Matrix factorizations and pentagon maps

We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order N=2 matrices of this specific class are equivalent to the homogeneous normalization map. From order N=3 matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.


Introduction
The interplay of matrix factorization problems and discrete integrable systems respectively in one, two or three independent variables has been introduced and studied in various seminal papers, see respectively [1,2,3], [4,5] or [6,7].Furthermore, there exists an intrinsic relation between matrix factorization problems and discrete integrability and it stands as an active area of research as the recent developments [8,9,10,11,12] suggest.In this paper, we study a particular class of matrices that participate in specific matrix re-factorization problems.These re-factorization problems turn out to be equivalent to pentagon, reverse-pentagon, or Yang-Baxter maps.
Pentagon maps serve as set theoretical solutions of the pentagon equation 12 S (3) 13 S (5) (2) that is the first non-trivial example of the so-called polygon equations c.f. [13,14].In relation (1), the superscripts denote objects (S (q) ) that might differ f.i.these objects might be operators or maps; so we have respectively the "operator" and the "set-theoretic" version of (1).In the set-theoretic version of (1) the subscripts denote the sets where the maps S (q) act.For detailed interpretation of (1) and for the definition of the reverse-pentagon and the Yang-Baxter equation, see Section 2.
The operator form of (1) first appeared in [15] in relation with conformal field theory.In [16] it was considered inside the context of three-dimensional integrable systems, while the set-theoretical version of (1) was studied in [17] in connection with Poisson maps, see also [18].Furthermore, the pentagon relation (1) itself, as it was shown in [19], serves as a manifestation of the 3 ↔ 2 Pachner move [20], where three tetrahedra with a common edge are replaced by two tetrahedra with a common face on a triangulation of a piecewise-linear 3-manifold.Pentagon maps appeared in the context of Roger's dilogarithm [21,22] and are also related to the closure relation of the Lagrangian multiform theory in the setting of discrete integrable systems [23,24], as well as to cluster algebras [25,26].For further connections and interrelations of the pentagon equation and maps with various areas of Mathematics and Physics we refer to [13].While for recent developments on pentagon maps we refer to [17,18,27,22,28,11,29,30].
This article is organized as follows.We begin with a brief introduction followed by Section 2 where we present the definitions and the basic mathematical notions used in this article.In Section 3, we introduce a special class of matrices that participate in matrix factorization problems that turn equivalent to pentagon, reverse-pentagon and Yang-Baxter maps.In particular, via these matrix re-factorization problems, when the order N of the matrices is N = 2, we recover a well known pentagon map that is the homogeneous normalization map [28,11].When N = 3, we obtain an extension of the homogeneous normalization map, along with novel entwining Yang-Baxter and pentagon maps.Note that the proof that the obtained mappings are pentagon maps, follows from direct computation.Alternatively, since the local (N − 1)−gon equation determines an N −gon map [13], proof of an anticipated result that the class of matrices introduced here, satisfies the local tetragon (2−gon) equation, would serve as an alternative proof of the pentagonal property.In appendix A, we explicitly provide the tetrahedron and 4-simplex maps associated with the presented extension of the normalization map.In the concluding Section 4, we summarize our results and we briefly discuss the case where the order of the matrices that participate in factorizations is N = 4. Finally note that all mappings presented in this article are expressed in totally non-commutative variables; that is no commutation relation is assumed among them.

Preliminaries on pentagon maps and more
Let X be a set.We proceed with the following definitions.
The maps S(q) : X × X → X × X, will be called entwining (non-constant) reverse-pentagon map if they satisfy the entwining (non-constant) reverse-pentagon equation 23 .The maps T (q) : X × X × X → X × X × X, will be called entwining (non-constant) tetrahedron maps if they satisfy the entwining (non-constant) tetrahedron equation [34] T 123 .Alternatively, tetrahedron maps are called 3-simplex or Zamolodchikov maps.The maps P (q) : X 4 → X 4 , where X 4 = X × X × X × X will be called entwining (non-constant) 4-simplex maps if they satisfy the entwining (non-constant) 4-simplex equation  In the definitions above, when the maps R (q) do not differ so we have we say that R serves as a solution of the constant Yang-Baxter equation or just the Yang-Baxter equation and the map R will be referred as constant Yang-Baxter map or simply Yang-Baxter map.In a similar manner, the constant pentagon, reverse-pentagon, tetrahedron and 4-simplex maps are defined.The term non-constant indicates the cases where at least one of the maps R (q) differs.For studies on entwining Yang-Baxter maps see [36,37,38,39], at the same time for recent investigations on constant Yang-Baxter maps see [40,41,42,43,38,44,45], whilst for non-commutative Yang-Baxter maps we refer to [46,47,48,49].
Additionally, for recent developments on tetrahedron and 4-simplex maps we refer to [50,11,51,12], while for permutation type maps that serve as solutions to higher simplex equations see [52].
Remark 2.1.Let τ : X × X → X × X the transposition map i.e. τ : (x, y) → (y, x).For pentagon and reverse-pentagon maps it is easy to verify the following: • τ is referred to as the opposite pentagon map and it is denoted as S op .Similar statements hold for reverse-pentagon maps.
Definition 3 (Ten-term relation [27]).The maps W q and W q that map X × X to itself, will be said to satisfy the non-constant or entwining ten-term relation if they satisfy 12 . (3) The ten-term relation in its entwining form (3) appeared in [53].

Proposition 2.3 ([27]
).Let S (q) respectively S(q) be a pentagon respectively a reverse-pentagon map that satisfy the ten-term relation ( 3), then the maps 13 , define non-constant tetrahedron maps, whereas the maps 13 , define non-constant a 4-simplex maps, where τ : X × X → X × X the transposition map i.e. τ ij stands for the transposition of the i−th and j−th arguments of X N , N ∈ N.
Let A be an associative algebra over a field F, with multiplicative identity that we denote with 1.Throughout this paper we consider In this general setting, A × could be a division ring for instance bi-quaternions.More generally, A × could stand for the subgroup of invertible matrices of the algebra A of N × N matrices.

Matrix factorizations pentagon and reverse-pentagon maps
Let I N the order N identity matrix and let p k , k ∈ {1, . . ., K}, be a set of k distinct positive integers l j i.e.
We consider the matrices A p k (x), which are defined by substituting the lines l 1 , . . ., l k of the identity matrix I N with the vectors V 1 (x), . . ., V k (x), respectively.Similarly we define the matrices A q λ (x).The matrices A p k (x) and A q λ (x) participate in the following re-factorization problems Unless otherwise stated, among the re-factorisation problems (4) we concentrate to the ones that: (1) when N is odd.When the three conditions above are satisfied, without loss of generality we have that turns equivalent to the map φ p,q : (x, y) → (u, v) = (u(x, y), v((x, y)).
The following statements are in order.
(1) The map φ p,q is a pentagon map.
The statements above can be easily proven.For instance let us sketch the proof of statement (1).For the proof we have to distinguish two cases, that is when the order N of the matrices is even or odd.For even N (N = 2k), we have p = {1, . . ., k} and q = {k + 1, . . ., 2k} so where 0 k the order k zero matrix, I k the order k identity matrix and x i , i = 1, 2, stands for k × k block matrices.Then it is clear that the re-factorization problem A {1,...,k} (u)A {k+1,...,2k} (v) = A {k+1,...,2k} (y)A {1,...,k} (x), coincides with the N = 2 re-factorization problem when we consider the variables x 1,j , y 1,j , u 1,j , v 1,j , j = 1, 2, that participate in (7) to be k × k matrices with entries in A × .So what is valid for the order N = 2 re-factorization problem (7), also holds for the re-factorization problem of order N = 2k.In the following Section we prove that ( 7) is equivalent to a pentagon map and that together with a similar analysis on odd-order re-factorization problems of the form (5), suffices as a proof of statement (1).Note that a consistency result such as the 3-factorization property [8,10] that applies to matrices associated with Yang-Baxter maps, is not developed yet for the matrix factorizations associated with pentagon maps that we proposed above.Since the local (N − 1)−gon equation determines an N −gon map [13,54], by re-formulating the matrix factorizations presented above as local tetragon (2-gon) equations we expect the anticipated consistency to follow.
Proposition 3.1.The map φ is a pentagon map.
Similarly we can prove that the left hand side of ( 14) applied on x (i) , y (i) , i = 1, 2, 3, agrees with the application of the right hand side of ( 14) to the same components.
Note that the mapping φ {1},{2} is of triangular form, with its components x (j) , y (j) , j = 1, 2, varying as in the N = 2 case, while the non-trivial coupling appears in the x (3) and y (3) components.The associated map with the components x (j) , y (j) , j = 1, 2, coincides with the normalization map (9).

2.3, the map
is a tetrahedron map, whereas the map is a 4-simplex map.For their explicit form see Appendix A. For recent developments on 4-simplex maps, see [55].

Conclusions
In this article we have proposed a specific class of matrices that their associated re-factorization problems turned equivalent either to pentagon, reverse-pentagon, or Yang-Baxter maps.We thoroughly examined the order N = 2 and N = 3 matrices and explicitly obtained the associated maps.For higher order matrices of the proposed form, further studies are required.