Electric network and Hirota type $4$-simplex maps

Bazhanov--Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of $n$-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov--Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov--Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.


Introduction
The n-simplex equations are generalisations of the famous Yang-Baxter equation and are fundamental equations of Mathematical Physics.The most celebrated members of the family of n-simplex equations are the Yang-Baxter (2-simplex) equation, the Zamolodchikov tetrahedron (3-simplex) equation [30] and the Bazhanov-Stroganov (4-simplex) equation [3].
The popularity of n-simplex equations is due to the fact that they appear in a wide range of areas of Mathematics and Physics, and they are strictly related to integrable systems of differential and difference equations.In particular, the n-simplex equations have applications in statistical mechanics, quantum field theories, combinatorics, low-dimensional topology, the theory of integrable systems, as well as a plethora of other fields of science (see, e.g., [1,2,27,9,11,13,14,23,28]).Therefore, the construction and classification of n-simplex maps is quite relevant and constitutes a very active area of research.
There are several methods that associate n-simplex maps (set theoretical solutions to the n-simplex equation) with integrable systems.We indicatively refer to the relation between n-simplex maps and integrable lattice equations via symmetries [24,14] as well as integrable nonlinear PDEs via Darboux and Bäcklund transformations [4,20,21].Thus, the development of methods for constructing interesting n-simplex maps is a significant task which may give rise to important integrable models.
This paper is concerned with the development of a simple scheme for construction of Bazhanov-Stroganov 4-simplex maps as extensions of Zamolodchikov tetrahedron maps.The proposed scheme is a generalisation of the method presented in [16] for constructing 4-simplex maps, and its methodology involves working with simple matrix refactorisation problems and makes use of straightforward computational algebra.The advantage of the proposed generalised scheme versus the method presented in [16] is that it derives more interesting 4-simplex maps which are noninvolutive extensions of involutive 3-simplex maps.This is a quite interesting phenomenon, since involutive maps possess trivial dynamics.
As an illustrative example for our method, we first use the Hirota tetrahedron map which has various important applications [5,26], however it lacks the property of being noninvolutive.One of its 4-simplex extensions, which we present in this paper, has equally elegant form and preserves all the properties of the original map (Lax representation, first integrals etc.); nonetheless, it has the extra significant property of being noninvolutive.Furthermore, we apply our method to Kashaev's electric network tetrahedron map [10,11], and we construct its 4-simplex extensions.The Kashaev electric network transform is the equivalency condition of two electric devices, each made of three resistors and with three outer contacts, with star and triangle diagrams, respectively [10].Moreover, it is related to the well-celebrated Miwa's integrable equation, which at a certain continuous limit reduces to the KP (Kadomtsev-Petviashvili) equation [7,10].
The rest of the text is organised as follows: In the next section, we provide all the required definitions and statements for the text to be self-contained.In particular, we give the definitions of Zamolodchikov's tetrahedron maps and Bazhanov-Stroganov 4-simplex maps, and we explain their relation with the local Yang-Baxter and tetrahedron equation, respectively.Section 3 deals with the development of a simple scheme for constructing Bazhanov-Stroganov maps as extensions of Zamolodchikov tetrahedron maps.We apply this scheme to a Sergeev type map and construct new 4-simplex maps.In Section 4, using the method presented in Section 3, we derive novel Bazhanov-Stroganov maps which are 4-simplex extensions of the famous Hirota tetrahedron map [5,26].In Section 5, we construct new 4-simplex maps which can be restricted to the famous electric network tetrahedron map at certain limits.Finally, in Section 6 we discuss the obtained results and list possible directions for future research.

Preliminaries
In this section, we give the definitions of tetrahedron and Bazhanov-Stroganov maps, and we explain how to derive such maps using matrix refactorisation problems.

Local Yang-Baxter equation and tetrahedron maps
Let X be a set.We denote X n = X × . . .× X n .A map T : X 3 → X 3 , namely T : (x, y, z) → (u(x, y, z), v(x, y, z), w(x, y, z)), is called a 3-simplex map or Zamolodchikov map or tetrahedron map if it satisfies the functional tetrahedron or Zamolodchikov's tetrahedron equation [29,30] Functions T ijk : X 6 → X 6 , i, j = 1, . . .6, i < j < k, in (1) are maps that act as map T on the ijk terms of the Cartesian product X 6 and trivially on the others.For instance, T 356 (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ) = (x 1 , x 2 , u(x 3 , x 5 , x 6 ), x 4 , v(x 3 , x 5 , x 6 ), w(x 3 , x 5 , x 6 )).Now, let L = L(x) be a matrix depending on a variable x ∈ X of the form , where its entries a, b, c and d are scalar functions of x.Let L 3 ij , i, j = 1, 2, 3, i = j, be the 3 × 3 extensions of is the Maillet-Nijhoff equation [23] in Korepanov's form, which appears in the literature as the local Yang-Baxter equation.If a map of T : X 3 → X 3 satisfies the local Yang-Baxter equation (2), then this map is possibly a tetrahedron map, and equation ( 2) is called its Lax representation [6].
The local Yang-Baxter ( 2) is a generator of Zamolodchikov tetrahedron maps.Tetrahedron maps are related to pentagon maps [5,12], and, generally, n-simplex maps are related to solutions of the n-gon equations [6].The most popular tetrahedron maps appear in the works of Sergeev [26] and Kashaev-Korepanov-Sergeev [11].

4-simplex extension scheme
Let T : X 3 → X 3 be a tetrahedron map with Lax representation (2) for some matrix L(x).The aim is to construct systematically a 4-simplex map S : X 4 → X 4 , with Lax representation (5), such that map S implies T at a certain limit.This can be achieved using a simple scheme which is summarised in Figure 1.
In particular: Step I: Consider a tetrahedron map, T : X 3 → X 3 , generated by L(x) via ( 2).If we consider the 3 × 3  , and substitute it to the local tetrahedron equation ( 5), we will obtain a trivial extension of map T as a solution to the 4-simplex equation [16].
Step II: In order to construct an nontrivial extension of map T , we introduce an auxiliary variable x 2 , namely we consider the following matrix such that for x 2 → 1, we have K(x 1 , x 2 ) → M(x 1 ).Substitute K(x 1 , x 2 ) to the local tetrahedron equation Step III: Solve (7) for u i , v i , w i and r i , i = 1, 2; we aim to obtain a 4-simplex map or a correspondence which, for particular values of the free variables, will define 4-simplex maps.Remark 3.1.This extends the method presented in [16] by allowing the elements a, b, c and d of matrix K(x 1 , x 2 ) in ( 6) to depend on the auxiliary variable x 2 .We will demonstrate that the latter assumption implies more interesting 4-simplex maps with more interesting dynamics than the maps derived via the method in [16].

Example: A new Sergeev type 4-simplex map
Here, we demonstrate that, employing the extension scheme of Section 3, one may construct more interesting 4-simplex extensions of Sergeev's maps than the ones presented in [16].Specifically: Step I: Consider the k-parametric family of tetrahedron maps C 3 → C 3 [26]: with Lax representation (2), where and substitute it to the local tetrahedron equation ( 7), one will obtain the 4-simplex map S : (x, y, z, t) → xy y+xz , xz k , y+xz x , t , which is a trivial extension of (8).
Step II: Consider, instead, matrix −k and substitute it to the local tetrahedron equation Step III: We solve equation ( 9) for (u 1 , u 2 , v 1 , v 2 , w 1 , w 2 , r 1 , r 2 ), and we see that equation ( 9) is equivalent to the following correspondence between C 8 and C 8 The above system does not define a 4-simplex map C 8 → C 8 for arbitrary w 2 .However, for the choices w 2 = t 2 and w 2 = y 2 we obtain the following Bazhanov-Stroganov 4-simplex maps Maps (10) and (11) are more interesting than maps ( 27) and ( 28) in [16].This can be done for all maps derived in [16].It can be proven that maps (10) and (11) are birational, and their inverses are also 4-simplex maps.

Hirota type Bazhanov-Stroganov map
In this section, we apply the 4-simplex extension scheme of the previous section to the well-known Hirota tetrahedron map.We construct novel 4-simplex maps which can be restricted to the Hirota map at certain limit.
Proposition 4.1.The following maps are eight-dimensional 4-simplex maps which share the same functionally independent invariants Moreover, map S 1 admits the invariant I 6 = x 2 z 2 t 1 , whereas S 2 admits the invariant Proof.Maps ( 15) and ( 16) follow after substitution of w 2 = t 2 and w 2 = z 2 , respectively, to (14).The 4simplex property can be verified by straightforward substitution to the 4-simplex equation ( 3).The invariants are obvious.

Electric network type Bazhanov-Stroganov map
Here, we apply the 4-simplex extension scheme presented in section 3 to the famous electric network tetrahedron map [10,11].We construct novel 4-simplex maps which can be restricted to the electric network map at certain limit.
The electric network tetrahedron map reads [5,26] T and it possesses a Lax representation (2) for L( In order to construct a 4-simplex extension of Hirota map (12), we consider the 3 × 3 extension of L(x): Then, we introduce a generalisation of matrix M(x), namely matrix Let us consider the 6 × 6 extensions of matrix K(x 1 , x 2 ), namely the following , and substitute them to the local tetrahedron equation The above is equivalent to the following correspondence between C 8 and C 8 : For particular choices of the free variables the above correspondence defines 4-simplex maps.Specifically, we have the following.
Proof.Maps (23) and (24) are obtained for the following choices of the free parameters u 1 , v 1 , w 1 and r 1 of correspondence (22): and respectively.Moreover, by construction, they satisfy the matrix refactrorisation problem (7), for K namely S 2 2 = id, which means that map (24) is noninvolutive.Now, in view of ( 23) and ( 24) he have that The matrices with rows (∇I i ) t and (∇J i ) t , i = 1, 2, 3, 4, have rank 4, thus invariants I i are functionally independent, and so are invariants J i , i = 1, 2, 3, 4.

Conclusions
In this paper, we presented a scheme for constructing interesting 4-simplex extensions of Zamolodchikov tetrahedron maps; this scheme generalises the method presented in [16].We demonstrated the advantage of this scheme using a Sergeev type tetrahedron map as an illustrative example.Moreover, we employed this scheme for the construction of novel Hirota type 4-simplex extensions, namely Bazhanov-Stroganov maps which are reduced to the famous Hirota map (12) at a certain limit.Similarly, we contructed new 4-simplex maps which can be reduced to the electric network transform (20) at a certain limit.
The results of this paper can be extended in the following ways.
• All maps presented in this paper admit enough functionally independent first integrals which indicates their integrability.Their Liouvile interability is an open problem.
• We derived maps using particular 3 × 3 matrices.One could study all the possible 3 × 3 matrices that generate 4-simplex maps via the local tetrahedron equation.Certain classification results on the solutions to the local tetrahedron equation which derive 4-simplex maps will appear in our future publication.
• Since the Hirota map ( 12) is related to Desargues lattices [5], one could study the relation of the latter to the 4-simplex generalisations of the Hirota map ( 15), ( 16) and (19).
• Study the relation of the derived Hirota type and electric network Bazhanov-Stroganov maps to 4D-lattice equations employing similar methods which relate Yang-Baxter and tetrahedron maps to 2D and 3D lattice equations.For instance, the symmetries of the associated lattice equations [25,14], the existence of integrals in separable variables [15], lifts and squeeze downs [22,24,19].
• Find which additional matrix conditions must be satisfied so that the solutions to the local tetrahedron equation define 4-simplex maps (see [17] for similar results for tetrahedron maps).
• Derive new 4-simplex extensions of all the tetrahedron maps which were considered in [16], including the Kadomtsev-Petviashvili tetrahedron map [6] map and the NLS type tetrahedron maps [18], using the scheme presented in Section 3.
• The method presented in Section 3 can be extended by allowing matrix (6) depend on other auxiliary variables.For example, in [8], the tetrahedron map (x, y, z) x 3 0 0