Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
V.M. Buchstaber, S. Igonin, S. Konstantinou-Rizos, M.M. Preobrazhenskaia
aa r X i v : . [ n li n . S I] A ug Yang–Baxter maps, Darboux transformations, and linearapproximations of refactorisation problems
V.M. Buchstaber ∗ , S. Igonin † , S. Konstantinou-Rizos ‡ , and M.M. Preobrazhenskaia § Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Russia Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, RussiaSeptember 2, 2020
Abstract
Yang–Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang–Baxter equation.For a set X = Ω × V , where V is a vector space and Ω is regarded as a space of parameters, a linearparametric YB map is a YB map Y : X × X → X × X such that Y is linear with respect to V andone has πY = π for the projection π : X × X → Ω × Ω. These conditions are equivalent to certainnonlinear algebraic relations for the components of Y . Such a map Y may be nonlinear with respectto parameters from Ω.We present general results on such maps, including clarification of the structure of the algebraicrelations that define them and several transformations which allow one to obtain new such maps fromknown ones. Also, methods for constructing such maps are described. In particular, developing anidea from [Konstantinou-Rizos S and Mikhailov A V 2013 J. Phys. A: Math. Theor. 46 425201],we demonstrate how to obtain linear parametric YB maps from nonlinear Darboux transformationsof some Lax operators using linear approximations of matrix refactorisation problems correspondingto Darboux matrices. New linear parametric YB maps with nonlinear dependence on parameters arepresented. PACS numbers:
Mathematics Subject Classification:
Keywords:
Yang–Baxter equation, parametric Yang–Baxter maps, Darboux transformations of Laxoperators, integrable PDEs of NLS type
The quantum Yang–Baxter equation is one of the most fundamental equations in mathematical physicsand has applications in many diverse branches of physics and mathematics, including statistical mechan-ics, quantum field theories, the theory of knots and braids. Set-theoretical solutions of the quantum ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] V be a set. A YB map Y : V × V → V × V is a set-theoretical solution to the quantum YBequation Y ◦ Y ◦ Y = Y ◦ Y ◦ Y . (1)Here Y ij : V × V × V → V × V × V , i, j = 1 , , i < j , is the map acting as Y on the i th and j th factorsof V × V × V and acting as identity on the remaining factor.A parametric YB map Y a,b : V × V → V × V, Y a,b ( x, y ) = (cid:0) u a,b ( x, y ) , v a,b ( x, y ) (cid:1) , x, y ∈ V, a, b ∈ Ω , (2)depends on parameters a, b ∈ Ω from some parameter set Ω and obeys the parametric YB equation Y a,b ◦ Y a,c ◦ Y b,c = Y b,c ◦ Y a,c ◦ Y a,b for all a, b, c ∈ Ω . (3)The precise definition of such maps is given in Section 2.1.In this paper we mostly study the case when V is a vector space over a field K and for all a, b ∈ Ωthe map Y a,b given by (2) is K -linear. Note that usually Ω is an open subset of another vector space, andthe dependence of Y a,b on the parameters a, b is nonlinear. (See also Remark 2.1 on the case when Ω isan algebraic variety.)Examples of such maps related to integrable PDEs of Kadomtsev–Petviashvili (KP) and nonlinearSchr¨odinger (NLS) types can be found in [6, 16]. These examples are discussed in Example 2.12 and inSection 4.4 of the present paper. Remark 1.1.
A parametric YB map (2) with parameters a, b ∈ Ω can be interpreted as the followingYB map Y without parameters Y : (Ω × V ) × (Ω × V ) → (Ω × V ) × (Ω × V ) , Y (cid:0) ( a, x ) , ( b, y ) (cid:1) = (cid:0) ( a, u a,b ( x, y )) , ( b, v a,b ( x, y )) (cid:1) , (4)which satisfies π Y = π for the projection π : (Ω × V ) × (Ω × V ) → Ω × Ω. Thus one can say that westudy YB maps of the form (4) which are linear with respect to V and may be nonlinear with respectto Ω. However, often it is useful to keep a, b as parameters and to work with Y a,b .Note that we do not impose any nondegeneracy conditions on YB maps. In particular, the maps arenot required to be invertible.For a vector space V , a parametric map (2) is linear if and only if it is of the form Y a,b : V × V → V × V, Y a,b ( x, y ) = (cid:0) A a,b x + B a,b y, C a,b x + D a,b y (cid:1) , x, y ∈ V, a, b ∈ Ω , (5)for some linear maps A a,b , B a,b , C a,b , D a,b from V to V , which may depend nonlinearly on a, b ∈ Ω.2n Proposition 2.7 we prove that a linear parametric map (5) satisfies the parametric YB equation (3)(i.e., (5) is a parametric YB map) if and only if A a,b , B a,b , C a,b , D a,b obey the algebraic relations (19a)–(19e) for all values of the parameters a, b, c ∈ Ω. The case when A a,b , B a,b , C a,b , D a,b do not depend on a, b was studied in [4, 8] (see Remark 1.4 for more details).Let YB Ω ( V ) be the set of linear parametric YB maps (5). An element Y a,b ∈ YB Ω ( V ) given by (5) iswritten as Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ). In Theorem 2.8 we describe several transformationswhich allow one to obtain new such maps from a given one.From Theorem 2.8 we deduce Corollary 2.9, namely, if ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ), then forany nonzero constant l ∈ K one has ( lA a,b , l − D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ). Thus, from a given linearparametric YB map (5), we derive a family of linear parametric YB maps (29) depending on l .Theorem 2.10 contains a number of identities for linear parametric YB maps. As discussed in Re-mark 2.11, this clarifies the structure of relations (19a)–(19e).Applying Corollary 2.9 to the YB map (36) from [6], for each nonzero l ∈ C we obtain a linearparametric YB map (37), which for l = 1 is new.It is known that one can sometimes construct rational parametric YB maps from matrix refactorisationproblems corresponding to Darboux matrices for Lax operators (see, e.g., [16]). However, usually such arefactorisation problem gives an algebraic variety [16], and it is often difficult to represent the variety asthe graph of a rational map; namely, one cannot always solve the corresponding polynomial equations inorder to define a map. Then one can try to find a linear approximation to the latter. Developing an ideafrom [16] (see Remark 1.2 below), in Section 4 we study linear approximations of such refactorisationproblems and present examples where this gives linear parametric YB maps. We present several explicitexamples of this procedure for Lax operators of NLS type. It would be interesting to apply the describedmethod to Lax operators of other types. Remark 1.2.
The idea to use linear approximations of Darboux matrix refactorisation problems forobtaining linear parametric YB maps appeared in [16], where one example of such a map was obtained(the map (80) in the present paper).Developing this idea, in Section 4 we present a detailed description of a method to obtain such maps.In particular, in Remarks 4.1, 4.2, 4.3, using Proposition 3.2, we explain why the linear parametric mapsderived from the considered linear approximations of Darboux matrix refactorisation problems satisfythe parametric YB equation. In [16] this is not explained.As said above, the map (80) was obtained in [16]. Applying Corollary 2.9 to the map (80), for eachnonzero constant l ∈ C we derive the linear parametric YB map (81), which for l = 1 is new.In Section 3, after recalling the well-known relation between matrix refactorisation problems and YBmaps (see, e.g., [31, 29, 19]), in Proposition 3.1 we present a straightforward generalisation of this relationto the case of refactorisation problems in arbitrary associative algebras. Using Proposition 3.1, we proveProposition 3.2, which explains why the linear parametric maps arising from the linear approximationsof the matrix refactorisation problems considered in Section 4 obey the parametric YB equation (3) (seeRemarks 4.1, 4.2, 4.3).As shown in the proof of Proposition 3.2, for a given associative algebra B , this proposition dealswith refactorisation in the associative algebra of formal sums m + εm , where m , m ∈ B and ε is aformal symbol assumed to satisfy ε = 0.In Section 5 we present the following example of constructing linear parametric YB maps (withnonlinear dependence on parameters) from a nonlinear nonparametric YB map.For any group G , one has the nonlinear nonparametric YB map (82) from [7]. Taking G = GL n ( K ) ⊂ Mat n ( K ) for a positive integer n , for any abelian subgroup Ω ⊂ G and any nonzero constant l ∈ K we3onstruct from (82) a new linear parametric YB map (86). The construction consists of three steps: • First assume that K is either R or C . Then the set G × G = GL n ( K ) × GL n ( K ) is a manifold, theYB map F given by (82) is an analytic diffeomorphism, hence we can consider the differential d F of F , which is given by (83), where M = Mat n ( K ). Since F is a YB map, its differential d F is aYB map as well. • Let Ω ⊂ G be an abelian subgroup of G . Denote by Y the restriction of the map (83) to the subset(Ω × M ) × (Ω × M ) ⊂ ( G × M ) × ( G × M ). Computing Y , we derive the YB map (84), whichcan be interpreted as the linear parametric YB map (85) with parameters a, b ∈ Ω. To obtain (84)from (83), we use the fact that F ( a, b ) = ( a, aba − ) = ( a, b ) for any a, b ∈ Ω ⊂ G , because ab = ba . • For each nonzero l ∈ K , applying Corollary 2.9 to the map (85), we get the linear parametric YBmap (86) with nonlinear dependence on the parameters a, b . In the described construction of (86)we have assumed that K is either R or C , but one can check that (86) is a parametric YB map forany field K .Thus, (86) represents an infinite collection of new linear parametric YB maps corresponding to abeliansubgroups Ω of the matrix group GL n ( K ). A detailed description of this construction is given in Section 5.In Section 6, in the framework of fibre bundles and vector bundles, we discuss certain generalisationsof some notions and constructions considered in this paper.Section 7 concludes the paper with some remarks and comments on how the results of this paper canbe extended. Remark 1.3.
Recall that, for a set S , a map F : S → S is said to be involutive if F = id.In the study of the dynamics of a map ˜ F : S → S , one considers its powers ˜ F k : S → S for nonnegativeintegers k . From this point of view, noninvolutive maps are more interesting than involutive ones.Known examples of YB maps very often turn out to be involutive. The maps (37), (81), (86) arenoninvolutive. Remark 1.4.
As a preparation for the study of linear parametric YB maps, in Section 2.3 we recallsome results (mostly from [4]) on linear YB maps without parameters. In particular, the result ofProposition 2.2 on the structure of linear YB maps without parameters appeared indepedently in [4]and [8] in different notation described in Remark 2.3. As noticed in [8], the result of Proposition 2.2 canalso be easily deduced from formulas in [9].As discussed in Remark 2.13 and in Section 2.3, in the case when A a,b , B a,b , C a,b , D a,b do not dependon a, b some results of Section 2.4 reduce to ones from [4]. Let V be a set. A Yang–Baxter ( YB ) map is a map Y : V × V → V × V, Y ( x, y ) = ( u ( x, y ) , v ( x, y )) , x, y ∈ V, satisfying the YB equation (1). The terms Y , Y , Y in (1) are maps V × V × V → V × V × V defined as follows Y ( x, y, z ) = (cid:0) u ( x, y ) , v ( x, y ) , z (cid:1) , Y ( x, y, z ) = (cid:0) x, u ( y, z ) , v ( y, z ) (cid:1) , (6)4 ( x, y, z ) = (cid:0) u ( x, z ) , y, v ( x, z ) (cid:1) , x, y, z ∈ V. (7)A parametric YB map Y a,b is a family of maps (2) depending on two parameters a, b ∈ Ω from someparameter set Ω and satisfying the parametric YB equation (3). The terms Y a,b , Y a,c , Y b,c in (3) aremaps V × V × V → V × V × V defined similarly to (6), (7), adding the parameters a, b, c . For instance, Y a,c ( x, y, z ) = (cid:0) u a,c ( x, z ) , y, v a,c ( x, z ) (cid:1) , x, y, z ∈ V. In general, V and Ω can be arbitrary sets. See also Remark 2.1 on the case when Ω is an algebraic variety. Remark 2.1.
Suppose that Ω is an algebraic variety and consider the Zariski topology on the algebraicvariety Ω × Ω. Quite often one meets the following situation. A parametric map Y a,b depends rationallyon parameters a, b from Ω, and there is an open dense subset W ⊂ Ω × Ω such that Y a,b is defined for all( a, b ) ∈ W ⊂ Ω × Ω.If such Y a,b satisfies the parametric YB equation Y a,b ◦ Y a,c ◦ Y b,c = Y b,c ◦ Y a,c ◦ Y a,b for all points( a, b ), ( a, c ), ( b, c ) from W ⊂ Ω × Ω, then Y a,b can be called a parametric YB map. (Although Y a,b maybe undefined for some points ( a, b ) ∈ (Ω × Ω) \ W .) Results and methods of this paper are valid for suchmaps.For example, the YB map (80) from [16] is of this type, because (80) is undefined for ( a, b ) satisfying a + b = 0. Let V be a vector space over a field K . By End( V ) we denote the space of linear maps V → V . Thespace End( V ) is an associative algebra with respect to composition of maps.Let A, B, C, D ∈ End( V ). Consider the vector space V × V ∼ = V ⊕ V . The linear map Y : V × V → V × V, Y ( x, y ) = ( Ax + By, Cx + Dy ) , x, y ∈ V, is written as Y = ( A, D ; B, C ) ∈ End( V × V ), where End( V × V ) is the space of linear maps V × V → V × V . Z > is the set of positive integers. Let n ∈ Z > . For any commutative algebra A , we denote byMat n ( A ) the associative algebra of n × n matrices with entries from A . Let V be a vector space over a field K . In this subsection, we study the YB equation (1) for a linear map Y : V × V → V × V given by Y ( x, y ) = ( Ax + By, Cx + Dy ) , x, y ∈ V, A, B, C, D ∈ End( V ) . (8)So here we consider linear YB maps without parameters. Many examples of such maps can be foundin [4, 8]. Results presented in this subsection serve as a preparation for the study of linear parametricYB maps in Subsection 2.4.Let YB( V ) ⊂ End( V × V ) be the subset of maps Y ∈ End( V × V ) satisfying (1). An element Y ∈ YB( V ) given by (8) is written as Y = ( A, D ; B, C ) ∈ YB( V ).The result of Proposition 2.2 is presented in [4, 8] in different notation described in Remark 2.3 below.As noticed in [8], this result can also be easily deduced from formulas in [9].5 roposition 2.2 ([4, 8]) . A map Y ∈ End( V × V ) given by (8) satisfies (1) if and only if the maps A, B, C, D ∈ End( V ) in (8) obey the following relations C = C − DCA, B = B − ABD, (9a) DC − CD = DCB, AB − BA = ABC, (9b) CA − AC = BCA, BD − DB = CBD, (9c) DA − AD = BCB − CBC. (9d)
Thus, for
A, B, C, D ∈ End( V ) we have ( A, D ; B, C ) ∈ YB( V ) if and only if A , B , C , D obey rela-tions (9a) – (9d) . Remark 2.3.
In [4, 8] the following braid relation˜ Y ◦ ˜ Y ◦ ˜ Y = ˜ Y ◦ ˜ Y ◦ ˜ Y (10)is studied instead of the YB equation (1).Consider the permutation map P ∈ End( V × V ), P ( x, y ) = ( y, x ). A map Y ∈ End( V × V ) givenby (8) satisfies (1) if and only if the map ˜ Y = P Y satisfies (10).From (8) one obtains
P Y ( x, y ) = ( Cx + Dy, Ax + By ) for x, y ∈ V . Because of this, formulas (9a)–(9d)appear in [4, 8] in different notation: A is interchanged with C , and B is interchanged with D .Equations (11), (12), (13) below and the results of Proposition 2.6 appear in [4] with the same notationchange.The following relations are presented in [4] as consequences of relations (9a)–(9d)[ C + B − CB, D ] = 0 , [ C + B − BC, A ] = 0 , (11)[ D − BDC, A ] = [
BC, C + B ] , [ A − CAB, D ] = [
CB, C + B ] , (12) (cid:18) C DA B (cid:19) (cid:18)
C BDCA B (cid:19) = (cid:18) C BDCA B (cid:19) , (13)where [ · , · ] denotes the commutator of maps in End( V ). Note that relations (11) are equivalent to[( C − id)( B − id) , D ] = 0 , [( B − id)( C − id) , A ] = 0 , where id : V → V is the identity map.The result of Proposition 2.4 below is presented in [4], using the notation described in Remark 2.3.This result follows from Proposition 2.3. Proposition 2.4 ([4]) . For any vector space V over a field K , the set YB( V ) is invariant under thefollowing transformations ( A, D ; B, C ) ( D, A ; C, B ) , (14)( A, D ; B, C ) ( LA, DL − ; B, C ) for any invertible L ∈ End( V ) that commutes with B , C , AD. (15)
That is, if ( A, D ; B, C ) ∈ YB( V ) then ( D, A ; C, B ) ∈ YB( V ) and ( LA, DL − ; B, C ) ∈ YB( V ) .Let V = K n for some n ∈ Z > . The space End( K n ) is identified with Mat n ( K ) , and we have YB( K n ) ⊂ (cid:0) Mat n ( K ) (cid:1) . The set YB( K n ) is invariant also under the transformation ( A, D ; B, C ) ( D T , A T ; B T , C T ) , (16) where T denotes the transpose operation in Mat n ( K ) . emark 2.5. Taking the composition of the transformations (16), (14), we see that the set YB( K n ) isinvariant also under the transformation ( A, D ; B, C ) ( A T , D T ; C T , B T ).For completeness, we present a proof for the following result from [4], which is stated in [4] withoutproof as a consequence of (13). Proposition 2.6 ([4]) . Let n ∈ Z > . Let A, B, C, D ∈ Mat n ( K ) such that ( A, D ; B, C ) ∈ YB( K n ) .Then every nonzero column of the matrix (cid:18) C BDCA B (cid:19) is an eigenvector of the matrix (cid:18)
C DA B (cid:19) witheigenvalue .Consider the vector space W = (cid:26)(cid:18) xy (cid:19) ∈ K n × K n (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) C DA B (cid:19) (cid:18) xy (cid:19) = (cid:18) xy (cid:19)(cid:27) . We have dim W ≥ max(rank B, rank C ) .Furthermore, if (cid:18) C DA B (cid:19) = (cid:18) n n (cid:19) , then det (cid:18) C BDCA B (cid:19) = 0 . Here n is the n × n identitymatrix.Proof. The first statement follows immediately from (13).Thus every column of the matrix (cid:18)
C BDCA B (cid:19) belongs to W . This implies the second statement,since rank (cid:18) C BDCA B (cid:19) ≥ max(rank B, rank C ).If the matrix (cid:18) C BDCA B (cid:19) is invertible, then (13) yields (cid:18)
C DA B (cid:19) = (cid:18) n n (cid:19) . Therefore, if (cid:18) C DA B (cid:19) = (cid:18) n n (cid:19) , then det (cid:18) C BDCA B (cid:19) = 0.
Let V be a vector space over a field K . Let Ω be a set. Consider a family of linear maps Y a,b ∈ End( V × V )depending on parameters a, b ∈ Ω. One has Y a,b ( x, y ) = (cid:0) u a,b ( x, y ) , v a,b ( x, y ) (cid:1) , x, y ∈ V,u a,b ( x, y ) = A a,b x + B a,b y, v a,b ( x, y ) = C a,b x + D a,b y, (17)for some linear maps A a,b , B a,b , C a,b , D a,b ∈ End( V ) depending on a, b . As said in Section 2.1, Y a,b iscalled a parametric YB map if it satisfies the parametric YB equation (3).Let YB Ω ( V ) be the set of such linear parametric YB maps. An element Y a,b ∈ YB Ω ( V ) given by (17)is written as Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ).Substituting (17) in (3), we see that equation (3) is equivalent to the following system u a,b ( u a,c ( x, v b,c ( y, z )) , u b,c ( y, z )) = u a,c ( u a,b ( x, y ) , z ) , (18a) v a,b ( u a,c ( x, v b,c ( y, z )) , u b,c ( y, z )) = u b,c ( v a,b ( x, y ) , v a,c ( u a,b ( x, y ) , z )) , (18b) v a,c ( x, v b,c ( y, z )) = v b,c ( v a,b ( x, y ) , v a,c ( u a,b ( x, y ) , z )) , x, y, z ∈ V. (18c)Proposition 2.7 below can be regarded as a generalisation of Proposition 2.2 to the case of linearparametric YB maps. 7 roposition 2.7. A parametric family of maps Y a,b ∈ End( V × V ) given by (17) satisfies (3) if and onlyif A a,b , B a,b , C a,b , D a,b in (17) obey the following relations for all values of the parameters a, b, c ∈ Ω C b,c C a,b = C a,c − D b,c C a,c A a,b , B a,b B b,c = B a,c − A a,b B a,c D b,c , (19a) D a,c C b,c − C b,c D a,b = D b,c C a,c B a,b , A a,c B a,b − B a,b A b,c = A a,b B a,c C b,c , (19b) C a,b A a,c − A b,c C a,b = B b,c C a,c A a,b , B b,c D a,c − D a,b B b,c = C a,b B a,c D b,c , (19c) D a,b A b,c − B b,c C a,c B a,b = A b,c D a,b − C a,b B a,c C b,c , (19d)[ A a,b , A a,c ] = 0 , [ D a,c , D b,c ] = 0 . (19e) That is, for maps A a,b , B a,b , C a,b , D a,b ∈ End( V ) depending on a, b , we have ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) if and only if A a,b , B a,b , C a,b , D a,b obey relations (19a) – (19e) .Proof. According to the left-hand side of (18a), u a,b ( u a,c ( x, v b,c ( y, z )) , u b,c ( y, z )) = A a,b u a,c ( x, v b,c ( y, z )) + B a,b u b,c ( y, z ) == A a,b ( A a,c x + B a,c v b,c ( y, z )) + B a,b ( A b,c y + B b,c z ) == A a,b ( A a,c x + B a,c ( C b,c y + D b,c z )) + B a,b ( A b,c y + B b,c z ) == A a,b A a,c x + ( A a,b B a,c C b,c + B a,b A b,c ) y + ( A a,b B a,c D b,c + B a,b B b,c ) z. (20)On the other hand, from the right-hand side of (18a), we obtain u a,c ( u a,b ( x, y ) , z ) = A a,c u a,b ( x, y ) + B a,c z == A a,c ( A a,b x + B a,b y ) + B a,c z = A a,c A a,b x + A a,c B a,b y + B a,c z. (21)Comparing the coefficients of x , y , z in the right-hand sides of equations (20) and (21), we deduce thefirst relation of (19e), the second relation of (19b), and the second relation of (19a). Similarly, one canshow that the rest of relations (19a)-(19e) are equivalent to (18b)-(18c). Theorem 2.8.
Let V be a vector space over a field K and Ω be a set. Consider a linear parametric YBmap Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) given by (17) with parameters a, b ∈ Ω .Then ( D b,a , A b,a ; C b,a , B b,a ) ∈ YB Ω ( V ) . That is, the set YB Ω ( V ) is invariant under the transformation ( A a,b , D a,b ; B a,b , C a,b ) ( D b,a , A b,a ; C b,a , B b,a ) . (22) This means that, if we set ˜ A a,b = D b,a , ˜ B a,b = C b,a , ˜ C a,b = B b,a , ˜ D a,b = A b,a , (23) the maps ˜ A a,b , ˜ B a,b , ˜ C a,b , ˜ D a,b obey relations (19a) – (19e) .Furthermore, the set YB Ω ( V ) is invariant under the transformation ( A a,b , D a,b ; B a,b , C a,b ) ( LA a,b , D a,b L − ; B a,b , C a,b ) for any invertible L ∈ End( V ) that commutes with A a,b , B a,b , C a,b , D a,b for all a, b ∈ Ω . (24) Let V = K n for some n ∈ Z > . The space End( K n ) is identified with Mat n ( K ) . The set YB Ω ( K n ) isinvariant also under the transformations ( A a,b , D a,b ; B a,b , C a,b ) ( A T a,b , D T a,b ; C T a,b , B T a,b ) , (25)( A a,b , D a,b ; B a,b , C a,b ) ( D T b,a , A T b,a ; B T b,a , C T b,a ) , (26) where T denotes the transpose operation. roof. According to Proposition 2.7, for ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) we have (19a)–(19e). To provethat YB Ω ( V ) is invariant under the transformation (22), we need to show ( ˜ A a,b , ˜ D a,b ; ˜ B a,b , ˜ C a,b ) ∈ YB Ω ( V )for the maps (23).Since (19a) is valid for ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ), using (19a), (23), we obtain˜ B c,b ˜ B b,a = ˜ B c,a − ˜ A c,b ˜ B c,a ˜ D b,a , ˜ C b,a ˜ C c,b = ˜ C c,a − ˜ D b,a ˜ C c,a ˜ A c,b . (27)Since equations (27) are valid for all values of a, b, c , we can make the change a c, c a in (27) andget ˜ B a,b ˜ B b,c = ˜ B a,c − ˜ A a,b ˜ B a,c ˜ D b,c , ˜ C b,c ˜ C a,b = ˜ C a,c − ˜ D b,c ˜ C a,c ˜ A a,b . (28)Equations (28) say that ˜ A a,b , ˜ B a,b , ˜ C a,b , ˜ D a,b satisfy relations (19a). In much the same way, one canshow that ˜ A a,b , ˜ B a,b , ˜ C a,b , ˜ D a,b given by (23) satisfy all relations (19a)–(19e), i.e., ( ˜ A a,b , ˜ D a,b ; ˜ B a,b , ˜ C a,b ) ∈ YB Ω ( V ).The other statements of the theorem are proved similarly. Note that the transformation (26) is equalto the composition of the transformations (25), (22). Corollary 2.9.
Let ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) . Then for any nonzero l ∈ K one has ( lA a,b , l − D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) . (29) Thus we obtain a family of linear parametric YB maps (29) depending on l .Proof. Consider the identity map id : V → V . Using (24) for L = l · id ∈ End( V ), one gets (29).Since the values of the parameters a, b, c ∈ Ω are arbitrary, we are allowed to make any permutationof a, b, c in equations (19a)-(19e). Making the permutation a c , b a , c b in the first equationfrom (19a) and in the first equation from (19c), we obtain C a,b C c,a = C c,b − D a,b C c,b A c,a , (30) C c,a A c,b − A a,b C c,a = B a,b C c,b A c,a . (31) Theorem 2.10.
For any Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) given by (17) , we have (cid:18) C a,b D a,b A a,b B a,b (cid:19) (cid:18) C c,a B a,c D b,c C c,b A c,a B b,c (cid:19) = (cid:18) C c,b B b,c D a,c C c,a A c,b B a,c (cid:19) , (32) (cid:18) C b,c D b,c C a,c A c,a B c,b B c,a (cid:19) (cid:18) C a,b D a,b A a,b B a,b (cid:19) = (cid:18) C a,c D a,c C b,c A c,b B c,a B c,b (cid:19) . (33) Consider the map P ∈ End( V × V ) , P ( x, y ) = ( y, x ) , and the maps H a,b,c ∈ End( V × V ) , H a,b,c ( x, y ) = ( C c,a x + B a,c D b,c y, C c,b A c,a x + B b,c y ) , x, y ∈ V, ˜ H a,b,c ∈ End( V × V ) , ˜ H a,b,c ( x, y ) = ( C b,c x + D b,c C a,c y, A c,a B c,b x + B c,a y ) , x, y ∈ V, depending on parameters a, b, c ∈ Ω . Equations (32) , (33) say that P Y a,b H a,b,c = H b,a,c , (34)˜ H a,b,c P Y a,b = ˜ H b,a,c . (35)9 roof. Equation (32) follows from (19a), (19c), (30), (31). Equation (33) is proved similarly.Clearly, equations (32), (33) are equivalent to (34), (35).
Remark 2.11.
Equations (32), (33) are equivalent to (19a), (19b), (19c), up to permutations of a, b, c .Thus, the rather cumbersome equations (19a), (19b), (19c) can be replaced by equations (32), (33), whichhave more clear structure, since they are of the form (34), (35).
Example 2.12.
Let K = C , V = C , and Ω = C . In [6] one can find the following linear parametric YBmap Y a,b : C × C → C × C with a = ( a , a ) ∈ C and b = ( b , b ) ∈ C Y a,b (cid:18) xy (cid:19) = a − b a − b b − b a − b a − a a − b a − b a − b ! (cid:18) xy (cid:19) , x, y ∈ C , a = ( a , a ) , b = ( b , b ) . (36)In [6] the parameters ( a , a ), ( b , b ) are denoted by ( p , q ), ( p , q ). Note that Remark 2.1 is applicableto this map.Let l ∈ C , l = 0. Applying Corollary 2.9 to Y a,b = (cid:0) a − b a − b , a − b a − b ; b − b a − b , a − a a − b (cid:1) ∈ YB Ω ( C ), we obtainthe linear parametric YB map Y la,b : C × C → C × C , Y la,b (cid:18) xy (cid:19) = l ( a − b ) a − b b − b a − b a − a a − b a − b l ( a − b ) ! (cid:18) xy (cid:19) , a = ( a , a ) , b = ( b , b ) . (37)For l = 1 the map (37) is new. Remark 2.13.
Theorem 2.8 can be regarded as a generalisation of Proposition 2.4 and Remark 2.5 tothe case of linear parametric YB maps. However, if Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) given by (17)does not depend on the parameters a, b ∈ Ω, then the requirements on L ∈ End( V ) in (15) are less strictthan in (24), because in (15) the map L is not required to commute with A and D . (In (15) the map L commutes with B , C , and the product AD .)Note also that, if Y a,b = ( A a,b , D a,b ; B a,b , C a,b ) ∈ YB Ω ( V ) does not depend on a, b , then equation (32)reduces to (13). In this section, after recalling the well-known relation between matrix refactorisation problems and YBmaps (see, e.g., [31, 29, 19]), in Proposition 3.1 we present a straightforward generalisation of this relationto the case of refactorisation problems in arbitrary associative algebras. Using Proposition 3.1, we proveProposition 3.2, whose applications are discussed in Section 4. Namely, Proposition 3.2 explains whythe linear parametric maps arising from the linear approximations of the matrix refactorisation problemsconsidered in Section 4 satisfy the parametric YB equation (3) (see Remarks 4.1, 4.2, 4.3).Let V , Ω, Λ be sets and n ∈ Z > . Let L ( x ; a, λ ) be an n × n matrix depending on x ∈ V , a ∈ Ω, λ ∈ Λ. Here a , λ are regarded as parameters, and λ is called a spectral parameter. To simplify notation,we set L a ( x ) = L ( x ; a, λ ), so λ is not written explicitly in this notation.Consider a family of maps Y a,b : V × V → V × V, Y a,b ( x, y ) = (cid:0) u a,b ( x, y ) , v a,b ( x, y ) (cid:1) , x, y ∈ V, (38)depending on parameters a, b ∈ Ω. Suppose that u = u a,b ( x, y ) and v = v a,b ( x, y ) obey the equation L a ( u ) L b ( v ) = L b ( y ) L a ( x ) (39)10or all values of x, y, a, b, λ . Then, following [29], we say that L a ( x ) = L ( x ; a, λ ) is a Lax matrix for theparametric map (38). Equation (39) is called the matrix refactorisation problem corresponding to L a ( x ).Suppose that the equation L a (ˆ x ) L b (ˆ y ) L c (ˆ z ) = L a ( x ) L b ( y ) L c ( z ) for all a, b, c ∈ Ω (40)implies ˆ x = x , ˆ y = y , ˆ z = z . Then, if L a ( x ) is a Lax matrix for a parametric map (38), this map satisfiesthe parametric YB equation (3) (see [31, 19]).We need the following generalisation, where matrices are replaced by elements of an associativealgebra. Proposition 3.1.
Let V , Ω be sets and A be an associative algebra. Consider maps Q a : V → A , u a,b : V × V → V, v a,b : V × V → V depending on parameters a, b ∈ Ω . Suppose that • the equation Q a (ˆ x ) Q b (ˆ y ) Q c (ˆ z ) = Q a ( x ) Q b ( y ) Q c ( z ) for all a, b, c ∈ Ω (41) implies ˆ x = x , ˆ y = y , ˆ z = z , • we have Q a (cid:0) u a,b ( x, y ) (cid:1) Q b (cid:0) v a,b ( x, y ) (cid:1) = Q b ( y ) Q a ( x ) for all a, b ∈ Ω , x, y ∈ V. (42) Then the parametric map (38) satisfies the parametric YB equation (3) .Proof.
In the case when A is the algebra of n × n matrices depending on a spectral parameter, a proofof this statement is presented in [19]. The same proof works for arbitrary associative algebras. Proposition 3.2.
Let V , Ω be sets and B be an associative algebra. Consider maps T a : V → B , u a,b : V × V → V, v a,b : V × V → V depending on parameters a, b ∈ Ω . For each a ∈ Ω , let S a ∈ B such that S a S b = S b S a for all a, b ∈ Ω . (43) Suppose that • the equation T a (ˆ x ) S b S c + S a T b (ˆ y ) S c + S a S b T c (ˆ z ) = T a ( x ) S b S c + S a T b ( y ) S c + S a S b T c ( z ) for all a, b, c ∈ Ω(44) implies ˆ x = x , ˆ y = y , ˆ z = z , • we have S a T b (cid:0) v a,b ( x, y ) (cid:1) + T a (cid:0) u a,b ( x, y ) (cid:1) S b = S b T a ( x ) + T b ( y ) S a for all a, b ∈ Ω , x, y ∈ V. (45)11 hen the parametric map (38) satisfies the parametric YB equation (3) .Proof. Consider the formal symbol ε . Let A be the set of formal sums m + εm , where m , m ∈ B .The set A is an associative algebra with the following operationsfor all m , m , ˜ m , ˜ m ∈ B ( m + εm ) + ( ˜ m + ε ˜ m ) = ( m + ˜ m ) + ε ( m + ˜ m ) , ( m + εm )( ˜ m + ε ˜ m ) = ( m ˜ m ) + ε ( m ˜ m + m ˜ m ) . (46)Note that, for any m, ˜ m ∈ B , one has ( εm )( ε ˜ m ) = 0 in A . Thus essentially we assume ε = 0.For each a ∈ Ω, consider the map Q a : V → A , Q a ( w ) = S a + εT a ( w ) , w ∈ V. (47)Using (46), (47), we obtainfor all x, y ∈ V Q b ( y ) Q a ( x ) = S b S a + ε (cid:0) S b T a ( x ) + T b ( y ) S a (cid:1) ,Q a (cid:0) u a,b ( x, y ) (cid:1) Q b (cid:0) v a,b ( x, y ) (cid:1) = S a S b + ε (cid:0) S a T b (cid:0) v a,b ( x, y ) (cid:1) + T a (cid:0) u a,b ( x, y ) (cid:1) S b (cid:1) . Hence equations (43), (45) are equivalent to (42). Similarly, equation (44) is equivalent to (41). Therefore,we can use Proposition 3.1.
It is well known that YB maps are closely related to quad-graph equations (namely, partial differenceequations defined on an elementary square of the two-dimensional lattice), see, e.g., [2, 10, 25] andreferences therein. Recall that Darboux and B¨acklund transformations can be employed in order toderive quad-graph equations [1, 10, 17, 23, 27]. In particular, the Bianchi permutability of Darbouxtransformations yields integrable partial difference equations. Since the former permutability conditionof Darboux transformations can be regarded as a matrix refactorisation problem similar to those describedin Section 3, this suggests to consider matrix refactorisation problems for particular Darboux matricesin order to construct YB maps.In [16, 18], Darboux matrix refactorisation problems related to NLS-type partial differential equationswere considered, and new birational parametric YB maps were constructed. In this section, developingan idea from [16], we demonstrate how to obtain linear parametric YB maps (with nonlinear dependenceon parameters), using linear approximations of such matrix refactorisation problems. Relations of ourresults with those of [16] are discussed in Remark 1.2 and in Subsection 4.4.
The Darboux transformations which were employed in [16] are associated with AKNS-type Lax operatorsof the form L = D x + U , where U = U ( p, q ; λ ) is a 2 × p = p ( x, t )and q = q ( x, t ) are potential functions, which are solutions to a certain NLS-type equation, and λ ∈ C isa parameter called the spectral parameter.In the considered Lax operators, U depends rationally on λ , so U can be viewed as a function of p, q with values in the Lie algebra sl ( C ( λ )), where C ( λ ) is the commutative algebra of rational functionsof λ . 12ollowing [16, 17], we say that, in this case, a Darboux transformation is determined by an invertible2 × M (called a Darboux matrix ) such that M L M − = M (cid:0) D x + U ( p, q ; λ ) (cid:1) M − = D x − D x ( M ) M − + M U M − = D x + U (˜ p, ˜ q ; λ ) , (48)where functions ˜ p, ˜ q are also solutions of the same NLS-type equation. The matrix M may depend onthe potential functions p, q, ˜ p, ˜ q , the parameter λ , and some auxiliary functions.Since the matrix U has zero trace, from (48) we see that the trace of D x ( M ) M − is zero, which yields D x (det( M )) = 0. This condition implies some relations between the potential functions p, q, ˜ p, ˜ q and theauxiliary functions that appear in the derivation of the Darboux matrix. Usually these relations allowone to express the auxiliary functions in terms of the potential functions. Depending on the complexityof the relations, sometimes one cannot derive a map from the Darboux matrix refactorisation problem,but a correspondence. However, in such cases, one can often derive a linear approximation to the map.This will be clear in the following subsections where we construct examples of linear parametric YB mapsassociated with Darboux transformations for the derivative NLS (DNLS) equation and a deformation ofthe derivative NLS (DDNLS) equation which first appeared in [22].All the Darboux transformations that are being used in the next subsections were constructed in [17]and are associated with the following Lax operators L DNLS = D x + λ (cid:18) − (cid:19) + λ (cid:18) p q (cid:19) , L DDNLS = D x + λ (cid:18) − (cid:19) + λ (cid:18) p q (cid:19) + 1 λ (cid:18) q p (cid:19) − λ (cid:18) − (cid:19) , i.e., the spatial parts of the Lax pairs for the DNLS and DDNLS equations. A Darboux matrix associated to the DNLS equation is the following M = f (cid:18) λ (cid:18) (cid:19) + λ (cid:18) p ˜ q (cid:19)(cid:19) + (cid:18) (cid:19) , (49)where p, ˜ q are potential functions, solutions to the DNLS equation, and f is an arbitrary function whichappears in the derivation of the Darboux matrix M . Moreover, p, ˜ q, f satisfy a system of differential-difference equations [17], which admits the following first integralΦ = f p ˜ q − f. (50)That is, D x (Φ) = 0, which is equivalent to D x (det( M )) = 0.Therefore, we can impose the relation Φ = const, which allows us to determine the function f . Weset Φ = − a , where a ∈ C , and replace ( p, ˜ q ) → ( εx , εx ). Then (50) becomes − a = f x x ε − f. (51)Expanding in ε around 0, we consider equation (51) up to O ( ε ) and take f = a + O ( ε ).The matrix M in (49) now becomes M ( x , x , a ) = λ (cid:18) a
00 0 (cid:19) + (cid:18) (cid:19) + ελ (cid:18) ax ax (cid:19) + O ( ε ) . (52)13or the matrix M a ( x , x ) ≡ M ( x , x , a ) in (52), consider the matrix refactorisation problem M a ( u , u ) M b ( v , v ) = M b ( y , y ) M a ( x , x ) up to O ( ε ) . (53)After expanding (53) in ε , we compute the coefficients of ε (i.e., the terms of degree 1 in ε ), which givethe system of equations au + bv = ax + by , v = x , u = y , au + bv = ax + by . (54)It is easy to check that the terms of degree 0 in ε in equation (53) cancel. Therefore, equation (53)(considered up to O ( ε )) is equivalent to (54).System (54) can uniquely be solved for u , u , v , v , which gives the following map Y a,b x x y y = u u v v a,b ≡ − ba ba
00 0 0 11 0 0 00 ab − ab x x y y . (55)It is easy to check that the corresponding matrices A a,b = (cid:18) − ba
00 0 (cid:19) , B a,b = (cid:18) ba
00 1 (cid:19) ,C a,b = (cid:18) ba (cid:19) , D a,b = (cid:18) − ab (cid:19) obey relations (19a)–(19e). Therefore, (55) is a linear parametric YB map. In Remark 4.1 we explainthis by means of Proposition 3.2. Remark 4.1.
Let V = C , Ω = C , and B = Mat ( C [ λ ]). For a ∈ Ω = C , one can rewrite formula (52)as M ( x , x , a ) = S a + εT a ( x , x ) + O ( ε ), where S a = λ (cid:18) a
00 0 (cid:19) + (cid:18) (cid:19) ∈ B , T a : C → B , T a ( x , x ) = λ (cid:18) ax ax (cid:19) , x , x ∈ C . (56)Since we make computations up to O ( ε ), we can use formula (46). Proposition 3.2 is applicable here,which explains why the map (55) obtained from (53), where M a ( x , x ) = M ( x , x , a ), satisfies theparametric YB equation (3).Equation (53) is studied up to O ( ε ) and is equivalent to linear equations (54). This allows one to saythat (53) can be regarded as a linear approximation of the matrix refactorisation problem correspondingto the matrix-function (49).Recall that the formula M ( x , x , a ) = S a + εT a ( x , x ) + O ( ε ) with S a , T a given by (56) is obtainedfrom (49), using the substitution ( p, ˜ q ) → ( εx , εx ). We use also the formula f = a + O ( ε ) suggestedby equation (51), which is obtained from the equation Φ = − a by this substitution. The substitution ischosen so that the resulting S a , T a obey the conditions of Proposition 3.2. In particular, since the matrix S a in (56) is diagonal, we have (43).The map (55) can be derived from the map (36) as follows. Substituting a = b = 0 in (36) anddenoting a = a , b = b , we obtain the map˜ Y a,b : C × C → C × C , ˜ Y a,b (cid:18) xy (cid:19) = a − ba ba ! (cid:18) xy (cid:19) , x, y ∈ C . (57)14ubstituting a = b = 0 in (36) and denoting a = a , b = b , we obtain the mapˆ Y a,b : C × C → C × C , ˆ Y a,b (cid:18) xy (cid:19) = ab b − ab ! (cid:18) xy (cid:19) , x, y ∈ C . (58)The map (55) is equal to the direct sum of the maps (57), (58). A Darboux matrix associated to the DDNLS equation is given by M = f (cid:18) λ λ − (cid:19) + λ (cid:18) f pf ˜ q (cid:19) + f g (cid:18) (cid:19) + 1 λ (cid:18) f ˜ qf p (cid:19) , (59)where p, ˜ q, f, g satisfy a system of differential-difference equations [17], which admits two first integralsΦ , Φ Φ = f ( g − p ˜ q ) , Φ = f ( g + 1 − p − ˜ q ) , (60)i.e., D x (Φ i ) = 0, i = 1 ,
2. The latter guarantees that D x (det( M )) = 0.Hence we can impose the relations Φ = c and Φ = c for constants c , c ∈ C , which allows usto determine the functions f , g . As noticed in [16], it is convenient to consider the case c = − k , c = k for a constant k ∈ C . In the obtained relations Φ = − k , Φ = k and in (59) we replace( p, ˜ q ) → ( εx , εx ), which gives f ( g − x x ε ) = 1 − k , f ( g + 1 − x ε − x ε ) = 1 + k , (61) M = f (cid:18) λ λ − (cid:19) + λ (cid:18) f εx f εx (cid:19) + f g (cid:18) (cid:19) + 1 λ (cid:18) f εx f εx (cid:19) . (62)Expanding in ε around 0, we solve equations (61) for f , g up to O ( ε ) and obtain 4 cases f = 1 − k O ( ε ) , g = 1 + k − k + O ( ε ) , (63) f = k −
12 + O ( ε ) , g = 1 + k − k + O ( ε ) , (64) f = 1 + k O ( ε ) , g = 1 − k k + O ( ε ) , (65) f = − k O ( ε ) , g = 1 − k k + O ( ε ) . (66)Let us study first the case (63). Substituting (63) in (62) and denoting the obtained matrix by M k ( x , x ), one derives the formula M k ( x , x ) = 1 − k (cid:18) λ λ − (cid:19) + 1 + k (cid:18) (cid:19) + ε − k (cid:18) λx + λ − x λx + λ − x (cid:19) + O ( ε ) . (67)For the matrix M k ( x , x ) in (67), consider the following matrix refactorisation problem M a ( u , u ) M b ( v , v ) = M b ( y , y ) M a ( x , x ) up to O ( ε ) . (68)15fter expanding (68) in ε , we consider the coefficients of ε , which give the system of equations u = y , v = x , (69)( a − b − u − ( a − b + 1) u − ( a + 1)( b − v == ( a − b − y − ( a + 1)( b − y − ( b − a + 1) x , (70)( a − b + 1) u +( a + 1)( b − v − ( a − b − v == ( a + 1)( b − y + ( a − b + 1) x − ( a − b − x . (71)The above system can uniquely be solved for u , u , v , v , which gives the map Y a,b x x y y = u u v v a,b ≡ ( a +1)( a − b )( a − a + b ) a − ba + b a ( b − a − a + b ) − a − ba + b a − ba + b a − b ( b − a + b ) − a − ba + b − ( a − b )( b +1)( b − a + b ) x x y y . (72)The matrix representing the linear map (72) satisfies the relations of Proposition 2.7. Namely, thecorresponding matrices A a,b = ( a +1)( a − b )( a − a + b ) a − ba + b ! , B a,b = a ( b − a − a + b ) − a − ba + b ! ,C a,b = a − ba + b a − b ( b − a + b ) ! , D a,b = − a − ba + b − ( a − b )( b +1)( b − a + b ) ! obey relations (19a)–(19e). Hence (72) is a linear parametric YB map. As shown in Remark 4.4, themap (72) is equivalent to the YB map (80), which appeared in [16]. Remark 4.2.
Similarly to Remark 4.1, the fact that the map (72) satisfies the parametric YB equation (3)is explained by Proposition 3.2.Equation (68) is studied up to O ( ε ) and is equivalent to linear equations (69), (70), (71). This allowsone to say that (68) can be regarded as a linear approximation of the matrix refactorisation problemcorresponding to the matrix-function (59).So in the case (63), using equation (68), we have obtained the map (72). The case (64) is obtainedfrom (63) by the change f
7→ − f . Since the matrix M in (62) is of the form M = f N ( x , x , g, λ, ε )for some matrix N ( x , x , g, λ, ε ), the change f
7→ − f does not affect equation (68), hence the case (64)gives the same map (72).The case (65) is obtained from (63) by the change k
7→ − k , hence in this case we need to make thechange k
7→ − k in the right-hand side of (67). Then the above procedure gives the map (72) with a, b replaced by − a, − b .The case (66) is obtained from (63) by the changes f
7→ − f and k
7→ − k . By the above arguments,in this case we get the map (72) with a, b replaced by − a, − b . Following [16], consider again the relations Φ = − k , Φ = k for a constant k ∈ C , where Φ i , i = 1 , = − k , Φ = k and in (59) we replace ( f p, f ˜ q ) → ( εx , εx ),16hich gives f g − x x ε = 1 − k , f g + f − x ε − x ε = 1 + k , (73) M = f (cid:18) λ λ − (cid:19) + λ (cid:18) εx εx (cid:19) + f g (cid:18) (cid:19) + 1 λ (cid:18) εx εx (cid:19) . (74)Expanding in ε around 0, we now solve equations (73) for f , f g up to O ( ε ) and obtain 4 cases f = 1 + k O ( ε ) , f g = 1 − k O ( ε ) , (75) f = − k O ( ε ) , f g = k −
12 + O ( ε ) , (76) f = 1 − k O ( ε ) , f g = 1 + k O ( ε ) , (77) f = k −
12 + O ( ε ) , f g = − k O ( ε ) . (78)Consider the case (75). Substituting (75) in (74) and denoting the obtained matrix by M k ( x , x ),we derive the formula M k ( x , x ) = 1 + k (cid:18) λ λ − (cid:19) + 1 − k (cid:18) (cid:19) + ε (cid:18) λx + λ − x λx + λ − x (cid:19) + O ( ε ) . (79)By the same procedure as in Subsection 4.3, considering equation (68) for the matrix (79), one obtainsthe following linear parametric YB map Y a,b x x y y = u u v v a,b ≡ ( a − a − b )( a +1)( a + b ) a − ba + b aa + b − ( a +1)( a − b )( b +1)( a + b ) a +1 b +1 b +1 a +1 ( a − b )( b +1)( a +1)( a + b ) 2 ba + b − a − ba + b − ( a − b )( b − b +1)( a + b ) x x y y , (80)which appeared in [16]. Remark 4.3.
As said above, the linear map (80) is obtained from equation (68) for (79). Similarly toRemarks 4.1, 4.2, the fact that the map (80) satisfies the parametric YB equation (3) is explained byProposition 3.2.In the case (76) one obtains the same map. In the cases (77), (78) one derives the map (80) with a, b replaced by − a, − b .Let l ∈ C , l = 0. Applying Corollary 2.9 to the map (80), we get the map Y la,b : C × C → C × C , Y la,b x x y y = l ( a − a − b )( a +1)( a + b ) l ( a − b ) a + b aa + b − ( a +1)( a − b )( b +1)( a + b ) a +1 b +1 b +1 a +1 ( a − b )( b +1)( a +1)( a + b ) 2 ba + b − a − bl ( a + b ) − ( a − b )( b − l ( b +1)( a + b ) x x y y . (81)For l = 1 the YB map (81) is new. 17 emark 4.4. The following observation was brought to us by A.V. Mikhailov. Consider the matrix M a,b = ( a +1)( a − b )( a − a + b ) a − ba + b a ( b − a − a + b ) − a − ba + b a − ba + b a − b ( b − a + b ) − a − ba + b − ( a − b )( b +1)( b − a + b ) from (72) and the diagonal matrix U a,b = diag( a − , a − , b − , b − U a,b M a,b U − a,b is equalto the 4 × a
7→ − a , b
7→ − b . This implies that the linear parametricYB maps (72) and (80) are equivalent up to a change of the basis in C × C and the change a
7→ − a , b
7→ − b . Let G be a group. One has the following YB map [7] F : G × G → G × G , F ( x, y ) = ( x, xyx − ) , x, y ∈ G . (82)Assume that K is either R or C . Let n ∈ Z > and consider the group G = GL n ( K ) ⊂ Mat n ( K ).Then G is a manifold, and for each x ∈ G = GL n ( K ) one has the tangent space T x G ∼ = Mat n ( K ).Set M = Mat n ( K ). The tangent bundle of the manifold G can be identified with the trivial bundle G × M → G .For G = GL n ( K ), the YB map (82) is an analytic diffeomorphism of the manifold G × G . Thedifferential d F of this diffeomorphism F can be identified with the following mapd F : ( G × M ) × ( G × M ) → ( G × M ) × ( G × M ) , d F (cid:0) ( x, M ) , ( y, M ) (cid:1) = (cid:18)(cid:16) x, M (cid:17) , (cid:16) xyx − , ∂∂ε (cid:12)(cid:12)(cid:12) ε =0 (cid:0) ( x + εM )( y + εM )( x + εM ) − (cid:1)(cid:17)(cid:19) , (83) x, y ∈ G = GL n ( K ) , M , M ∈ M = Mat n ( K ) . Since F is a YB map, its differential d F is a YB map as well.Let Ω ⊂ G be an abelian subgroup of G . Denote by Y : (Ω × M ) × (Ω × M ) → (Ω × M ) × (Ω × M )the restriction of the map d F to the subset (Ω × M ) × (Ω × M ) ⊂ ( G × M ) × ( G × M ). As d F is a YBmap, Y is a YB map as well.Let a, b ∈ Ω. Since ab = ba , computing (83) for x = a and y = b , we obtain Y : (cid:0) Ω × Mat n ( K ) (cid:1) × (cid:0) Ω × Mat n ( K ) (cid:1) → (cid:0) Ω × Mat n ( K ) (cid:1) × (cid:0) Ω × Mat n ( K ) (cid:1) , Y (cid:0) ( a, M ) , ( b, M ) (cid:1) = (cid:0) ( a, M ) , ( b, aM a − − bM a − + M ba − ) (cid:1) . (84)Similarly to Remark 1.1, the YB map (84) can be interpreted as the following linear parametric YB map Y a,b : Mat n ( K ) × Mat n ( K ) → Mat n ( K ) × Mat n ( K ) ,Y a,b ( M , M ) = ( M , aM a − − bM a − + M ba − ) , (85)with parameters a, b ∈ Ω. 18et l ∈ K , l = 0. Applying Corollary 2.9 to the map (85), we obtain the linear parametric YB map Y la,b : Mat n ( K ) × Mat n ( K ) → Mat n ( K ) × Mat n ( K ) ,Y la,b ( M , M ) = ( lM , l − aM a − − bM a − + M ba − ) , (86) a, b ∈ Ω , Ω is an abelian subgroup of GL n ( K ) . In the above construction of (86) we have assumed that K is either R or C , in order to use tangent spacesand differentials. Now one can check that (86) is a parametric YB map for any field K . In this section we introduce certain generalisations of some notions and constructions considered in thepaper.Let E , Ω be topological spaces and ψ : E → Ω be a fibre bundle. Consider a YB map Y : E × E → E × E satisfying π Y = π , where π = ψ × ψ : E × E → Ω × Ω . (87)Such YB maps generalise parametric YB maps discussed in Remark 1.1, which correspond to the case ofthe trivial bundle Ω × V → V , where Ω and V are topological spaces.Suppose that ψ : E → Ω is a vector bundle and consider the vector bundle ψ × ψ : E × E → Ω × Ω.Suppose further that a YB map Y : E × E → E × E satisfies π Y = π for π given by (87), and Y is linearalong the fibres of ψ × ψ . Such YB maps generalise linear parametric YB maps, which correspond to thecase of the trivial vector bundle Ω × V → Ω, where V is a vector space.Let M be a (smooth or complex-analytic) manifold and Y : M × M → M × M be a (smooth orcomplex-analytic) YB map. Consider the tangent bundle τ : T M → M of M and the differentiald Y : T M × T M → T M × T M of the map Y . Since Y is a YB map, its differential d Y is a YB map as well. (This follows from generalproperties of the differentials of smooth and analytic maps.)For any subset S ⊂ M satisfying Y ( S × S ) ⊂ S × S , one hasd Y (cid:0) τ − ( S ) × τ − ( S ) (cid:1) ⊂ τ − ( S ) × τ − ( S ) , and the restriction of the YB map d Y to the subset τ − ( S ) × τ − ( S ) ⊂ T M × T M is a YB map as well.This generalises the construction from Section 5, which corresponds to the case when M = G = GL n ( K ),where K is R or C , the map Y = F is given by (82), and S = Ω is an abelian subgroup of the group M = GL n ( K ).We plan to study these generalisations in future works. In this paper we have presented a number of general results on linear parametric YB maps, including clar-ification of the structure of the nonlinear algebraic relations that define them and several transformationswhich allow one to obtain new such maps from known ones. Also, methods to construct such maps havebeen described. In particular, we have demonstrated how to obtain linear parametric YB maps (with19onlinear dependence on parameters) from nonlinear Darboux transformations of some Lax operators,using linear approximations of matrix refactorisation problems corresponding to Darboux matrices.As illustrative examples, the following new linear parametric YB maps with nonlinear dependence onparameters have been presented. • For each nonzero constant l ∈ C , we have the linear parametric YB maps (37), (81) with parame-ters a, b . For l = 1 these maps are new. For l = 1 the map (37) appeared in [6] and (81) appearedin [16]. • The map (86) is new for each nonzero constant l ∈ K , where K is any field (e.g., K = R or K = C ).Note that (86) actually represents an infinite collection of linear parametric YB maps correspondingto abelian subgroups Ω of the matrix groups GL n ( K ), n ∈ Z > .As discussed in Remark 1.3, in terms of dynamics, noninvolutive maps are more interestingthan involutive ones, but known examples of YB maps very often turn out to be involutive. Themaps (37), (81), (86) are noninvolutive.In Section 6, using fibre bundles and vector bundles, we have introduced certain generalisations ofsome notions and constructions considered in the paper. We plan to study these generalisations in futureworks.Also, motivated by the results of this paper, we suggest the following directions for future research: • Develop similar approaches for entwining Yang–Baxter maps which are set-theoretical solutions tothe parametric, entwining Yang–Baxter equation which reads S a,b ◦ R a,c ◦ T b,c = T b,c ◦ R a,c ◦ S a,b . (88)Here S , R , T are maps from V × V to V × V depending on two parameters from Ω for some sets V , Ω. The maps S a,b , R a,c , T b,c from V × V × V to V × V × V with parameters a, b, c ∈ Ω areconstructed from S , T , R in the standard way (see, e.g., [20, 12, 18]). If S = R = T ≡ Y , thenequation (88) becomes the parametric YB equation (3). • Extend the methods of this paper to the case of the functional (Zamolodchikov’s) Tetrahedronequation, which can be regarded as a higher-dimensional generalisation of the (parametric) YBequation. A number of interesting results on relations of the functional Tetrahedron equation tointegrable systems are known (see, e.g., [6, 11, 14, 15, 30] and references therein); however, it hasnot yet attracted the same attention as the YB equation.
Acknowledgements
This work is supported by the Russian Science Foundation (grant No. 20-71-10110).We would like to thank A.V. Mikhailov and D.V. Talalaev for useful discussions.
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