Construction of an algebra corresponding to a statistical model of the square ladder (square lattice with two lines)

In this paper we define infinite-dimensional algebra and its representation, whose basis is naturally identified with semi-infinite configurations of the square ladder model. We also extrapolate the ideas for the cyclic 3-leg triangular ladder model. All of these propose a way for generalization, which leads to representations of N = 2, ... algebras. Keywords: 2D lattice, square ladder, triangular ladder, conformal algebra, semi-infinite forms, fermions, quadratic algebra, superfrustration, graded Euler characteristic, cohomology, deformation, Jacobi triple product, superalgebras, operator algebras, N = 2, ... algebras.


Introduction
For each graph Γ we can construct a statistical model in which the set of configurations is the set of arrangements of particles at graph vertices such that at each vertex at most one particle is located and two particles cannot be located at vertices joined by an edge.
The previous paper [1] discussed combinatorial properties of the set of configurations of the 2 × square lattice (or simply the square ladder) graph: Studies of integrable models of statistical mechanics show a close connection between the set of configurations of the corresponding lattice graph and the representation of some infinite-dimensional algebra [2][3] [4][5] [6][7] [8]. Based on this, the paper yields the following results: In Section 2, based on idea of semi-infinite forms, the set of configurations is defined for the square ladder model, which is infinite in both directions. A bigraduation is introduced and statistical sum is calculated.
In Section 3, such a deformation of the fermion algebra for the graph of the square ladder model, which is infinite in both directions, is determined (the obtained algebra is close to conformal algebras) that the character of its representation is equal to the statistical sum from Section 2.
In Section 4, the cohomology of complexes, constructed from finite-dimensional quotient algebras of the deformed algebra from Section 3, are calculated. The corresponding complexes are either acyclic, or their cohomology is one-dimensional. The deformation was selected by the latter, among other things.
In Section 5, further generalization is discussed.

Set of semi-infinite configurations
Instead of the square ladder model, Fig. 1, it is convenient to consider an equivalent combinatorial problem (the results of paper [1] remain valid), the twisted square ladder model: Figure 2: twisted square ladder model According to [8], the set Ω ⊂ Z is called a Dirac set if Ω = Ω ∩ Z ≥0 and Ω = Z <0 ∖ Ω are finite. The value (Ω) = count(Ω ) − count(Ω ) is called the charge of the set Ω, and (Ω) = ∑︀ ∈Ω ∪Ω | | is called its energy.
The set of such configurations {. . . , −1 , , +1 , . . . , 1 } of the graph of the twisted square ladder model, which is infinite in both directions, that the set of indexes Ω = { } is a Dirac set, is called the set of semi-infinite configurations Δ ∞ 2 . The location of particle is given by or depending on whether the particle is in the upper or lower row, where is the corresponding column number. It follows from the definition of Dirac set that a configuration has "tail" (see Fig. 3 Moreover, it follows from the definitions of set of configurations, Dirac set and the type of the graph under consideration (Fig. 2) that all − +1 are elements of one row.

Calculation of statistical sum
In this subsection, our goal will be to calculate the statistical sum of the subset of configurations Δ ∞ 2 with the fixed type of tail ( Fig. 3) relative to and , i.e.
Proof. Let's consider Proof. One can readily see that the following is true as we consider all possible options for particle arrangement at the column no. 0. Then, ). Therefore, we obtain functional equation: Going to the limit −→ ∞, we obtain the required. ▽ Remark.
due to the equality of generating functions for splitting into odd and different summands [9].

SqL algebra
The fermion algebra for the infinite in both directions graph as per Fig. 2 is the following algebra of anti-commuting elements and , ∈ Z: Let's determine the deformation of this algebra, namely the SqL algebra, generated by anti-commuting elements , , ∈ Z, satisfying the relations below (let's denote it as REL): with additional action of two operators and : [ , ] = 0; The relations REL are infinite; therefore, formally, the algebra with such relations has no sense. However, if we consider its representation in a graduated space, with peak limiting for the graduation, as is customary in the theory of conformal algebras, then everything will be determined.
Accordingly, for any integer let's define induced representations with extreme vectors, i.e. spaces ϒ( ) spawned by elements , , ∈ Z, from the vectors , for which the following is true: In other words, There are mappings This result is derived from the existence of monomial basis (Lemma 2) in ϒ( ). Now we are going to discuss it. But, to begin with, let's construct an auxiliary representation of the SqL algebra by using the Clifford algebra.

Constructing representation of SqL algebra
The algebra is generated by elements −2 −1 , ∈ N ∪ {0}, , * , ∈ Z, for which the following is true: Moreover, additional action of and is defined: Let's introduce generating functions: Let's define: The relations REL are fulfilled for ( ) and ( ): Let's suppose there exists extreme vector 0 : Let's set an action for and on 0 :
Proof. As is well known, the character of the space of polynomials in anti-commuting variables , * −1 , = 0, −1, . . . , is equal to according to the Jacobi triple product [9]. Moreover, as is well known, the character of the space of polynomials in commuting variables 2 −1 , = 0, −1, . . . , is equal to Proof. Using REL, as well as skew symmetry, any monomial̃︀ 1 . . .̃︀̃︀ 1 . . .̃︀ can be expressed in the required form as above (see [7] and [8]): each of newly appeared monomials will be lexicographically less than the original (deg( ) = deg( ) = , all follow after all ). Of course, newly appeared monomials may contain pairs breaking the conditions | − | ≥ 2, but the lexicographical order allows to develop an iterative procedure.
Let's demonstrate the linear independence. For this purpose, let's use SqL representation defined above. Let's set the homomorphism of algebras: from here we obtain the mapping of spaces: By definition of , the latter is Then, from the definition of , we obtain the following isomorphism: Therefore, Thereafter, there are no additional relationships for 1 2 . . .   So, we obtain the representation, which can be naturally identified with the set of semiinfinite configurations with the fixed type of tail.
Let's set an action of and on extreme vectors : The character of ϒ is defined as ϒ ( ). Therefore, algebra is a finite-dimensional quotient algebra of SqL. It follows from Lemma 2 that the dimension of is identical to the dimension of the fermion algebra of the twisted square ladder with columns. Moreover, they have the same monomial basis.
Symbols or also denote the operator of multiplication by the corresponding element. Let's construct a complex ∼ = with differential 0 + 0 . Let's introduce elements from 2 +1 , which are defined recursively as follows: Proposition 4. The cohomology of the complexes 2 +1 , 2 +2 is one-dimensional. The element ℎ 2 +1 is a representative of the corresponding cohomoloogy class in both cases.  2 +1 are acyclic. The cohomology of the complexes K , 2 +1 , K , 2 +1 is one-dimensional in case of odd , and elements ℎ 2 +1 , ℎ 2 +1 are a representatives of the corresponding cohomology class, respectively. If is even, then the complexes are acyclic.
The cohomology of the complexes K , 2 +1 , K , 2 +1 is one-dimensional in case of even , and elements ℎ 2 +1 , ℎ 2 +1 are a representatives of the corresponding cohomology class, respectively. If is odd, then the complexes are acyclic.
We prove Lemma 3 and Proposition 4 for the odd case by induction. The reader is encouraged to check the induction base for As is well known, a long exact sequence of cohomologies is associated with exact triple. But due to the inductive assumption, we obtain the next exact sequence for 2 +3 : Similar exact sequences can also be written for all , 2 +3 . Because we know Euler characteristics of all , 2 +1 and 2 +1 itself for any (absolute values do not exceed 1) from paper [1] and thanks to Lemma 2, then, to prove the simultaneous induction, we just need to show that Up to a change of basis, we have: Using the latter equations, it is possible to rewrite − ℎ 2 −1 ( − ℎ 2 −1 ) as an element of the monomial basis, therefore, it is not equal to zero, see Lemma 2.
The statement for 2 follows from the statement for 2 −1 and Lemma 3, as it is possible to write exact triple for 2 with 2 −1 and

Appendix: a generalization
Conjecture is that it is possible to universalize the ideas for some m-leg ladder models. That way, we obtain representations of = 2, . . . algebras.

Graded Euler characteristic
Let's recall basic definitions. For each graph Γ we can construct a statistical model in which the set of configurations is the set of arrangements of particles at graph vertices such that at each vertex at most one particle is located and two particles cannot be located at vertices joined by an edge.
To any graph Γ there corresponds the fermion algebra (Γ) defined as follows.
The statistical sum of the model is the sum over the space of all configurations. The contribution of each configuration depends on parameters. For some parameters values, the contribution of any configuration to the statistical sum equals ±1, depending on the parity of the number of particles in the given configuration. This statistical sum is naturally interpreted as the Euler characteristic of the complex (Γ). By a weight system { } on the graph Γ we mean a function on assigning a number ( ) to each vertex . For each configuration = { 1 , 2 , . . . , } we define its energy We introduce the statistical sum ∑︀ ( ), where the summation is over all configurations. We refer to this as the graded Euler characteristic and denote it by E w (Γ).  Proof. It is easy to see that such relation holds:

Cyclic 3-leg triangular ladder model
as any non-empty arrangement of particles in second column of Fig. 5 partitions the graph into two disconnected graphs, one of which is a one-point space. The Euler characteristic of a disconnected graph is the product of the Euler characteristics of its connected components. In addition, 3 1 = −2. ▽ Let's consider the weight system (which we denote by ), where each fermion in the column number , = 1, . . . , , has weight [ 2 ] + − 1.
To the cyclic 3-leg triangular ladder with columns we assign a table of weights and numbers of fermions; namely, each cell of this table contains the number of all admissible arrangements of a given number of fermions with given weights.
The number of a column in the table corresponds to the number of arranged fermions. The first column corresponds to 0 fermions; it is impossible to arrange more than fermions. The rows correspond to weights. We assume that the configuration with no fermions has weight 0.
Below we give examples of such tables. We leave a cell empty if there exist no configurations with given weight and given number of fermions. Proof. This can be proved in the same manner as in paper [1]. ▽ All definition connected with semi-infinite configurations are the same. The reader is encouraged to check it.
Let Θ denote such set of weight systems, for which is assumed that each fermion in the column number 1 on Fig. 7 has weight 0.