Non-commutative rational Yang-Baxter maps

Starting from multidimensional consistency of non-commutative lattice modified Gel'fand-Dikii systems we present the corresponding solutions of the functional (set-theoretic) Yang-Baxter equation, which are non-commutative versions of the maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel'fand-Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular we recover known non-commutative version of Hirota's lattice sine-Gordon equation, and we present an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.


Introduction
Let X be any set, a map R : X × X satisfying in X × X × X the relation where R ij acts as R on the i-th and j-th factors and as identity on the third, is called Yang-Baxter map [12,29]. If additionally R satisfies the relation where R 21 = τ • R • τ and τ is the transposition, then it is called reversible Yang-Baxter map.
In this paper we study properties of a non-commutative version of the maps arising from geometric crystals [18,13,28]. In particular we will demonstrate the following result. It is easy to see that the products α = x 1 . . . x L and β = y 1 . . . y L are conserved by the map R. This can be used to reduce the number of variables. For example, in the simplest case L = 2 define x = x 1 , y = y 1 to get a parameter dependent reversible Yang-Baxter map R(α, β) : (x, y) → (x,ỹ) which in the commutative case is equivalent to the F III map in the list given in [2]. In recent studies on discrete integrable systems the property of multidimensional consistency [1,24] is considered as the main concept of the theory. Roughly speaking, it is the possibility of extending the number of independent variables of a given nonlinear system by adding its copies in different directions without creating this way inconsistency or multivaluedness. It is known [2,27] how to relate three dimensional consistency of integrable discrete systems with Yang-Baxter maps. There is also well known connection between Yang-Baxter maps and the braid relations. Figure 1. The discrete KP map and its three dimensional consistency Non-commutative versions of integrable maps or discrete systems [25,22,4,26,7] are of growing interest in mathematical physics. They may be considered as a useful platform to more thorough understanding of integrable quantum or statistical mechanics lattice systems, where the quantum Yang-Baxter equation [3,20] plays a role.
In Section 2 we use three dimensional consistency of non-commutative Kadomtsev-Petviashvilii (KP) map to construct corresponding Yang-Baxter maps following ideas of [18,19] applied there in the commutative case. It turns out that we can construct the solutions under periodicity and centrality (of certain products of the variables) assumptions. Then in Section 3 we consider implication of the centrality assumption on the level of the non-commutative modified lattice Gel'fand-Dikii equations. In the simplest case we recover non-autonomous version of non-commutative Hirota's sine-Gordon equation [4]. We present also an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.
Remark. Throughout the paper we will work with division rings of (non-commutative) rational functions in a finite number of (non-commuting) variables. This approach is intuitively accessible, see however [5] for formal definitions.
2. Non-commutative rational realization of the symmetric group 2.1. KP maps. Consider the linear problem of the non-commutative KP hierarchy [19,26,9] here φ k : Z N → D M , and D is a division ring, and ε i ∈ Z N has 1 at i-th place and all other zeros. The potentials u i,k : Z N → D satisfy then the compatibility conditions where we write u i,k(j) (n) instead of u i,k (n + ε j ), and we skip the argument n. In consequence we obtain the transformation rule which can be written as a non-commutative discrete KP map Proposition 2.1. [9, 10] The non-commutative discrete KP map is three-dimensionally consistent, i.e. both ways to calculate u i(jl) give the same result, see Figure 1.
Remark. Three dimensional consistency of the non-commutative discrete KP map is a consequence [9] of the four dimensional consistency of the so called Desargues maps [8].
To make connection with the Yang-Baxter maps consider N -cube graph, whose vertices are identified with binary sequences of length N with two vertices connected by an edge if their sequences differ at one place only. The shortest paths from the initial vertex (0, 0, . . . , 0) to the terminal one (1, 1, . . . , 1) can be identified with permutations: a permutation σ ∈ S N corresponds to the path with subsequent steps in directions (ε σ(1) , ε σ(2) , . . . , ε σ(N ) ). The symmetric group acts then on paths by the left natural action ρ.π σ = π ρσ . Given initial weights u i , i = 1, . . . , N , on edges connecting the initial vertex (0, 0, . . . , 0) with the vertex ε i , by the KP map we attach a weight to each edge of the cube graph. Each such path gives then a sequence of weights w σ , for example w Id = (u 1 , u 2(1) , . . . , u N (1,2,...,N −1) ). We are interested in maps r σ from the reference weights w Id to weights w σ . In particular, we study maps r i , i = 1, . . . , N − 1, which correspond to transpositions σ i = (i, i + 1) generating the symmetric group S N and satisfying the Coxeter relations [17] In order to find such maps r i we have to find the so called first companion map where we use variation of the terminology of [2,27] where Yang-Baxter maps were studied in relation to multidimensionally consistent edge-field maps, 2.2. The first companion map and the centrality assumption. We will concentrate on deriving the first companion map, which we temporarily denote by r : which satisfy the recurrence relations where by definition P  We group such unwanted terms into (disjoned) parts depending on the smallest p. Such a part has the structure which due to the induction assumption and equations (2.4) is equal to its primed version, therefore both cancel out.
From now on we assume L-periodicity condition: x k+L = x k , y k+L = y k . Define P k = P (L−1) k , then Lemma 2.2 and recurrence relations (2.5) imply Notice that if we would impose the additional normalization condition x k+i , then equations (2.4) could be solved as . However, equations (2.7) and condition (2.6) are not compatible for general non-commuting variables. The above procedure of getting solutions works if we make additional centrality assumptions which state that α = L i=1 x i and β = L i=1 y i commute with other elements of the division ring.
which means that the above expression is central and independent of index k as well. In particular P k x k commutes with y k P k+1 .
Proof. The first part follows from identities where we used also the periodicity assumption. The second part is implied by equations (2.5). Proof. Notice that by Lemma 2.3 x ′ k and y ′ k given by (2.7) satisfy the normalization condition. Then also both expressions vanish due to Lemma 2.3.
Corollary 2.6. The problem of finding the first companion of the KP map in the periodic reduction can be considered as a refactorization problem A(x)A(y) = A(x ′ )A(y ′ ), where the matrix with the central spectral parameter λ is the L-periodic reduction of the discrete non-commutative KP hierarchy (Gel'fand-Dikii system) linear problem (2.1) studied in [9].

Realization of Coxeter relation under the centrality assumption.
Consider again a sequence (w 1 , w 2 , . . . , w N ) o weights along a shortest path in N -cube from the initial to the terminal vertex, each weight is a sequence w j = (w j,1 , . . . w j,L ) of non-commuting variables satisfying the centrality assumption that the product α j = w j,1 w j,2 . . . w j,L commutes with all w j,k . As we already have mentioned the symmetric group S N acts in natural way on the paths and thus on the weights. To make use of results of Section 2.2 define polynomials (2.10) where the second index should be considered modulo L.
If we assume centrality of the products α j = L i=1 w j,i then the maps r j satisfy the Coxeter relations Proof. The commutativity part is clear from definition of the maps. Involutivity part comes from Corollary 2.5, and the braid relations follow from the path-interpretation and uniqueness of the companion map subject to normalization conditions (2.6). Equivalently, we can use the unique refactorization interpretation given in Corollary 2.6, and follow the argumentation presented in [29].
Corollary 2.9. We can consider the division ring D as a division algebra over a fixed subfield k of its center. Therefore we can state the centrality condition as α j ∈ k.

Non-commutative Gel'fand-Dikii systems with the centrality condition
In [9,10] we studied periodic reductions of the discrete KP hierarchy under two extreme assumptions about non-commutativity/commutativity of dependent variables. Results of Section 2.2 suggest to consider an analogous centrality condition on the level of equations (2.2). By simple calculation we obtain the following result.
Proposition 3.1. In the L-periodic reduction u i,k+L = u i,k of the non-commutative KP system (2.2) assume centrality of the products U i = u i,1 u i,2 . . . u i,L . Then the products u i,k u i,k+1 . . . u i,k+L−1 do not depend on index k, and U i is a function of n i only Using the above result one can obtain non-autonomous non-commutative discrete equations of the modified Gel'fand-Dikii type. It is known [9] that the first part of equations(2.2) implies existence of potentials ρ k such that u i,k = ρ −1 k ρ k(i) , while the second part gives the corresponding vertex form of the non-commutative discrete KP hierarchy Then we replace one of the functions ρ i by others and the central non-autonomous factors. To make connection with known results it is convenient to define central functions F i = (U i ) 1/L of the corresponding single variables n i , and then consider the central function G defined by compatible system G (i) = F i G.
We remark that such G is a product of functions of single variables.
In the simplest case L = 2 define, like in [9], a function x by ρ 1 = xG. Then 2) produces the non-commutative Hirota (or discrete sine-Gordon or lattice modified Korteweg-de Vries) equation studied in [14,15,4] Remark. To recover the equation in the form studied in [4] notice that after extracting x −1 (j) the expressions in brackets commute, and use inverses of the non-autonomous factors F i . For L = 3 define unknown functions x and y by equations Making such substitution in (3.2) for k = 1 and k = 3 we obtain the following non-commutative integrable two-component system (equation for k = 2 is then its consequence) Next, by elimination of the field x we obtain integrable non-commutative and non-autonomous (with central non-autonomous coefficients F i ) version of the lattice modified Boussinesq [23] equation .

Concluding remarks
We presented a non-commutative rational Yang-Baxter map obtained from the non-commutative discrete KP hierarchy subject to periodicity and centrality constraints. The corresponding integrable systems, which generalize the non-commutative non-autonomous Hirota's sine-Gordon equation [4] have been also considered. In particular we have obtained an integrable non-commutative and nonautonomous lattice modified Boussinesq equation. We remark, see [9,10], that three dimensional consistency of the equations considered here is a consequence of four dimensional compatibility of the non-commutative Hirota's discrete KP system [8], where the counterpart of the functional Yang-Baxter equation is the functional pentagon equation [11]. Since the solutions of the pentagon equation presented in [11] allow for quantization (understood as a reduction from the non-commutative case by adding certain commutation relations preserved by the integrable evolution), we expect that also the non-commutative rational Yang-Baxter map obtained above can be quantized in such a way also. It would be instructive to understand various applications of the Hirota discrete KP systems and its reductions reviewed in [21] from that perspective.