On the $3D$ consistency of a Grassmann extended lattice Boussinesq system

In this paper, we formulate a"Grassmann extension"scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems of P$\Delta$Es, based on the ideas presented in \cite{Sokor-Kouloukas}. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its $3D$-consistent limit. We show that some systems retain their $3D$-consistency property in their Grassmann extension.


Introduction
Over the past few decades, there has been an increasing interest in the study of noncommutative extensions of integrable equations or systems of equations (indicatively we refer to [7,8,11,12,18,24]), due to their numerous applications in Physics. Famous examples include noncommutative analogues of the KdV, the NLS, the sine-Gordon and other well-celebrated equations of Mathematical Physics. Therefore, it is quite important to develop methods for solving such -noncommutative -systems.
On the other hand, in the commutative case, plenty of methods have been discovered for solving discrete integrable systems (see [13] and the references therein). One of the most well-studied and important class of such systems are the so-called "quad-graph systems", namely systems of difference equations defined on an elementary quadrilateral of the two-dimensional lattice. For those quad-graph systems which possess the "3D consistency" property, Bäcklund transformations can be derived automatically, and therefore interesting solutions can be constructed starting from trivial ones. Due to the useful properties of 3D consistent quad-graph systems and the availability of simple algebraic schemes for contructing solutions to them, they can be used as good models for studying their continuous analogues, i.e. systems of nonlinear PDEs, via continuum limits. At the same time, 3D consistent quad-graph systems are strongly related to Yang-Baxter maps, namely solutions to the set-theoretical Yang-Baxter equation, one of the most fundamental equation of Mathematical Physics. This is a quite important connection, and a lot of work has been done in this direction (indicatively we refer to [1,6,14,26,27]).
The importance of noncommutative extensions of integrable systems from a Physics perspective, and the innovating results that have already been obtained in the continuously developing field of Discrete Integrable Systems, motivates us to extend to the noncommutative case the already existing methods for constructing solutions to integrable systems in the commutative case. Towards this direction, a few steps have been made over the past few years. In particular, in the recent work of Grahovksi and Mikhailov [10], integrable discretisations were found for a class of NLS equations on Grassmann algebras. This motivated the construction of Grassmann extended systems of differential-difference and difference-difference equations [31,32,33], as well as the consideration of continuum limits of Grassmann extended difference equations (see, for instance, [19,20]). Furthermore, the latter results and the aforementioned strict relation between quad-graph systems and Yang-Baxter maps motivated the beginning of the extension of the theory of Yang-Baxter maps on Grassmann algebras [9,16]. In addition, a Grassmann extension of the discrete potential KdV equation together with its associate Yang-Baxter map were constructed in [15].
In this paper, motivated by the above-mentioned developments and the results obtained in [15], we formulate a scheme for constructing noncommutative (Grassmann) extensions of quad-graph systems together with their associated Grassmann extended Yang-Baxter maps. Moreover, we answer the main question which arose in [15] on whether the noncommutativity "kills" the 3D consistency property for all quad-graph systems. In particular, the Grassmann extended discrete potential KdV system which was constructed in [15] does not have the 3D consistency property. However, this in not the case for all the Grassmann extended integrable systems; in fact, in this paper, we construct a Grassmann extension of a Boussinesq system which retains the 3D consistency of its original, commutative version.
As an illustrative example for the description of our scheme, we consider a discrete Boussinesq system. The Boussinesq equation, in both its continuous and its discrete (lattice Boussinesq) version, has been studied extensively over the past few decades, earning its place on the list of fundamental equations of Mathematical Physics. It owes its popularity to its quite interesting and, also, simple form, with a number of applications in Fluid Dynamics and in the theory of Integrable Systems.

Main results
This paper is concerned with the formulation of a scheme for constructing Grassmann extensions of quadgraph systems and their associated Yang-Baxter maps. The methods in this scheme are demonstrated via the following Boussinesq system of difference equations for p n,m = p(n, m), q n,m = q(n, m), n, m ∈ N: where a, b ∈ C (see various forms of this system [5,21,29]). In fact, we construct and study the integrability of its noncommutive extension not only in terms of possessing a Lax representation, but also as a 3D consistent system [4,22]. We also derive the associated Yang-Baxter map.
To conclude, we state what is new in this paper:

Notation
Here, we explain the notation we shall be using throughout the text.

Functions of discrete variables and shifts
Let f be a function of two discrete variables n and m, i.e f = f (n, m). Let also S and T be the shift operators in the n and m direction of a two-dimensional lattice, respectively. We adopt the notation: for example, f 10 = f (n + 1, m), f 01 = f (n, m + 1) and f 11 = w(n + 1, m + 1) as represented in Figure 1. Furthermore, if our field f lives on the three-dimensional lattice, namely f = f (n, m, k), and Z is the shift operator in the k-direction, then we shall be using three indices to determine the position of f on the lattice. That is, For instance, f 101 = w(n + 1, m, k + 1) as in Figure 1.

Commutative and anticommutative variables
We shall be using Latin letters for all commuting variables, whereas all the anticommutative variables will be denoted by Greek letters. For instance, pq = qp, whereas τ θ = −θτ . As an exception, the spectral parameter, λ ∈ C, is a commuting variable.

3D consistency VS the Yang-Baxter equation
"Quad-graph" equations (or systems) and "Yang-Baxter maps" constitute the two sides of the same coin. In this section, we explain the relation between the 3D consistency property and the Yang-Baxter equation.

Quad-graph equations and parametric Yang-Baxter maps
Using the notation introduced in section 2.1.1, let the fields (f , f 10 , f 01 , f 11 ) lie on the vertices of the square in Figure 1. Let us also consider the following equation where the parameters a, b ∈ C and Q is a linear function in every field f ij . Equation (2) is called equation on quad-graph and can be interpreted as in Figure 1-(a). That is, knowing any 3 of the fields f ij on the vertices, one can uniquely identify the fourth, using (2). Now, by the term "parametric Yang-Baxter map" we understand set-theoretical solutions of the parametric Yang-Baxter equation, namely maps Y a,b ∈ End(V × V ), where V is algebraic variety, i.e.
satisfying the parametric Yang-Baxter equation The Y ij ∈ End(V ×V ×V ) are defined as: where π 12 is the involution defined by π 12 ((x; a), (y; b), (z; c)) = ((y; b), (x; a), (z; c)). The geometric interpretation of these maps, can be understood in a similar way as for quad-graph equations, but with the values being considered on the edges of the quad, as in Figure 1- Similarly to quad-graph systems, we can also interpret Yang-Baxter maps on the square, but considering the values on the edges instead of the vertices (see Figure 2-(b)).

Lax representations & integrability
where S and T are shift operators, as defined in section 2.1.1.
Similarly, for Yang-Baxter maps, Lax matrix is a matrix L = L(x, a; λ) ≡ L a (x) that satisfies the following matrix refactorisation problem [28] If equation (5) defines a map (3), then it is called Lax representation of the map. An alternative way to verify that a map satisfies the Yang-Baxter equation is to consider the following matrix trifactorisation problem where L a (x) is the same matrix satisfying (5). In particular, if the above trifactorisation problem implies that u = x, v = y and w = z, then map (3) defined by (5) satisfies the parametric Yang-Baxter equation (4) [17,30].
In the case of quad-graph equations or systems as (2), the possession of Lax representation is usually used as working definition of integrability. However, a stronger integrability criterion is that of 3Dconsistency [4,22] which implies integrability in the sense of Lax representation.
From the analysis-point-of-view, 3D-consistency is the property of equation (2) to be consistently generalisable in three dimensions, by "adding" a third discrete variable k in the field f , namely considering f = f (n, m, k). Geometrically, it means that a quad-graph system-as interpreted in Figure 1-(a)-can be generalised and "written" in a consistent way on the faces of the cube of Figure 1-(b). That is, we first rewrite our system (2) on the bottom, front and left side of the cube, respectively, as follows: Then, considering f , f 100 , f 010 and f 001 as initial values on the cube in Figure 1-(b), there are three ways to calculate f 111 : 1) Using the first equation of (6), determine f 110 ; 2) Using the second equation, determine f 101 ; 3) With use of the last equation of (2), determine f 011 . Consequently, having f 110 , f 101 and f 011 at our disposal, we can determine f 111 , using any of the top, back or right side of the cube. 3D-consistency means that, independently of which of the former sides we use, we obtain exactly the same value f 111 .
The strict relation between the 3D-consistency property and the Yang-Baxter equation can be demonstrated in Figure 3. In fact, one can consider three initial values x, y and z taken on the sides of the cube as in Figure 3. Now, acting on (x, y, z) with the left part of the Yang-Baxter equation, that is, using the bottom, back and left side of the cube, we obtain new values (x,ŷ,ẑ). On the other hand, acting on (x, y, z) with the right part of the Yang-Baxter equation, namely via the left, front and top side of the cube, we obtain the values (x,ỹ,z). The Yang-Baxter equation is satisfied when the "hats" coincide with the "tildes" and vice versa.

Grassmann algebra
The elements of G that belong either to G 0 or to G 1 are called homogeneous, the ones in G 1 are called odd (or fermionic), while those in G 0 are called even (or bosonic).
By definition, the parity |a| of an even homogeneous element a is 0, while it is 1 for odd homogeneous elements. The parity of the product |ab| of two homogeneous elements is a sum of their parities: |ab| = |a| + |b|. Now, for any homogeneous elements a and b, Grassmann commutativity means that ba = (−1) |a||b| ab . This implies that if α ∈ G 1 , then α 2 = 0, and αa = aα, for any a ∈ G 0 .
The notions of the determinant and the trace of a matrix in G are defined for square matrices, M , of the following block-form The blocks P and L are matrices with even entries, while Π and Λ possess only odd entries (note that the block matrices are not necessarily square matrices). In particular, the superdeterminant of M , which is usually denoted by sdet(M ), is defined to be the following quantity where det(·) is the usual determinant of a matrix.
In this section, we gave all the definitions related to Grassmann algebras that are essential for this paper. However, for more information on Grassmann analysis one can consult [3].

Grassmann extension scheme
Here, we demonstrate a scheme for constructing Grassmann extensions of discrete integrable systems together with their associated Grassmann extended Yang-Baxter maps. We formulate the ideas presented in [15] which constitute a combination of the methods introduced in [25] and [9].

Grassmann extension
Grassmann extension of Yang-Baxter Lift to Yang-Baxter Squeeze down Figure 4: Grassmann extension scheme The scheme consists of three steps: I. Starting from an integrable quad-graph equation, Q a,b (f, f 10 , f 01 , f 11 ) = 0, derive a Yang-Baxter map using the symmetries of the equation in order to transfer from a one-field equation (with field f ) to a two-field map (with fields u and v) [25], namely a map (x, y) Note that this method is reversible, when applied to equations of certain form. II. The method was introduced in [9, 10]. Starting from map Y a,b obtained in Step I, construct its . This extension is applicable to Yang-Baxter maps which admit Lax matrix, and it is based on the consideration of a more general Lax matrix which includes anticommutative variables. That is, we consider an augmented Lax matrix, L a = L a (x, χ), which contains the old (bosonic) "x" and the new (fermionic) "χ" variables. Our demand is that this matrix satisfies two conditions: 1. Its bosonic limit is equal to the original Lax matrix (with only bosonic elements); that is, lim χ→0 L a (x, χ) = L a (x). 2. Its superdeterminant is equal to the determinant of the original Lax matrix, i.e. sdet L a = det L a . Note that the augmented matrix L a must be in the block-form (7) in order to be able to define its determinant. This method will be demonstrated in the next section for our map.

III. Since
Step I is reversible, we can apply the reverse idea to the map S a,b obtained in Step II in order to "squeeze it down" to a lattice equation Q a,b (f, f 10 , f 01 , f 11 , φ, φ 10 , φ 01 , φ 11 ) = 0, such that lim (φ,φ 10 ,φ 01 ,φ 11 )→0 Q a,b = Q a,b . To do so, we use some symmetries of map S a,b . The derived Grassmann extended quad-graph system, Q a,b = 0, is by definition integrable, since it has a Lax representation. Its Lax representation can be derived from the matrix refactorisation problem associated with the Grassmann extended Yang-Baxter map S a,b , by relabeling the variables (matrix entries).

Boussinesq system and a lift to a Boussinesq type Yang-Baxter map
In this section, starting from a Boussinesq lattice system, we construct its associated Yang-Baxter lift.

Step I : Lift to a Yang-Baxter map
Our aim is to derive a Yang-Baxter map starting from (8). The idea is to move from the fields (p, q, r) (functions of two discrete variables n, m ∈ N) to elements of an algebraic variety V . The right change of variables is indicated by the Lax representation (10) itself.
In particular, comparing (9) to the following matrix refactorisation problem we set x 1 = p, x 2 = q, x 3 = Sq and x 4 = Sr, namely we consider the following 3 × 3 matrix Here, we understand x i , i = 1, . . . , 4, as elements of an algebraic variety V , and we substitute (12) to (11). Then, (11) implies a correspondence given by This correspondence is a solution of (11) for any v 1 . For a particular value of v 1 , the above correspondence defines the following eight-dimensional map; in fact, we have the following.
given by is an eight-dimensional parametric Yang-Baxter map with invariants Proof. For the particular choice v 1 = x 1 , the correspondence (13) implies the eight-dimensional map (14)- (15). The Yang-Baxter property of the latter can be shown by straightforward substitution of (15) into the Yang-Baxter equation.
Regarding the invariants, the trace of the monodromy matrix tr(L b (y)L a (x)) = 1 + I 2 − I 3 , where L a is given in (12). Thus, I 2 and I 3 are invariants. The rest, I 1 and I 4 , are found from the characteristic Remark 3.2.2. The above procedure is reversible. That is, starting from map (14)-(15), we can retrieve the Boussinesq lattice system (14)- (15). This follows from the observation that x 3 = y 2 , in (15), implies u 2 = v 3 , in combination with a certain change of variables.

3.3
Step II : Grasmann extended Yang-Baxter map of Boussinesq type In this section, we construct a Grassmann extension of Boussinesq type Yang-Baxter map (14)- (15). In order to do that, we consider the Lax matrix (10) augmented with two additional fermionic fields χ 1 , χ 2 , such that the conditions described in step II of the scheme are satisfied.
In particular, we consider following 4 × 4 matrix which is matrix L a in (12) augmented with two additional fields χ i ∈ G 1 , i = 1, 2. The above generalisation respects the following conditions 1. Bosonic limit lim 2. Determinant sdet(L a ) 2 = det(L a ) = a − λ.
2 By sdet(.) we denote the "superdeterminant" in the Grassmann case [3]. The superdeterminant is defined on matrices of the block form M = P Π Λ L , where P and L are square matrices of even entries, whereas Π and Λ are matrices with odd entries, not necessarily square.
where L a = L a (x, χ χ χ) is given by (17), is equivalent to the following correspondence: and Proof. Equation (20) implies equation (23) for u 4 and v 1 , as well as the following system of equations for the rest of the variables u 1 , u 2 , u 4 , ξ 1 , v 3 and ξ 1 . From (24d) we obtain ξ 1 ψ 2 = ψ 1 ψ 2 + (y 2 − u 2 )χ 1 ψ 2 . Substituting the latter to (24g) and using (23), we obtain u 2 as given in (21b). With use of u 2 , (24a) and (24c) imply v 3 and u 1 as given in (22b) and (21a), respectively. Subsequently, with use of (22b), from equation (24b) follows that v 4 is given by (22c), whereas equations (24d) and (24b) imply the following expressions for ξ 1 and η 2 . Multiplying both the numerator and the denominator of the fractions in the above equations by the conjugate expression of the denominator, it follows that ξ 1 and η 2 are given by (21d) and (22e), respectively.

Step III : Squeeze down to Grassmann extension of the Boussinesq lattice equation
Here, we shall construct our Grassmann extended system of lattice equations using the symmetries of the Yang-Baxter map which was constructed in the previous section. That said, using the observation that, in (26), y 2 = x 3 implies u 2 = v 3 , we shall construct a Grassmann extension of the Boussinesq lattice equation (8). In particular, we have the following.

3D consistency of a Grassmann extended Boussinesq-type system
Now, conservation law (32) indicates to seek a function f = f (n, m) such that namely, seek f = f (n, m) satisfying the following system difference equations f 01 − f = p 01 + qq 01 − r.
The above equation implies f = C(n+m). We restrict ourselves to the case where C(n+m) = const. = 0, and from system (35) follows that which are equivalent to (34a) and (34c).
With the use of (39), we prove the following.
Remark 4.0.2. The bosonic limit of system (40) is the Boussinesq lattice system as it appears in [5].
Although system (40) is integrable in the sense that it possesses Lax representation, we cannot claim integrability in the sense of 3D-consistency, since the term "τ 11 " is missing. Since one of the main purposes of this paper is to prove that the 3D-consistency is preserved in the Grassmann extension of some systems, we shall prove this property for the bosonic limit of system of system (40), as τ → 0. In particular, we have the following.
The proof of this theorem is presented in Appendix B. It is worth mentioning that, since the "111" values depend on the initial p and q (see Appendix B), the system (4.0.3) does not have the "tetrahedron property" 3 . For the proof the following Lemma is needed.  A(a 100 , a 010 , a 001 , b 100 , b 010 where a and b can be either odd or even variables, is invariant under simultaneous cyclic permutations of (a 100 , a 010 , a 001 ) and (b 100 , b 010 , b 001 ). A(a 100 , a 010 , a 001 , b 100 , b 010  The Lax representation of system (4.0.3) is given by (30)-(31) for τ → 0, namely by L a (p 01 , q 01 , q 11 , r 11 , θ 11 )L b (p, q, q 01 , r 01 , θ 01 ) = L b (p 10 , q 10 , q 11 , r 11 , θ 11 )L a (p, q, q 10 , r 10 , θ 10 ),
On the other hand, this paper answers an important question regarding the 3D consistency of systems, when they are extended to the Grassmann case. That is, not all systems lose their property to be 3D consistent in their noncommutative extension, which can be demonstrated by system (42). The 3D-consistency property of systems like (42) is a very important, since for systems with such property we can: • algorithmically construct its Lax representation [4,5,13,23]; • obtain a Bäcklund tranformation [2,13].
In our case the Lax representation is already known.
Regarding the 3D-consistency of the system (40), we would like to stress out the following. The demand that quad-graph equations (2) need to be linear in every variable is because we need to be able to solve uniquely for any of the fields f , f 10 , f 01 and f 11 . This is essential for the 3D-consistency property. Nevertheless, this is not quite the case for systems with anticommutative variables: In our system, (40), all equations are linear in all variables, which is obvious if one rewrites the equations in polynomial form. However, this does not imply unique solvability. For instance, equation (40c) cannot be solved for neither τ , nor θ 10 , nor θ 01 .
Our results can be extended in several ways. We list a couple of problems for future work.
1. The complete (Liouville) integrability of maps (25)- (26) and (14)- (15) is an open problem. We conjecture that there is a suitable Poisson bracket with respect to which the maps' invariants are in involution.
2. Study the solutions of system (42). In particular, knowing that system (42) has the 3D-consistency property, we can derive a Bäcklund transformation by setting (p 001 , q 001 , r 001 , θ 001 ) ≡ (u, v, w, φ), and rewrite (62), (63) as a Bäcklund tranformation between (p, q, r, θ) and (u, v, w, φ), 4 namely: 3. Continuum limits. Using the above Bäcklund transformation to derive solutions of system (42) and, then, considering the continuum limits of these solutions, we can study the behaviour of the solutions of the corresponding Boussinesq-type system of PDEs.
A Proof of theorem 3.3.2 Throughout this proof we shall be using the following "tilde-hat" notation.