + ǫ ( − [ a ][ b ]+[ a ]+[ b ] G ba = 0 , G ab ≡ ε ac G cb . (4.7)In particular, it follows from (4.3), (4.7) that the matrix G is traceless0 = K (cid:16) G + ( − ) G ( − ) (cid:17) K = 2( ε ab G ac ¯ ε cb ) K = 2 ǫ str( G ) K . emark 5. The characteristic identity (4.5) is equivalent to the equation K (cid:16) βG + G ( − ) G ( − ) (cid:17) = (cid:16) βG + ( − ) G ( − ) G (cid:17) K , provided that the relations (4.3) and (4.6) are satisfied. Remark 6.
Comparing the
RLL -relations (4.1) and the graded Yang-Baxter equation(3.14), one finds that the latter can be written in the form of the
RLL -relation with the L -operator represented as an operator in V ⊗ N | M ) L ( u ) = 1 u ( − ) R ( u )( − ) = + 1 u (cid:16) β + ( e K − ǫ P ) (cid:17) − ǫβu P . (4.8)Then the operators L ( u ) and L ( v ) in (4.1) should be understood as u ( − ) R ( u )( − ) and u ( − ) R ( u )( − ) , respectively. Taking into account the term proportional to u − in (4.8), we represent the traceless generators G a c which satisfy (4.6) as G a a c c ≡ T a c ( G a c ) = ( e K a a c c − ǫ P a a c c ) = ¯ ε a a ε c c − ǫ ( − [ a ][ a ] δ a c δ a c , (4.9)i.e., this formula gives the defining representation T of the generators G ab of osp with thestructure relations (4.6) and the conditions (4.3), (4.7). We note that the choice of thebasis of osp in (4.9) differs from the choice of the basis of osp in (2.16) by sign factors T a c ( G a c ) = ( − [ a ][ a ]+[ c ][ c ] ( e G a c ) a c . This is consistent with the equality e G = G = ( − ) G ( − ) , where G and e G aredefined in (4.9) and (2.16) (compare eqs. (2.20), (2.24) with (4.3)). Y ( osp ) In this section we intend to construct an explicit representation of Y ( osp ) in which thegenerators of osp ⊂ Y ( osp ) satisfy the quadratic characteristic equation (4.5). We followthe approach of [2] and introduce a generalized algebra A of super-oscillators that consistsof both bosonic and fermionic oscillators simultaneously.Consider the super-oscillators c a ( a = 1 , , . . . , N + M ) as generators of an associativealgebra A with the defining relation[ c a , c b ] ǫ ≡ c a c b + ǫ ( − [ a ][ b ] c b c a = ¯ ε ab , (5.1)where the matrix ¯ ε ab is defined in (2.3) and (2.5). In view of (2.1), for ǫ = −
1, the super-oscillators c a with [ a ] = 0 (mod2) are bosonic and with [ a ] = 1 (mod2) are fermionic. For ǫ = +1 the statistics of the super-oscillators c a is unusual and we will discuss this in moredetail in Remark at the end of this section. Nevertheless, we assume the grading tobe standard grad( c a ) = [ a ] in both cases ǫ = ± c a → c ′ a = U ac c c of the super-group OSp with the elements U ∈ Osp (see [2]).With the help of convention (2.6) for lowering indices one can write relations (5.1) inthe equivalent forms[ c a , c b ] ǫ ≡ c a c b + ǫ ( − [ a ][ b ] c b c a = ε ba ⇔ c a c b + ǫ ( − [ a ][ b ] c b c a = δ ba . (5.2)10he super-oscillators c a satisfy the following contraction identities: c a c a = ¯ ε ab ε ad c b c d = ǫ ( − [ a ] c a c a , c a c a = ¯ ε ab ε ad c d c b = ǫ ( − [ a ] c a c a . So, we have c a c a = ¯ ε ab ( c b c a + ǫ ( − [ a ] c a c b ) = ¯ ε ab ε ab = ω ,c a c a = ¯ ε ab ( c a c b + ǫ ( − [ a ] c b c a ) = ¯ ε ab ε ba = D , D ≡ N + M . (5.3)Further we need the super-symmetrised product of two super-oscillators: c ( a c b ) := 12 (cid:0) c a c b − ǫ ( − [ a ][ b ] c b c a (cid:1) = − ǫ ( − [ a ][ b ] c ( b c a ) ∈ A , (5.4)and define the operators F ab ≡ ǫ ( − [ b ] c ( a c b ) , F ab = ε bc F ac . (5.5)In [2] we have proved the following statement. Proposition 4.
The operators F ab ∈ A defined in (5.5) are traceless and possess thesymmetry property (4.3), (4.7): str( F ) = ( − [ a ] F aa = 0 , F ab = − ǫ ( − [ a ][ b ]+[ a ]+[ b ] F ba . (5.6) In addition they satisfy the supercommutation relations (4.6) for the generators of osp ( − ) F ( − ) F − F ( − ) F ( − ) = [ ǫ P − e K , F ] , (5.7) and obey the quadratic characteristic identity (4.5): F ab F bc + βF ac − ǫω str( F ) δ ac = 0 , (5.8) where β = 1 − ω/ . Thus, the elements F ab = ǫ ε bd ( − [ b ] c ( a c d ) ∈ A given in (5.5) form a set of tracelessgenerators of osp which satisfy all conditions of Proposition and it means that thefollowing statement holds. Proposition 5.
The L -operator (4.2) in the super-oscillator representation (5.1): L ab ( u ) = ( u + α −
12 ) δ ab + ǫ ( − [ b ] c a c b ≡ ( u + α −
12 ) δ ab + B ab , (5.9) where we introduce for convenience B ab ≡ F ab + δ ab = ǫ ( − [ b ] c a c b , obey the RLL equation(4.1) which in the component form is given by ( − [ c ]([ b ]+[ c ]) R a a b b ( u − v ) L b c ( u ) L b c ( v ) = ( − [ a ]([ a ]+[ b ]) L a b ( v ) L a b ( u ) R b b c c ( u − v ) , (5.10) and the R -matrix (3.13) is R a a b b ( u ) = u ( u + β ) δ a b δ a b − ǫ ( u + β )( − [ a ][ a ] δ a b δ a b + u ¯ ε a a ε b b . roof. One can prove this Proposition directly. To simplify the notation, we write ( − a and ( − ab instead of ( − [ a ] and ( − [ a ][ b ] . After substituting the L -operator (5.9), theRLL equation (5.10) takes the form( u − v )( u − v + β ) (cid:16) ( − c ( a + c ) B a c B a c − ( − a ( a + c ) B a c B a c (cid:17) −− ( u − v + β ) ǫ ( − a a + a c + c c (cid:16) uδ a c B a c + vδ a c B a c − uδ a c B a c − vδ a c B a c (cid:17) ++( u − v )¯ ε a a (cid:16) ( − c ( c + b ) ε b b ( uδ b c + B b c ) B b c + vε b c B b c (cid:17) −− ( u − v ) ε c c (cid:16) ( − a ( a + b ) ¯ ε b b B a b ( uδ b c + B b c ) + v ¯ ε b a B a b (cid:17) = 0 . Taking into account the representation B ab = ǫ ( − b c a c b and defining relations (5.1), onechecks that the above relation is valid at arbitrary u and v . Remark 7.
The Quadratic Casimir operator C of the superalgebra osp ( N | M ) in thedifferential representation (5.5) is equal to the fixed number C = ( − [ a ] F ab F ba = ǫ ω ( ω − . (5.11)It means that this realization (5.5) corresponds to a limited class of representations ofthe superalgebra osp ( N | M ). This fact reflects the general statement of [27] that not allrepresentations of simple Lie algebras g of B, C and D types are the representations ofthe corresponding Yangians Y ( g ). Remark 8.
For ǫ = − M = 2 m the super-oscillator algebra (5.1) is representedin terms of m copies of the bosonic Heisenberg algebras c j = x j , c m + j = ∂ j , j = 1 , . . . , m ,and N fermionic oscillators c m + α = b α , α = 1 , , . . . , N , with the (anti)commutationrelations[ x i , ∂ j ] = − δ ij , [ b α , b β ] + := b α b β + b β b α = 2 δ αβ , [ x i , b α ] = 0 = [ ∂ i , b α ] , (5.12)which are equivalent to (5.1) with the choice of the metric ¯ ε ab as ( M + N ) × ( M + N )matrix ¯ ε ab = − I m I m I N ! ⇒ ε ab = I m − I m I N ! . (5.13)The fermionic variables b β with the commutation relations (5.12) generate the N -dimensionalClifford algebra. Let N be an even number N = 2 n . In this case, one can introduce thelongest element b ( N +1) = ( i ) n b b · · · b N which anticommutes with all generators b α andpossesses ( b ( N +1) ) = 1. Then, for ǫ = +1 and even numbers M = 2 m , N = 2 n , one canrealize the super-oscillator algebra (5.1) (with the metric (5.13)) in terms of the generators c j = x j · b ( N +1) , c m + j = ∂ j · b ( N +1) ( j = 1 , . . . , m ) , c m + α = b α ( α = 1 , , . . . , N ) , (5.14)where the operators x i , ∂ j and b α satisfy (5.12). Note that the super-oscillator algebra(5.1) for ǫ = +1 has an unusual property that generators c a and c b with gradings [ a ] = 0and [ b ] = 1 anticommute, which is not usual feature of bosons and fermions in field theories.The implementation (5.12) of algebra (5.1) suggests the rules of Hermitian conjugationfor the generators c a ( c j ) † = c j , ( c m + j ) † = − c m + j , j = 1 , . . . , m , ( c m + α ) † = c m + α , α = 1 , , . . . , N , (5.15)12hich follow from the commonly used properties of the Heisenberg and Clifford algebras:( x j ) † = x j , ( ∂ j ) † = − ∂ j , ( b α ) † = b α , ( b ( N +1) ) † = b ( N +1) . We shall apply the rules (5.15)below. Remark 9.
Consider the graded tensor product
A ⊗ A and denote the generators of thefirst and second factors in
A ⊗ A respectively as c a and c a . Since ⊗ is the graded tensorproduct, we have (cf. (5.1))[ c a , c b ] ǫ ≡ c a c b + ǫ ( − ab c b c a = 0 . (5.16)Any element of A ⊗ A can be written as a polynomial f ( c a , c b ) and its condition ofinvariance under the action of the group Osp is written as h A ba ( F ab + F ab ) , f ( c a , c b ) i = 0 , where (see (5.5)) F ab ≡ ǫ ( − b c ( a c b )1 , F ab ≡ ǫ ( − b c ( a c b )2 (5.17)are the generators of the osp algebras and A ba are the super- parameters (with grad( A ba ) =[ a ] + [ b ]). In the case of an even function f , when grad( f ) = 0, this invariance conditionis equivalent to h ( F ab + F ab ) , f ( c a , c b ) i = 0 . (5.18)Now we introduce the super-symmetrized product c ( a · · · c a k ) of any number of super-oscillators, which generalizes the super-symmetrized product of two super-oscillators (5.4).The general definition and properties of such super-symmetrized products are given inAppendix B . In [2] we have proved the following statement. Proposition 6.
The elements I k = ε a b . . . ε a k b k c ( a · · · c a k )1 c ( b k · · · c b )2 ∈ A ⊗ A , k = 1 , , . . . , (5.19) are invariant under the action (2.7) of the supergroup OSp : c a → U ab c b . It means thatthe elements (5.19) are invariant under the action of the Lie superalgebra osp and satisfythe invariance condition (5.18): h ε a b . . . ε a k b k c ( a · · · c a k )1 c ( b k · · · c b )2 , F ab + F ab i = 0 , (5.20) where F ab and F ab are the generators (5.17) of the Lie super-algebra osp (see Proposition ). It turns out that the invariants (5.19) are not functionally independent. Indeed, wehave the following statement.
Proposition 7.
The invariants (5.19) satisfy the recurrence relation I k I = I k +1 + k (cid:0) ( k − − ω (cid:1) I k − , ω = ǫ ( N − M ) , (5.21) where I = 1 and I = ε ab c a c b = c a c a . In the representations (5.12), (5.14) and (5.13)Hermitian conjugations of invariant elements (5.19) are I † k = I k , I † k +1 = − I k +1 . (5.22)13 roof. The derivation of the recurrence relation (5.21) is given in Appendix B . To prove(5.22), it is useful to define the invariants e I m = σ m I m , (5.23)where σ = −
1, i.e. σ = ± i . Then, the recurrence relation (5.21) for new invariants e I k has the form: e I k +1 = z e I k + k k − − ω ) e I k − , ω = ǫ ( N − M ) , (5.24)where e I = 1 and we introduce the operator z := e I = σI = σε ab c a c b = σ ( c · c ) = − σε ba c b c a = − σ ( c · c ) , (5.25)which is Hermitian z † = z in the representations (5.12), (5.14) and (5.13). One can provethe latter statement by making use of the rules (5.15) and commutation relations (5.16).In view of the recurrence relation (5.24) and initial conditions e I = 1 and e I = z allinvariant operators e I k are k -th order polynomials (with real coefficients) of the Hermitianoperator z . Therefore all e I k are the Hermitian operators e I † k = e I k , and therefore, takinginto account (5.23) and σ ∗ = − σ , we deduce (5.22).Now we introduce a generating function of the Hermitian invariant operators e I k : F ( x | z ) = ∞ X k =0 e I k x k k ! . (5.26)Since the invariants e I k are polynomials in z , the generating function (5.26) depends on x and z only. Proposition 8.
The generating function (5.26) is equal to F ( x | z ) = (cid:16) − x (cid:17) ω − z (cid:16) x (cid:17) ω + z . (5.27) Proof.
Using the recurrence relation (5.24) we obtain: ∞ X k =0 e I k +1 x k k ! = z ∞ X k =0 e I k x k k ! + 14 ∞ X k =2 e I k − x k ( k − − ω ∞ X k =1 e I k − x k ( k − . (5.28)Now changing the summation indices and using (5.26) one deduces: F x ( x | z ) = zF ( x | z ) + x F x ( x | z ) − xω F ( x | z ) , (5.29)where F x ( x | z ) ≡ ∂ x F ( x | z ) = P ∞ k =0 e I k +1 x k k ! . The general solution to this ordinary differen-tial equation is given in (5.27) up to an arbitrary constant factor c . The invariants e I k areextracted from the generating function (5.26) using the formula e I k ( z ) = ∂ kx F (0 | z ) = c ∂ kx (1 − x ω − z (1 + x ω + z (cid:12)(cid:12)(cid:12) x =0 , (5.30)from which we fix the constant c = F (0 | z ) = e I = 1.14 The construction of the R-operator in the super-oscillatorrepresentation
Let T be the defining representation of the Yangian Y ( osp ). In the previous section wehave considered the RLL -relation (4.1) and (5.10) that intertwines L -operators || L ab ( u ) || ∈ T ( Y ( osp )) ⊗ A (given in (5.9)) by means of the R -matrix (3.13) in the defining represen-tation, i.e., R ( u ) ∈ T ( Y ( osp )) ⊗ T ( Y ( osp )). In other words, the R -matrix in the RLL -relations (4.1) and (5.10) acts in the space V ⊗ N | M ) , where V ( N | M ) is the space of the definingrepresentation T of Y ( osp ( N | M )).There is another type of RLL -relations which intertwines the L -operators (5.9) bymeans of the R -matrix in the super-oscillator representation, i.e., ˆ R ( u ) ∈ A ⊗ A , where ⊗ is the graded tensor product. In components, this type of RLL relations has the formˆ R ( u ) L ab ( u + v ) L bc ( v ) = L ab ( v ) L bc ( u + v ) ˆ R ( u ) , (6.1)or after substitution of the L -operator (5.9) we haveˆ R ( u ) (cid:0) ( u + v ) δ ab + ǫ ( − b c a c b (cid:1)(cid:0) vδ bc + ǫ ( − c c b c c (cid:1) == (cid:0) vδ ab + ǫ ( − b c a c b (cid:1)(cid:0) ( u + v ) δ bc + ǫ ( − c c b c c (cid:1) ˆ R ( u ) . (6.2)Here for simplicity we fix α = 1 / L -operators and associate thefirst and second factors in A ⊗ A , respectively, with the algebras A and A generated bythe elements c a and c b such that [ c a , c b ] ǫ = 0 (see (5.16)).The RLL relation (6.2) is quadratic with respect to the parameter v . The termsproportional to v are cancelled, the terms proportional to v giveˆ R ( u )( c a c c + c a c c ) = ( c a c c + c a c c ) ˆ R ( u ) , (6.3)while the terms independent of v areˆ R ( u ) (cid:0) uδ ab + ǫ ( − b c a c b (cid:1) ( − c c b c c = ( − b c a c b (cid:0) uδ bc + ǫ ( − c c b c c (cid:1) ˆ R ( u ) . (6.4) R operator The relations (6.3) are nothing but the invariance conditions (5.18) with respect to theadjoint action of osp h ˆ R ( u ) , F ab + F ab i = 0 . (6.5)It means that one can search for the ˆ R ( u )-operator as a sum of osp -invariants (5.19)ˆ R ( u ) = X k r k ( u ) k ! I k = X k r k ( u ) k ! ε ~a,~b c ( a ...a k )1 c ( b k ...b )2 , (6.6)where we use the concise notation ε ~a,~b = ε a b . . . ε a k b k , c ( a ...a k )1 := c ( a · · · c a k )1 , c ( b k ...b )2 := c ( b k · · · c b )2 . Inserting this ansatz into the condition (6.4), we obtain (see [2]) the recurrence relationfor r k ( u ) r k +2 ( u ) = 4( u − k ) k + 2 + u − ω r k ( u ) , (6.7)15hich is solved in terms of the Γ-functions: r m ( u ) = ( − m Γ( m − u )Γ( m +1+ u − ω ) A ( u ) ,r m +1 ( u ) = ( − m Γ( m − u − )Γ( m +1+ u − ω +12 ) B ( u ) , (6.8)where the parameter ω = ǫ ( N − M ) was defined in (3.5) and A ( u ) , B ( u ) are arbitraryfunctions of u . Substituting (6.8) in (6.6) gives the expression for the osp -invariant R -matrix which intertwines two L operators in (6.1).The methods used in [2] (for derivation of (6.6) and (6.8)) require the introduction ofadditional auxiliary variables and are technically quite nontrivial and cumbersome. Belowin this paper, in Section , we give a simpler and more elegant derivation of conditions(6.7). This derivation is based on an application of the generating function (5.27) forthe invariants e I k , where the explicit form (5.27) is obtained by means of the recurrencerelation (5.21). R operator There is another form of R operators which intertwines the L operators in the RLL equa-tions (6.2) and are expressed as a ratio of Euler Gamma-functions. For the sℓ (2) casethis type of solutions for R operator was first obtained in [5] (see also [6] and [28]). Thegeneralization to the sℓ ( N ) case (for a wide class of representations of sℓ ( N )) was givenin [29]. For orthogonal and symplectic algebras (and a very special class of their repre-sentations) analogous solutions of (6.2) were recently obtained in [7]. Below we generalizethe results of [7] and find the solutions for the super-oscillator Faddeev-Takhtajan-Tarasovtype R -operator in the case of osp Lie superalgebras.
Proposition 9.
The R operator intertwining the super-oscillator L operators in the RLL equations (6.1), (6.2) obeys the finite-difference equation ˆ R ( u | z + 1) ( z − u ) = ˆ R ( u | z −
1) ( z + u ) , (6.9) where z = σ c a c a and σ = − . The solution of this functional equation is given by theratio of the Euler Gamma-functions ˆ R ( u | z ) = r ( u, z ) Γ (cid:0) ( z + 1 + u ) (cid:1) Γ (cid:0) ( z + 1 − u ) (cid:1) , (6.10) where r ( u, z ) is an arbitrary periodic function r ( u, z + 2) = r ( u, z ) which normalizes thesolution. Proof.
Taking into account the experience related to the orthogonal and symplectic cases(see [7]), we will look for a solution to the first equation (6.3) asˆ R ( u ) = ˆ R ( u | z ) , z = σc a c a = ǫσ ( − a c a c a = − σc a c a , (6.11)where σ is a numerical constant to be defined. In the last chain of equalities we haveused (5.16). In other words, the operator ˆ R ( u ) acting in V ⊗ V is given by a functionof an invariant z bilinear in super-oscillators c a and c a . Note that in the orthogonal andsymplectic cases [7] the conventional invariants I k (5.19) are in one-to-one correspondencewith polynomials of z of the order k . In the super-symmetric case of the algebras osp
16e prove this fact in Appendix B (see eq. (B.9) and comment after this equation). Tojustify the ansatz (6.11), we recall that the super-oscillators belonging to different factorsin A ⊗ A and acting in different auxiliary spaces V and V commute according to (5.16) c a c b = − ǫ ( − ab c b c a , (6.12)so we have zc b = σc a c a c b = − ǫ ( − ab σc a c b c a = ( − ab +1 σ (cid:0) ǫ ¯ ε ab − ( − ab c b c a (cid:1) c a = c b z − σc b , (6.13) zc b = σc a c a c b = c a σ ( δ ba − ǫ ( − ab c b c a ) = σc b − σǫ ( − ab c a c b c a = σc b + c b z. (6.14)Combining these relations we obtain z ( c a c b + c a c b ) = ( c a c b + c a c b ) z, (6.15)i.e. z commutes with the sum c a c b + c a c b , and hence an arbitrary function ˆ R ( u | z )depending on z satisfies the invariance conditions (6.3) and (6.5).Let us introduce c b ± := ( c b ± σc b ) , (6.16)and consider a linear combination of (6.13) and (6.14) zc b ± ≡ z ( c b ± σc b ) = c b ± z ± σ ( c b ∓ σ − c b ) = c b ± ( z ∓ , (6.17)where the last equation is obtained under the choice σ = − ⇒ σ = √− (cid:26) i , − i . (6.18)Taking into account (6.17), we haveˆ R ( u | z ) c b ± = c b ± ˆ R ( u | z ∓ , c b ± ˆ R ( u | z ) = ˆ R ( u | z ± c b ± . (6.19)Then multiplying (6.4) by c d ± ε da (or by c d ∓ ε da ) from the left and by c c ± from the right andcontracting oscillator vector indices, one obtains four independent scalar relations. Twoof them are c d ± ε da ˆ R ( u | z ) (cid:0) uδ ab + ǫ ( − b c a c b (cid:1) ( − c c b c c c c ± = (6.20)= c d ± ε da ( − b c a c b (cid:0) uδ bc + ǫ ( − c c b c c (cid:1) ˆ R ( u | z ) c c ± . Applying (6.19), (5.3), the definition (6.11) of z and c d ± c d = σ ( − z ± ω ) , c d ± c d = ω ∓ z, ( − c c c c c ± = ǫσ ( z ± ω ) , ( − c c c c c ± = ǫ ( ω ± z ) , these two relations (6.20) turn to be functional equations on ˆ R ( u | z ):ˆ R ( u | z ± c d ± ε da (cid:0) uδ ab + ǫ ( − b c a c b (cid:1) ( − c c b c c c c ± == c d ± ε da ( − b c a c b (cid:0) uδ bc + ǫ ( − c c b c c (cid:1) c c ± ˆ R ( u | z ∓ , ⇒ ˆ R ( u | z ± ǫ ( u ∓ z ) (cid:0) z − ω (cid:1) = ǫ ( − u ∓ z ) (cid:0) z − ω (cid:1) ˆ R ( u | z ∓ . ǫ (cid:0) z − ω (cid:1) in both sides we obtain a pair of equationsˆ R ( u | z ±
1) ( u ∓ z ) = ( − u ∓ z ) ˆ R ( u | z ∓ , (6.21)which are equivalent for both choices of signs to the one equation (6.9). In a similar fashionthe other pair of relations gives identities c d ∓ ε da ˆ R ( u | z ) (cid:0) uδ ab + ǫ ( − b c a c b (cid:1) ( − c c b c c c c ± == c d ∓ ε da ( − b c a c b (cid:0) uδ bc + ǫ ( − c c b c c (cid:1) ˆ R ( u | z ) c c ± , ⇒ (6.22)ˆ R ( u | z ∓ c d ∓ ε da (cid:0) uδ ab + ǫ ( − b c a c b (cid:1) ( − c c b c c c c ± == c d ∓ ε da ( − b c a c b (cid:0) uδ bc + ǫ ( − c c b c c (cid:1) c c ± ˆ R ( u | z ∓ , ⇒ ˆ R ( u | z ∓ ǫ ( u ± z ) (cid:0) z ± ω (cid:1) = ǫ ( u ± z ) (cid:0) z ± ω (cid:1) ˆ R ( u | z ∓ , which are satisfied automatically. Finally, the solution of the functional equations (6.9),(6.21) can be found immediately and is given in (6.10) by the ratio of the Euler Gamma-functions.We see that the scalar projections (6.20) and (6.22) of the RLL relation are exactlythe same as in the non-supersymmetric case [7], i.e. no signs related to grading appear.Moreover, we stress that the functional equation (6.9) is independent of the parameter ǫ ,which distinguishes the cases of the algebras osp ( N | M ) and osp ( M | N ). Remark 10.
We have two choices (6.18) of the parameter σ and therefore we have twoversions of the solution (6.10)ˆ R ( ± )12 ( u | z ) = r ( ± ) ( u, z ) Γ (cid:0) ( ± ic a c a + 1 + u ) (cid:1) Γ (cid:0) ( ± ic a c a + 1 − u ) (cid:1) . (6.23)In view of the identity Γ(1 − x )Γ( x ) = π/ sin( πx ), these two versions are equivalent to eachother up to a special choice of the normalization functions r ( ± ) ( u, z ). So one can consideronly one of the solutions (6.23). ˆ R operator in special cases of osp ( N | m ) In this section, we work out the explicit form of the solution (6.10) in a few particularcases. osp ( M | N ) = osp (1 | In this case, we have N = 2 and M = 1, and the superalgebra osp (1 |
2) is describedby the bosonic oscillator c ≡ a † , c ≡ a (in the holomorphic representation we have c ≡ x, c ≡ ∂ ) and by one fermionic variable c ≡ b with the commutation relations (5.1):[ x, ∂ ] = − , { b, b } = 2 , [ x, b ] = 0 = [ ∂, b ] , (7.1)where { b, b ′ } ≡ b · b ′ + b ′ · b denotes the anticommutator. To obtain (7.1) from (5.1) and(5.2), we fix there ǫ = − ε ab = − ! ⇒ ε ab = − / ! . b can be understood in the matrix representation as asingle Pauli matrix (say τ ). To define the operator z in (6.11) we need two copies of super-oscillator algebras A and A with the generators c a = ( x , ∂ , b ) and c a = ( x , ∂ , b )which act in two different spaces V and V . Then, the invariant operator z in (6.11) lookslike z = σc a c a = σε ab c a c b = σ ( x ∂ − x ∂ ) + σ b b ≡ x + b , (7.2) x = σ ( x ∂ − x ∂ ) , b = σ b b , σ = ± i, where b i satisfy b i = 1 in view of (7.1) and anticommute b b = − b b in order to ensure(6.12). The characteristic equation for the fermionic part of z : b = σ b b b b = − b b b b = 14 ( b ) ( b ) = 14 , (7.3)(here we took into account (7.1)) allows one to introduce the projection operators: P ± = 12 ± b , P i · P j = δ ij P i , i, j = ± , P + + P − = 1 . (7.4)Now any function f of b can be decomposed in these projectors b · P ± = ± P ± ⇒ f ( b ) = f ( b ) · ( P + + P − ) = f (cid:0) / (cid:1) P + + f (cid:0) − / (cid:1) P − . (7.5)Accordingly, the R -operator (6.10) can also be decomposed as:ˆ R ( u | z ) = ˆ R ( u | x + b ) = (cid:0)
12 + b (cid:1) · ˆ R (cid:0) u | x + 12 (cid:1) + (cid:0) − b (cid:1) · ˆ R (cid:0) u | x − (cid:1) . (7.6)and finally we haveˆ R osp (1 | ( u | z ) = r + ( u, x ) Γ (cid:0) ( x + + u ) (cid:1) Γ (cid:0) ( x + − u ) (cid:1) · P + + r − ( u, x ) Γ (cid:0) ( x + + u ) (cid:1) Γ (cid:0) ( x + − u ) (cid:1) · P − , (7.7)where r ± ( u, x ) = r ( u, x ± ) are periodic functions in x , i.e., the general osp (1 | R -operators consist of two independent terms acting on two invariant subspaces, corre-sponding to eigenvalues ± of the fermionic part b ≡ σb b of the invariant operator z .The coefficients in the expansion (7.7) in projectors P ± are the functions of the bosonicpart x = σ ( x ∂ − x ∂ ) of the invariant operator z . These coefficients are nothing butthe R -operators for the bosonic subalgebra sℓ (2) ≃ sp (2) ⊂ osp (1 | osp (2 | In this case, we have two bosonic c = x , c = ∂ and two fermionic c = b , c = b ,oscillators which we realize using even and odd variables with the commutation relations(5.1): [ x, ∂ ] = − , { b α , b β } = 2 δ αβ , [ x, b α ] = 0 = [ ∂, b α ] . (7.8)Here again we fix ǫ = − ε ab = − ⇒ ε ab = − / / .
19e introduce two super-oscillator algebras A and A with the generators { c a } and { c a } ,respectively. The invariant operator (6.11) is z = σε ab c a c b = σ ( x ∂ − x ∂ ) + σ b α b α ≡ x + b , (7.9)where σ = ± i .The fermionic oscillators b αi ∈ A i with commutation relations (7.8) and (5.16) generatethe 4-dimensional Clifford algebra. It is well known (see, e.g., [30]) that the generators ofthis Clifford algebra can be realized in terms the Pauli matrices τ α : b = τ ⊗ I , b = τ ⊗ I , b = τ ⊗ τ , b = τ ⊗ τ , where I is the unit (2 ×
2) matrix.The characteristic equation for the fermionic part b = σ b α b α of the operator (7.9) is b ( b − ) = . (7.10)The invariant subspaces spanned by the eigenvectors corresponding to eigenvalues 0, ± b are extracted by the projectors: P = 1 − b , P +1 = 12 ( b + b ) , P − = 12 ( b − b ) , P + P +1 + P − = 1 . (7.11)The R -operator is decomposed as follows:ˆ R ( u | z ) = ˆ R ( u | x + b ) == ˆ R ( u | x + b )( P + P +1 + P − ) = P ℓ =0 , ± ˆ R ( u | x + ℓ ) P ℓ . (7.12)Then (6.10) implies that the spinor-spinor R -operator invariant with respect to osp (2 | R osp (2 | ( u | z ) = X ℓ =0 , ± r ( u | x + ℓ ) Γ (cid:0) ( x + ℓ + 1 + u ) (cid:1) Γ (cid:0) ( x + ℓ + 1 − u ) (cid:1) P ℓ , r ( u | z + 2) = r ( u | z ) . (7.13)Note that in view of the periodicity condition r ( u | x −
1) = r ( u | x + 1) one can rewrite (7.13)as follows:ˆ R osp (2 | ( u | z ) = r ( u | x ) Γ (cid:0) ( x + 1 + u ) (cid:1) Γ (cid:0) ( x + 1 − u ) (cid:1) (1 − b ) + 12 r ( u | x + 1) Γ (cid:0) ( x + u ) (cid:1) Γ (cid:0) ( x − u ) + 1 (cid:1) ( xb + u b ) . (7.14)In the pure bosonic case of the orthogonal algebras so (2 k ), the general solution for the ˆ R -operator splits into two independent solutions corresponding to two nonequivalent chiralleft and right representations (see [8], [1], [7]). This does not happen here in the super-symmetric case, where the even and odd functions of b are not separated, due to thedependence on the bosonic operator x in the coefficients r ( u | x + ℓ ) which mixes the chiralrepresentations of so (2 k ). 20 .3 The case of osp ( n | This case is a generalization of the examples considered in the previous subsections (for n = 1 and n = 2 we respectively reproduce the results for the osp (1 |
2) and osp (2 | A with two bosonic c = x , c = ∂ and n fermionic generators c α = b α ( α = 1 , ..., n ) with the commutation relations (5.1):[ x, ∂ ] = − , { b α , b β } = 2 δ αβ , [ x, b α ] = 0 = [ ∂, b α ] , (7.15)where the fermionic elements b α are the generators of the n -dimensional Clifford algebra.This corresponds to the choice of the parameter ǫ = − ε ab = − I n ! ⇒ ε ab = − I n ! (7.16)where I n stands for the n × n unit matrix. The invariant operator z ∈ A ⊗ A is z = σε ab c a c b = σ ( x ∂ − x ∂ ) + σ b α b α ≡ x + b , (7.17)where { c a } and { c a } are the generators of the first and second factor in A ⊗ A , andthe cross-commutation relations of { c a } and { c a } are given in (5.16). The characteristicequation for the fermionic part b = σ b α b α of the invariant z has the order n + 1 (cf. (7.3)and (7.10)): n Y m =0 (cid:0) b − m + n (cid:1) = 0 . (7.18)One can prove (7.18) by noticing that the operator b is represented as b = ¯ z α z α − n/ z α ≡ ( b α − σb α ) = c α − and z α ≡ ( b α + σb α ) = c α + are respectivelythe creation and annihilation fermionic operators in the Fock space F which is created fromthe vacuum | i : z α | i = 0 ( ∀ α ). Then the operator in the left-hand side of (7.18) is equalto zero since it is zero on all basis vectors ¯ z α · · · ¯ z α m | i ∈ F (here 1 ≤ α < ... < α m ≤ n and m ≤ n ) which are the eigenvectors of b with eigenvalues ( m − n ).The projectors P ℓ on invariant subspaces in F spanned by the eigenvectors of b cor-responding to eigenvalues ( m − n ) ≡ ℓ , where ℓ = − n , − n + 1 , ..., n , are immediatelyobtained from (7.18): P ℓ = n/ Y m = ℓ m = − n/ b − mℓ − m , b · P ℓ = ℓ P ℓ , n/ X ℓ = − n/ P ℓ = 1 . (7.19) The case of even n = 2 k We see that eigenvalues of b are integer (or half-integer) when the number n is even (orodd). Thus, for the case osp ( n |
2) = osp (2 k | n = 2 k is even, the expansion of thesolution (6.10) goes over integer eigenvaluesˆ R osp (2 k | ( u | z ) = ˆ R osp (2 k | ( u | x + b ) = k P ℓ = − k ˆ R osp (2 k | ( u | x + ℓ ) P ℓ , (7.20)and it impliesˆ R osp (2 k | ( u | z ) = k X ℓ = − k r ( u | x + ℓ ) Γ (cid:0) ( x + ℓ + 1 + u ) (cid:1) Γ (cid:0) ( x + ℓ + 1 − u ) (cid:1) P ℓ , r ( u | z + 2) = r ( u | z ) . (7.21)21 he case of odd n = (2 k + 1)For the case osp ( n |
2) = osp (2 k +1 | n = 2 k +1 is odd, the expansion of the solution(6.10) goes over half-integer eigenvalues of b : − k +12 , − k − , . . . , − , , , . . . , k +12 , andwe have the expansionˆ R osp (2 k +1 | ( u | z ) = ˆ R osp (2 k +1 | ( u | x + b ) = k + X ℓ = − k − ˆ R osp (2 k +1 | ( u | x + ℓ ) P ℓ , (7.22)which for solution (6.10) impliesˆ R osp (2 k +1 | ( u | z ) = k + X ℓ = − k − r ( u | x + ℓ ) Γ (cid:0) ( x + ℓ + 1 + u ) (cid:1) Γ (cid:0) ( x + ℓ + 1 − u ) (cid:1) P ℓ , (7.23)where the periodic function r ( u | z + 2) = r ( u | z ) normalizes the solution. osp ( n | m ) We consider the osp ( n | m ) invariant super-oscillator algebra which is realized in termsof m pairs of the bosonic oscillators c j = x j , c m + j = ∂ j , j = 1 , . . . , m and n fermionicoscillators c m + α = b α , α = 1 , , . . . , n , with the commutation relations (5.12) deducedfrom (5.1) with the choice of the parameter ǫ = − z ∈ A ⊗ A defined in (6.11) is z = σε ab c a c b = σ m X j =1 ( x j ∂ j − x j ∂ j ) + σ n X α =1 b α b α ≡ x + b . (7.24)Here the operator b is the same as in the previous examples of Section . Thus, the R operator (6.10) in the case of the algebra osp ( n | m ) is expanded over the projectionoperators P ℓ like in the case of osp ( n | R osp (2 m | n )12 ( u | z ) willbe given by (7.21) or (7.23):ˆ R osp ( n | m )12 ( u | z ) = X ℓ ∈ Ω n r ( u | x + ℓ ) Γ (cid:0) ( x + ℓ + 1 + u ) (cid:1) Γ (cid:0) ( x + ℓ + 1 − u ) (cid:1) P ℓ , (7.25)where x = σ m X j =1 ( x j ∂ j − x j ∂ j ) , r ( u | z ) = r ( u | z + 2) , the projectors P ℓ are defined in (7.19) andΩ n = (cid:26) {− k, − k, . . . , k − , k } , n = 2 k, {− k − , . . . , k + } , n = 2 k + 1 k ∈ N . (7.26) In this section, we give a more direct and elegant derivation of the R matrix solution(6.6), (6.7), that does not require the introduction of additional auxiliary variables (as22t was done in [2]) and is based only on using the generating function (5.26), (5.27) ofthe invariant operators e I k . In addition, this derivation partially explains the relationshipbetween the two types of solutions (6.6), (6.7) and (6.10) for the R operator.Now we clarify the relation of the Shankar-Witten form of the R operator (6.6), (6.7)and Faddeev-Takhtajan-Tarasov type R operator given by the ratio of two Euler Gamma-functions in (6.10). First, we write (6.6) in the formˆ R ( z ) = ∞ X k =0 e r k ( u ) k ! e I k ( z ) , (8.1)where e r k ( u ) = ( − σ ) k r k ( u ) and e I k = σ k I k . Recall that e I k are the Hermitian invariantsintroduced in the proof of Proposition . Proposition 10.
The R operator (8.1) obeys (6.4) or equivalently the finite-differenceequation (6.9): W ≡ ( z − u ) ˆ R ( z + 1) − ( z + u ) ˆ R ( z −
1) = 0 , (8.2) (which was used to find the second solution (6.10)) if the coefficients e r k ( u ) satisfy e r k +2 ( u ) = − u − k ) k + 2 + u − ω e r k ( u ) , (8.3) that in terms of r k ( u ) is written as (6.7). Proof.
One can write (8.2) as W = ∞ X k =0 e r k ( u ) k ! (cid:16) ( z − u ) e I k ( z + 1) − ( z + u ) e I k ( z − (cid:17) = (8.4)= ∞ X k =0 e r k ( u ) k ! (cid:16) ( z − u ) ∂ kx F ( x | z + 1) − ( z + u ) ∂ kx F ( x | z − (cid:17) x =0 . We use the relation (5.29) in the form: zF ( x | z ) = (cid:20)(cid:0) − x (cid:1) ∂ x + ωx (cid:21) F ( x | z ) , and obtain W = ∞ X k =0 e r k ( u ) k ! ∂ kx (cid:16) ( z + 1 − u − F ( x | z + 1) − ( z − u + 1) F ( x | z − (cid:17) x =0 = (8.5)= ∞ X k =0 e r k ( u ) k ! ∂ kx (cid:16) [(1 − x ∂ x + ωx − u − F ( x | z +1) − [(1 − x ∂ x + ωx u +1] F ( x | z − (cid:17) x =0 , Then we use the equations F ( x | z + 1) = 1 + x − x F ( x | z ) , F ( x | z −
1) = 1 − x x F ( x | z ) , However we stress that generating function (5.27) is obtained by using of the recurrence relation (5.24)while the latter is derived in the Appendix B by means of auxiliary variables. W = ∞ X k =0 r k ( u ) k ! ∂ kx (cid:26) (1 − x x∂ x − u − ( u − ω + 2) x (cid:27) (cid:16) F ( x | z )1 − x (cid:17) x =0 . (8.6)Now we apply the identity ∂ kx x = x∂ kx + k∂ k − x to move derivatives ∂ x in (8.6) to the rightand obtain: W = ∞ X k =0 e r k ( u ) k ! (cid:20) ( k − u ) ∂ kx − k + u − ω k ( k − ∂ k − x (cid:21) (cid:16) F ( x | z )1 − x (cid:17) x =0 . (8.7)In the second term in square brackets we shift the summation parameter k → k + 2 anddeduce W = ∞ X k =0 k ! (cid:16) ( k − u ) e r k ( u ) − k + 2 + u − ω e r k +2 ( u ) (cid:17) ∂ kx (cid:16) F ( x | z )1 − x (cid:17) x =0 = 0 . The resulting expression vanishes due to (8.3). Thus, we prove that the finite-differenceequation (8.2) is valid if the coefficients e r k ( u ) satisfy (8.3). Remark 11.
We prove that both R operators (8.1), (8.3) and (6.10) satisfy the sameequation (8.2) and indeed obey the RLL relations (6.2). It is worth also to note thatthe differential operator in the curly brackets in (8.6) coincides (up to change of variable x = σλ ) with the differential operator in curly brackets presented in formula (6.43) of ourwork [2]. This suggests to regard the generating function F ( x | z )(1 − x ) − as a coherentstate in the super-oscillator space. Acknowledgments
The authors would like to thank S.Derkachov for useful discussions and comments. A.P.I.acknowledges the support of the Russian Science Foundation, grant No. 19-11-00131.The work of D.K. was partially supported by the Armenian State Committee of Sciencegrant 18T-132 and by the Regional Training Network on Theoretical Physics sponsoredby Volkswagenstiftung Contract nr. 86 260.24
Properties of operators P , K and relations for matrixgenerators of Brauer algebra We use here the concise matrix notation introduced in Sections
2, 3 (this convenientnotation was proposed in [31]). The matrices (2.21) satisfy the identities
AIf P = P , K = ( − ) K ( − ) , ( − ) K = ( − ) K , K ( − ) = K ( − ) , P P = , K K = ω K , K P = P K = ǫ K , (A.1)where ω = ǫ ( N − M ), the operator ( − ) is defined in (2.10) and we introduce the matrix( − ) i = ( − [ a i ] δ a i b i of super-trace in the i -th factor V ( N | M ) of the product V ⊗ N | M ) . Then,by making use definitions (3.1) and (3.2), we have( − ) P = P ( − ) , ( − ) P = P ( − ) , P ( − ) = ( − ) P , (A.2) P P = ( − ) P ( − ) P = P ( − ) P ( − ) = P ( − ) P ( − ) , P K = ( − ) K ( − ) P , K P = P ( − ) K ( − ) , (A.3) ǫ K P = K ( − ) K ( − ) , ǫ P K = ( − ) K ( − ) K , (A.4) K K = ǫ K ( − ) P ( − ) = ǫ ( − ) P ( − ) K , (A.5) K K = ǫ ( − ) P ( − ) K = ǫ K ( − ) P ( − ) . (A.6)The mirror counterparts of identities (A.3) – (A.6) are also valid. The identities (A.2),(A.3) follow from the representation (3.1): P = ( − ) P = P ( − ) , where P is theusual permutation operator. The identities (A.6) follow from the definitions (2.21), (3.1),(3.2) of the operators P and K . We prove only the first equality in (A.6) since the otheridentities in (A.4), (A.5) and (A.6) can be proved in the same way. We denote the incomingmatrix indices by a , a , a and the outcoming indices by c , c , c while summation indicesare b i and d i . Then we have( K K ) a a a c c c = ¯ ε a a ε c b ¯ ε b a ε c c = ¯ ε a a δ a c ε c c = δ a c δ a b ǫ ( − ) [ a ][ b ] ¯ ε a b ε c c == ǫ ( − [ a ][ a ] ( P ) a a b c ( − [ a ][ b ] ( K ) a b c c = ǫ (cid:0) P ( − ) ( − ) K (cid:1) a a a c c c , and in view of the relation ( − ) K = ( − ) K which follows from (2.4) and obviousidentity P ( − ) = ( − ) P we obtain the first equality in (A.6).By means of the relations (A.3) – (A.6) one can immediately check eqs. (2.25), (2.26)and also deduce P P P = P P P . (A.7) K K K = K , K K K = K , (A.8) P K K = P K , K K P = K P , (A.9) P K K = P K , K K P = K P . (A.10)The identity (A.7) follows from the relations in the first line of (A.3). We consider a fewrelations in (A.8) – (A.10) in detail. We start to prove the first relation in (A.8):( K K K ) a a a c c c = ¯ ε a a ε b b ¯ ε b a ε d c ¯ ε b d ε c c = ¯ ε a a δ a b δ b c ε c c = K a a c c δ a c . P K K ) a a a c c c = ( − [ a ][ a ] δ a b δ a b ¯ ε b a ε d c ¯ ε b d ε c c = ( − [ a ][ a ] ¯ ε a a δ a c ε c c == δ a c δ a b ( − [ a ][ c ] ¯ ε a b ε c c = δ a c δ a b ( − [ b ][ c ] ¯ ε a b ε c c = ( P K ) a a a c c c , and similarly one deduces other relations in (A.9) and (A.10). From the identities (A.7)– (A.10) we also deduce the following relations: K P K = ǫ K , K P K = ǫ K . (A.11) P K P = P K P . (A.12) P P K = K P P , K P P = P P K . (A.13)Indeed, if we act from the left on both sides of the first relation in (A.9) by K and use(A.1), (A.8) we obtain the first relation in (A.11). In the same way one can deduce fromthe first relation in (A.10) the second relation in (A.11). Now we act on both sides of(A.9) by P from the right and use the last equation in (A.10). As a result, we arrive atidentity (A.12). Finally, relations (A.13) trivially follow from eq. (A.12).At the end of this appendix, we stress that the identities (A.1), (A.7) – (A.10) are theimages of the defining relations (3.6), (3.7) for the Brauer algebra in the representation(3.4). The R -matrix (3.11) is the image of the element (3.8) and the Yang-Baxter equation(3.12) is the image of the identity (3.10). Thus, it follows from proposition that the R -matrix (3.11) is a solution of the braided version of the Yang-Baxter equation (3.12). B Supersymmetrized products of super-oscillators
The product of k super-oscillators is transformed under the action (2.7) of the group Osp as follows: c a c a · · · c a k → ∆ ( k − ( U ) a a ...a k b b ...b k c b c b · · · c b k , (B.1)where U ∈ Osp , and the tensor product of k defining representations of the group Osp isgiven by the formula∆ ( k − ( U ) a a ...a k b b ...b k = U a b ( − b a U a b ( − b b · · · ( − ( k − P j =1 b j ) a k U a k b k ( − ( k − P j =1 b j ) b k , or in the concise notation we have∆ ( k − ( U ) ...k = U ( − ) [1][2] U ( − ) [1][2] · · · ( − ) [ k ] k − P j =1 [ j ] U k ( − ) ( k − P j =1 [ j ])[ k ] . One can check that any element X ∈ B k ( ω ) of the Brauer algebra (3.6), (3.7) in therepresentation (3.4) commutes with the action of the Osp supergroup∆ ( k − ( U ) ...k · X = X · ∆ ( k − ( U ) ...k . (B.2)Define the super-symmetrized product of two super-oscillators c a , c b as c ( a c b ) ≡ (cid:16) c a c b − ǫ ( − [ a ][ b ] c b c a (cid:17) = 12 (cid:16) c a c b − ǫ P abde c d c e (cid:17) = ( A ) abde c d c e , (B.3)26here P is the super-permutation matrix (3.1) and ( A ) abde is the antisymmetrizer A = (1 − s ) in the representation (3.4). The direct generalization of (B.3) to the super-symmetrized product of any number of super-oscillators is the following: c ( a c a · · · c a k ) = ( A k ) a a ...a k b b ...b k c b c b · · · c b k , (B.4)where A k is the k -th rank antisymmetrizer in the representation (3.4). The antisym-metrizer A k can be defined via the recurrence relations (see, e.g., [30]) A k = k A k − (cid:0) − s k − + s k − s k − − ... + ( − k − s k − · · · s s (cid:1) == k (cid:0) − s k − + s k − s k − − ... + ( − k − s s · · · s k − (cid:1) A k − , and after substituting here the recurrence relations for A k − , A k − etc., we arrive at thefactorised formula A k = 1 k ! (cid:0) − s k − + ... + ( − k − s s · · · s k − (cid:1) · · · (cid:0) − s + s s (cid:1)(cid:0) − s (cid:1) . (B.5)We stress that in view of the relation (B.2) the super-symmetrized product (B.4) is trans-formed under the action of Osp as a usual product (B.1).Note that upon opening the parentheses, the element (B.5) equals the alternating sumof all k ! elements of the permutation group S k . Using this fact one can give a more explicitformula for super-symmetrized product (B.4) of a higher number of super-oscillators c ( a c a . . . c a k ) ≡ k ! X σ ∈ S k ( − ǫ ) p ( σ ) ( − ˆ σ c a σ . . . c a σk = 1 k ! ∂ a κ · · · ∂ a k κ ( κ · c ) k = (B.6)= ∂ a κ · · · ∂ a k κ exp( κ a c a ) | κ =0 , where p ( σ ) = 0 , σ . Here we introduce (see [2])auxiliary super-vector κ a such that the derivatives ∂ aκ = ∂∂κ a satisfy ∂ a κ c a = − ǫ ( − a a c a ∂ a κ , ∂ bκ ( κ a c a ) = c b + ( κ a c a ) ∂ bκ , ∂ aκ ∂ bκ = − ǫ ( − [ a ][ b ] ∂ bκ ∂ aκ , and (B.6) holds due to the Leibniz rule.Now we explain the notation ˆ σ in (B.6). Let s j ≡ σ j,j +1 be an elementary transpositionof the j -th and ( j + 1)-st site. For the transposition s j we define ˆ s j = [ a j ][ a j +1 ]. Then fora general permutation σ = s j s j . . . s j k − s j k ∈ S k , we haveˆ σ = [ a j k ][ a j k +1 ] + [ a s jk ( j k − ) ][ a s jk ( j k − +1) ] + · · · + [ a s j ··· s jk ( j ) ][ a s j ··· s jk ( j +1) ] . (B.7)As an example, according to the definition (B.6), we have the relation useful in practice: c ( a · · · c a i · · · c a j · · · c a k ) = ( − ǫ )( − [ a i ][ a j ]+ j − P l = i +1 ([ a i ]+[ a j ])[ a l ] c ( a · · · c a j · · · c a i · · · c a k ) . (B.8)In eq. (5.19) we have defined the supersymmetric invariants I m Using this definition, therepresentation (B.6) and the definition (6.11) of z , we obtain the recurrence relation − σ I m · z = I m I = ε a b . . . ε a m b m ∂ a κ . . . ∂ a m κ ∂ b m κ . . . ∂ b κ ε ab ( ∂ aκ + ǫ ( − a κ a )( ∂ bκ ++ ǫ ( − b κ b ) e κ · c + κ · c | κ i =0 = I m +1 + ε a b . . . ε a m b m ∂ a κ . . . ∂ a m κ ∂ b m κ . . . ∂ b κ ε ab ( − a ×× κ a κ b e κ · c + κ · c | κ i =0 = I m +1 − ǫ P mi =1 ε a b . . . ε a i − b i − ε a i +1 b i +1 . . . ε a m b m ∂ a κ . . . ∂ a i − κ ×× ∂ a i +1 κ . . . ∂ a m κ (cid:0) ωǫ − ǫκ d ∂ κ ,d (cid:1) ∂ b m κ . . . ∂ b i +1 κ ∂ b i − κ . . . ∂ b κ e κ · c + κ · c | κ i =0 = I m +1 + m (cid:0) ( m − − ω (cid:1) I m − . (B.9)27or a proof of (B.9) we refer to the papers [1] (see analogous calculation in eq. (5.6) there)and [2]. Taking into account the initial conditions I = 1 and I = − σz , we deduce from(B.9) that the invariants I m are polynomials in z of the order m . References [1] A. P. Isaev, D. Karakhanyan, R. Kirschner,
Orthogonal and symplectic Yangians andYang-Baxter R-operators , Nucl. Phys. B 904 (2016) 124147; arXiv:1511.06152v1.[2] J. Fuksa, A. P. Isaev, D. Karakhanyan and R. Kirschner,
Yangians and YangBax-ter R-operators for ortho-symplectic superalgebras , Nucl. Phys. B (2017) 44;[arXiv:1612.04713 [math-ph]].[3] R.Shankar and E.Witten,
The S -matrix of the kinks of the ( ¯ ψψ ) model , Nucl.Phys.B141 (1978) 349-363.[4] D. Chicherin, S. Derkachov and A. P. Isaev, Conformal group: R-matrix and star-triangle relation , JHEP, 04(2013)020; arXiv:1206.4150 [math-ph].[5] V.O.Tarasov, L.A.Takhtajan and L.D.Faddeev,
Local hamiltonians for integrablequantum models on a lattice , Theor. Math. Phys. 57 (1983) 163-181[6] L.D. Faddeev,
How Algebraic Bethe Ansatz works for integrable model , In: Quan-tum Symmetries/Symetries Quantiques, Proc.Les-Houches summer school, LXIV.Eds. A.Connes,K.Kawedzki, J.Zinn-Justin. North-Holland, 1998, 149-211, [hep-th/9605187][7] D. Karakhanyan and R. Kirschner,
Spinorial RRR operator and Algebraic BetheAnsatz
Nucl.Phys.B 951 (2020) 114905; e-Print: 1911.08385 [math-ph].[8] D. Chicherin, S. Derkachov and A. P. Isaev,
Spinorial R-matrix , Journal of Phys. A:Vol. 46, Number 48 (2013)485201; arXiv:1303.4929 [math-ph].[9] M.Karowski and H.J.Thun,
Complete S-matrix of the O (2 N ) Gross-Neveu model ,Nucl.Phys. B190[FS3] (1981) 61-92.[10] Al.B. Zamolodchikov,
Factorizable Scattering in Assimptotically Free 2-dimensionalModels of Quantum Field Theory , PhD Thesis, Dubna (1979), unpublished.[11] N.Yu. Reshetikhin,
Algebraic Bethe-Ansatz for SO ( N ) invariant transfer-matrices ,Zap. Nauch. Sem. LOMI, vol. 169 (1988) 122 (Journal of Math. Sciences, Vol.54, No.3 (1991) 940-951).[12] E. Ogievetsky, P. Wiegmann and N. Reshetikhin, The Principal Chiral Field in Two-Dimensions on Classical Lie Algebras: The Bethe Ansatz Solution and FactorizedTheory of Scattering , Nucl. Phys. B (1987) 45.[13] F.A. Berezin,
Introduction to algebra and analisis with anticommuting variables ,Moscow State University Press, Moscow (1983).2814] D. Arnaudon, J. Avan, N. Crampe, L. Frappat, E. Ragoucy,
R-matrix presentationfor (super)-Yangians Y(g) , J. Math. Phys. 44 (2003) 302, math.QA/0111325;D. Arnaudon, a. o.,
Bethe Ansatz equations and exact S matrices for the osp ( M | n ) open super spin chain , Nucl. Phys. B 687, Issue 3 (2004) 257.[15] J. Birman and H. Wenzl, Braids, link polynomials and a new algebra , Trans. Amer.Math. Soc. 313 (1989), 249273.[16] R. Brauer,
On algebras which are connected with the semisimple continuous groups ,Ann. of Math. (2) 38 (1937), no. 4, 857872.[17] H. Wenzl,
On the structure of Brauer’s centralizer algebras , Ann. Math. 128 (1988)173-193.[18] M. Nazarov,
Youngs Orthogonal Form for Brauers Centralizer Algebra , J. Algebra182 (1996) 664693.[19] A. P. Isaev, A. I. Molev,
Fusion procedure for the Brauer algebra , Algebra i analiz, :3 (2010), 142–154.[20] A.P. Isaev, A.I. Molev and O.V. Ogievetsky, A new fusion procedure for the Braueralgebra and evaluation homomorphisms , Int. Math. Res. Not. (2012), 25712606.[21] A.P.Isaev and M.A.Podoinitsyn,
D-dimensional spin projection operators for arbitrarytype of symmetry via Brauer algebra idempotents , e-print (2020), arXiv:2004.06096[hep-th].[22] P.P. Kulish and E.K. Sklyanin,
On solutions of the Yang-Baxter equation , J. Sov.Math. , Issue 5 (1982) 1596 [Zap. Nauchn. Semin. (1980) 129].[23] P.P. Kulish, Integrable graded magnets , J. Sov. Math. (1986) 2648 [Zap. Nauchn.Semin. (1985) 140].P.P. Kulish and N.Yu. Reshetikhin, Universal R Matrix Of The Quantum SuperalgebraOsp(2 | , Lett. Math. Phys. 18 (1989) 143.[24] A.B. Zamolodchikov and Al.B. Zamolodchikov, Relativistic factorized S-matrix in twodimensions having O(N) isotopic symmetry , Nucl. Phys., B133 (1978), p. 525[25] B. Berg, M. Karowski, P. Weisz, V. Kurak,
Factorized U ( n ) symmetric S-matrices intwo dimensions , Nuclear Physics B 134, Issue 1 (1978) 125-132.[26] A.P. Isaev, Quantum groups and Yang-Baxter equations , preprint MPIM (Bonn), MPI2004-132,(http://webdoc.sub.gwdg.de/ebook/serien/e/mpi mathematik/2004/132.pdf).[27] V.G.Drinfeld,
Hopf algebras and the quantum Yang-Baxter equation , Dokl.Akad.NaukSSSR, Volume 283,Number 5 (1985) 1060.[28] S.E.Derkachov,
Factorization of R-matrix. I.
Journal of Mathematical Sciences 143.1(2007) 2773; arXiv:math.QA/0503396.[29] S.E.Derkachov and A.N.Manashov,
Factorization of R-matrix and Baxter Q-operatorsfor generic sℓ ( N ) spin chains , Journal of Physics. A, Math. Theor., (2009) Theory Of Groups And Symmetries. Representations ofGroups and Lie Algebras, Applications.
World Scientific (2020) 600 pp.[31] L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan,
Quantization of Lie groups andLie algebras , (Russian) Algebra i Analiz (1989) no. 1, 178–206. English translationin: Leningrad Math. J.1