# On the Deformation Theory of Structure Constants for Associative Algebras

Authors: B. G. Konopelchenko
Journal: Advances in Mathematical Physics
 (2010)

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## Abstract

An algebraic scheme for constructing deformations of structure constants
for associative algebras generated by deformation driving algebras
(DDAs) is discussed. An ideal of left divisors of zero plays a central role in
this construction. Deformations of associative three-dimensional algebras
with the DDA being a three-dimensional Lie algebra and their connection
with integrable systems are studied.


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## Body

## 1. Introduction

An idea to study deformations of structure constants for associative algebras goes back to the classical works of Gerstenhaber [1, 2]. As one of the approaches to deformation theory he suggested “to take the point of view that the objects being deformed are not merely algebras, but essentially algebras with a fixed basis” and to treat “the algebraic set of all structure constants as parameter space for deformation theory” [2].Thus, following this approach, one chooses the basisP0,P1,…,PN for a given algebra A, takes the structure constants Cjkn defined by the multiplication table(1.1)PjPk=∑n=0NCjknPn,j,k=0,1,…,N,
and looks for their deformations Cjkn(x), where (x)=(x1,…,xM) is the set of deformation parameters, such that the associativity condition(1.2)∑m=0NCjkm(x)Cmln(x)=∑m=0NCklm(x)Cjmn(x)
or similar equation is satisfied.A remarkable example of deformations of this type withM=N+1 has been discovered by Witten [3] and Dijkgraaf et al. [4]. They demonstrated that the function F which defines the correlation functions 〈ΦjΦkΦl〉=∂3F/∂xj∂xk∂xl and so forth in the deformed two-dimensional topological field theory obeys the associativity equation (1.2) with the structure constants given by(1.3)Cjkl=∑m=0Nηlm∂3F∂xj∂xk∂xm,
where the constants are ηlm=(g-1)lm and glm=∂3F/∂x0∂xl∂xm where the variable x0 is associated with the units element. Each solution of the WDVV equations (1.2) and (1.3) describes a deformation of the structure constants of the N+1- dimensional associative algebra of primary fields Φj.The interpretation and formalization of the WDVV equation in terms of Frobenius manifolds proposed by Dubrovin [5, 6] provides us with a method to describe class of deformations of the so-called Frobenius algebras. An extension of this approach to general algebras and corresponding F-manifolds has been given by Hertling and Manin [7]. The beautiful and rich theory of Frobenius and F-manifolds has various applications from the singularity theory to quantum cohomology (see, e.g., [6, 8, 9]).An alternative approach to the deformation theory of the structure constants for commutative associative algebras has been proposed recently in [10–14]. Within this method the deformations of the structure constants are governed by the so-called central system (CS). Its concrete form depends on the class of deformations under consideration and CS contains, as particular reductions, many integrable systems like WDVV equation, oriented associativity equation, and integrable dispersionless, dispersive, and discrete equations (Kadomtsev-Petviashvili equation, etc.). The common feature of the coisotropic, quantum, discrete deformations considered in [10–14] is that for all of them elements pj of the basis and deformation parameters xj form a certain algebra (Poisson, Heisenberg, etc.). A general class of deformations considered in [13] is characterized by the condition that the ideal J=〈fjk〉 generated by the elements fjk=-pjpk+∑l=0NCjkl(x)pl  representing the multiplication table (1) is the Poisson ideal. It was shown that this class contains a subclass of so-called integrable deformations for which the CS has a simple and nice geometrical meaning.In the present paper we will discuss a purely algebraic formulation of such integrable deformations. We will consider the case when the algebra generating deformations of the structure constants, that is, the algebra formed by the elementspj of the basis and deformation parameters xk(deformation driving algebra (DDA)), is a Lie algebra. The basic idea is to require that all elements fjk=-pjpk+∑l=0NCjkl(x)pl are left divisors of zero and that they generate the ideal J of left divisors of zero. This requirement gives rise to the central system which governs deformations generated by DDA. This central system of equations for structure constants differs, in general, from the associativity condition. So, deformed algebras form families of commutative but not necessarily associative algebras.Here we will study the deformations of the structure constants for the three-dimensional algebra in the case when the DDA is given by one of the three-dimensional Lie algebras. Such deformations are parametrized by a single deformation variablex. Depending on the choice of DDA and identification of p1,p2, and x with the elements of DDA, the corresponding CS takes the form of the system of ordinary differential equations or the system of discrete equations (multidimensional mappings). In the first case the CS contains the third-order ODEs from the Chazy-Bureau list as the particular examples. This approach provides us also with the Lax form of the above equations and their first integrals.The paper is organized as follows. General formulation of the deformation theory for the structure constants is presented in Section2. Quantum, discrete, and coisotropic deformations are discussed in Section 3. Three-dimensional Lie algebras as DDAs are analyzed in Section 4. Deformations generated by general DDAs are studied in Section 5. Deformations driven by the nilpotent and solvable DDAs are considered in Sections 6 and 7, respectively.


## 2. Deformations of the Structure Constants Generated by DDA

So, we consider a finite-dimensional commutative algebraA with (or without) unit element P0 in the fixed basis composed by the elements P0,P1,…,PN. The multiplication table (1) defines the structure constants Cjkl. The commutativity of the basis implies that Cjkl=Ckjl. In the presence of the unit element one has Cj0l=δjl where δjl is the Kronecker symbol.Following Gerstenhaber's suggestion [1, 2] we will treat the structure constants Cjkl as the objects to deform and will denote the deformation parameters by x1,x2,…,xM.  For the undeformed structure constants the associativity conditions (1.2) are nothing else than the compatibility conditions for the table of multiplication (1.1). In the construction of deformations we should first specify a “deformed” version of the multiplication table and then require that this realization is self-consistent and meaningful.Thus, to define deformations one has the following.(1)
We associate a set of elementsp0,p1,…,pN,x1,x2,…,xM with the elements of the basis P0,P1,…,PN and deformation parameters x1,x2,…,xM.(2)
We consider the Lie algebraB of the dimension N+M with the basis elements e1,…,eN+M obeying the commutation relations:
(2.1)[eα,eβ]=∑γ=1N+MCαβγeγ,α,β=1,2,…,N+M.(3)
We identify the elementsp1,…,pN,x1,x2,…,xM with the elements e1,…,eN+M, thus defining the deformation driving algebra (DDA). Different identifications define different DDAs. We assume that the element p0 commutes with all elements of DDA and we put p0=1. The commutativity of the basis in the algebra Aimplies the commutativity between pj, and in this paper we assume the same property for all xk. So, we will consider the DDAs defined by the commutation relations of the type
(2.2)[pj,pk]=0,[xj,xk]=0,[pj,xk]=∑lαjlkxl+∑lβjklpl,
where αjlk and βjkl are some constants.(4)
We consider the elements(2.3)fjk=-pjpk+∑l=0NCjkl(x)pl,j,k=1,…,N
of the universal enveloping algebra U(B) of the algebra DDA(B). These fjk “represent” the table (1) in U(B).(5)
We require that allfjk are left zero divisors and have a common right zero divisor.In this casefjk generate the left ideal J of left zero divisors. We remind that non-zero elements a and b are called left and right divisors of zero if ab=0 (see e.g., [15]).Definition 2.1.
The structure constantsCjkl(x) are said to define deformations of the algebra A generated by given DDA if all fjk are left zero divisors with common right zero divisor.To justify this definition we first observe that the simplest possible realization of the multiplication table (1) in U(B) given by the equations fjk=0,j,k=1,…,N is too restrictive in general. Indeed, for instance, for the Heisenberg algrebra B [12] such equations imply that [pl,Cjkm(x)]=∂Cjkm/∂xl=0 and, hence, all Cjkm are constants. So, one should look for a weaker realization of the multiplication table. A condition that all fjk are left zero divisors is a natural candidate. The condition of compatibility of the corresponding equations fjk·Ψjk=0,j,k=1,…,N where Ψjk are right zero divisors requires that the l.h.s. of these equations and, hence, Ψjk should have a common divisor (see, e.g., [15]). We restrict ourselves to the case when Ψjk=Ψ·Φjk,j,k=1,…,N where Φjk are invertible elements of U(B). In this case one has the set of equations(2.4)fjk·Ψ=0,j,k=0,1,…,N;
that is, all left zero divisors fjk have common right zero divisor Ψ.These conditions impose constraints onCjkm(x). To clarify these constraints we will use the associativity of U(B). First we observe that due to the relations (2.2) one has the identity (p0=1)(2.5)[pl,Cjkm(x)]=∑t=0NΔjk,lmt(x)pt,
where  Δjk,lmt(x) are certain functions of x1,…,xM only. Then, taking into account (2.2) and associativity of U(B), one obtains(2.6)(pjpk)pl-pj(pkpl)=∑s,t=0NKkljst·fst+∑t=0NΩkljt(x)·pt,j,k,l=0,1,…,N,
where (2.7)Kkljst=12(δksδlt+δktδls)pj-12(δksδjt+δktδjs)pl+12(δjsCklt+δjtCkls)-12(δlsCkjt+δltCkjs)+Δkl,jst-Δkj,lst,Ωkljt(x)=∑sCjksClst-∑sClksCjst+∑s,n(Δkj,lsn-Δkl,jsn)Csnt.Thus, the identity (2.6) gives(2.8)∑s,t=0NKkljst·fst+∑t=0NΩkljt(x)·pt=0,j,k,l=0,1,…,N.
Due to the relations (2.4), (2.8) implies that(2.9)(∑t=0NΩkljt(x)·pt)Ψ=0.
These equations are satisfied if(2.10)Ωkljt(x)=∑sCjksClst-∑sClksCjst+∑s,n(Δkj,lsn-Δkl,jsn)Csnt=0,j,k,l,t=0,1,…,N.This system of equations plays a central role in our approach. IfΨ has no left zero divisors linear in pj, the relation (2.10) is the necessary condition for existence of a common right zero divisor for fjk since U(B) has no zero elements linear in pj (see e.g., [16]).AtN≥3 it is also a sufficient condition. Indeed, if Cjkm(x) are such that (2.10) is satisfied, then(2.11)∑s,t=0NKkljst·fst=0,j,k,l=0,1,…,N.Generically, it is the system of(1/2)N2(N-1) linear equations for N(N+1)/2 unknowns fst with noncommuting coefficients Kkljst. At N≥3 for generic (nonzeros, nonzero divisors) Kkljst(x,p) the system (2.11) implies that(2.12)αjkfjk=βlmflm,j,k,l,m=1,…,N,(2.13)γjkfjk=0,j,k=1,…,N,
where αjk,βlm, and γjk are certain elements of U(B) (see e.g., [17, 18]). Thus, all fjk are right zero divisors. They are also left zero divisors. Indeed, due to Ado's theorem (see e.g., [16]) finite-dimensional Lie algebra B and, hence, U(B) are isomorphic to matrix algebras. For the matrix algebras zero divisors (matrices with vanishing determinants) are both right and left zero divisors [15]. Then, under the assumption that all αjk and βlm are not zero divisors, the relations (2.12) imply that the right divisor of one of fjk is also the right zero divisor for the others.AtN=2 one has only two relations of the type (2.12) and a right zero divisor of one of f11,f12,f22 is the right zero divisor of the others. We note that it is not easy to control assumptions mentioned above. Nevertheless, (2.4) and (2.10) certainly are fundamental one for the whole approach.We will refer to the system (2.10) as the Central System (CS) governing deformations of the structure constants of the algebra Agenerated by a given DDA. Its concrete form depends strongly on the form of the brackets [pt,Cjkl(x)] which are defined by the relations (2.2) for the elements of the basis of DDA. For stationary solutions (Δjk,lmt=0) the CS (2.10) is reduced to the associativity conditions (1.2).


## 3. Quantum, Discrete, and Coisotropic Deformations

Coisotropic, quantum, and discrete deformations of associative algebras considered in [10–14] represent particular realizations of the above general scheme associated with different DDAs.For thequantum deformations one has M=N and the deformation driving algebra is given by the Heisenberg algebra [12]. The elements of the basis of the algebra A and deformation parameters are identified with the elements of the Heisenberg algebra in such a way that(3.1)[pj,pk]=0,[xj,xk]=0,[pj,xk]=ℏδjk,j,k=1,…,N,
where ℏ is the real constant (Planck's constant in physics). For the Heisenberg DDA(3.2)Δjk,lmt=ℏδ0t∂Cjkm(x)∂xl,
and consequently(3.3)Ωkljn(x)=ℏ∂Cjkn∂xl-ℏ∂Ckln∂xj+∑m=0N(CjkmCmln-CklmCjmn)=0,j,k,l,n=0,1,…,N.Quantum CS (3.3) governs deformations of structure constants for associative algebra driven by the Heisenberg DDA. It has a simple geometrical meaning of vanishing Riemann curvature tensor for torsionless Christoffel symbols Γjkl identified with the structure constants (Cjkl=ℏΓjkl) [12].In the representation of the Heisenberg algebra (3.1) by operators acting in a linear space H left divisors of zero are realized by operators with nonempty kernel. The ideal J is the left ideal generated by operators fjk which have nontrivial common kernel or, equivalently, for which equations(3.4)fjk|Ψ〉=0,j,k=1,2,…,N
have nontrivial common solutions |Ψ〉⊂H. The compatibility condition for (3.4) is given by the CS (3.3). The common kernel of the operators fjk forms a  subspace HΓ in the linear space H. So, in the approach under consideration the multiplication table (1) is realized only on HΓ, but not on the whole H. Such type of realization of the constraints is well known in quantum theory as Dirac's recipe for quantization of the first-class constraints [19]. In quantum theory context equation (3.4) serves to select the physical subspace in the whole Hilbert space. Within the deformation theory one may refer to the subspace HΓ as the “structure constants” subspace. In [12] the recipe (3.4) was the starting point for construction of the quantum deformations.Quantum CS (3.3) contains various classes of solutions which describe different classes of deformations. An important subclass is given by isoassociative deformations, that is, by deformations for which the associativity condition (1.2) is valid for all values of deformation parameters. For such quantum deformations the structure constants should obey the following equations:(3.5)∂Cjkn∂xl-∂Ckln∂xj=0,j,k,l,n=1,…,N.
These equations imply that Cjkn=∂2Φn/∂xj∂xk where Φn are some functions while the associativity condition (1.2) takes the following form:(3.6)∑m=0N∂2Φm∂xj∂xk∂2Φn∂xm∂xl=∑m=0N∂2Φm∂xl∂xk∂2Φn∂xm∂xj.
It is the oriented associativity equation introduced in [5, 20]. Under the gradient reduction Φn=∑l=0Nηnl(∂F/∂xl) equation (3.7) becomes the WDVV equations (1.2) and (1.3).Non-isoassociative deformations for which the condition (3.5) is not valid are of interest too. They are described by some well-known integrable soliton equations [12]. In particular, there are Boussinesq equation among them for N=2 and Kadomtsev-Petviashvili (KP) hierarchy for the infinite-dimensional algebra of polynomials in the Faa' de Bruno basis [12]. In the latter case the deformed structure constants are given by(3.7)Cjkl=δj+kl+Hj-lk+Hk-lj,j,k,l=0,1,2,…
with(3.8)Hkj=1ℏPk(-ℏ∂̃)∂logτ∂xj,j,k=1,2,3,…,
where τ is the famous tau-function for the KP hierarchy and Pk(-ℏ∂̃)≑Pk(-ℏ(∂/∂x1),(-1/2)ℏ(∂/∂x2),-(1/3)ℏ(∂/∂x3),…) where Pk(t1,t2,t3,…) are Schur polynomials defined by the generating formula exp(∑k=1∞λktk)=∑k=0∞λkPk(t).Discrete deformations of noncommutative associative algebras are generated by the DDA withM=N and commutation relations(3.9)[pj,pk]=0,[xj,xk]=0,[pj,xk]=δjkpj,j,k=1,…,N.
In this case(3.10)Δjk,lmt=δlt(Tl-1)Cjkm(x),j,k,l,m,t=0,1,2,…,N,
where for an arbitrary function φ(x) the action of Tj is defined by Tjφ(x0,…,xj,…,xN)=φ(x0,…,xj+1,…,xN). The corresponding CS is of the form(3.11)ClTlCj-CjTjCl=0,j,l=0,1,…,N,
where the matrices Cj are defined as (Cj)kl=Cjkl,j,k,l=0,1,…,N. The discrete CS (3.11) governs discrete deformations of associative algebras.  The CS (3.11) contains, as particular cases, the discrete versions of the oriented associativity equation, WDVV equation, Boussinesq equation, and discrete KP hierarchy and Hirota-Miwa bilinear equations for KP τ-function [13].Forcoisotropic deformations of commutative algebras [10, 11] again M=N, but the DDA is the Poisson algebra with pj and xk identified with the Darboux coordinates, that is,(3.12){pj,pk}=0,{xj,xk}=0,{pj,xk}=-δjk,j,k=0,1,…,N,
where {,} is the standard Poisson bracket. The algebra U(B) is the commutative ring of functions and divisors of zero are realized by functions with zeros. So, the functions fjk should be functions with common set Γ of zeros. Thus, in the coisotropic case the multiplication table (1) is realized by the following set of equations [10]:(3.13)fjk=0,j,k=0,1,2,…,N.
The compatibility condition for these equations is (see e.g., [10])(3.14){fjk,fnl}|Γ=0,j,k,l,n=1,2,…,N.
The set Γ is the coisotropic submanifold in R2(N+1). The condition (3.14) gives rise to the following system of equations for the structure constants:(3.15)[C,C]jklrm≑∑s=0N(Csjm∂Clrs∂xk+Cskm∂Clrs∂xj-Csrm∂Cjks∂xl-Cslm∂Cjks∂xr+Clrs∂Cjkm∂xs-Cjks∂Clrm∂xs)=0
while the equations Ωkljn(x)=0 have the form of associativity conditions  (1.2):(3.16)Ωkljn(x)=∑m=0N(Cjkm(x)Cmln(x)-Cklm(x)Cjmn(x))=0.Equations (3.15) and (3.16) form the CS for coisotropic deformations [10]. In this case Cjkl is transformed as the tensor of the type (1,2) under the general transformations of coordinates xj, and the whole CS of (3.15) and (3.16) is invariant under these transformations [14]. The bracket [C,C]jklrm has appeared for the first time in [21] where the so-called differential concomitants were studied. It was shown in [16] that this bracket is a tensor only if the tensor Cjkl obeys the algebraic constraint (3.16). In [7] the CS of (3.15) and (3.16) has appeared implicitly as the system of equations which characterizes the structure constants for F-manifolds. In [10] it has been derived as the CS governing the coisotropic deformations of associative algebras.The CS of (3.15) and (3.16) contains the oriented associativity equation, the WDVV equation, dispersionless KP hierarchy, and equations from the genus zero universal Whitham hierarchy as the particular cases [10, 11]. Yano manifolds and Yano algebroids associated with the CS of (3.15) and (3.16) are studied in [14].We would like to emphasize that for all deformations considered above the stationary solutions of the CSs obey the global associativity condition (1.2).


## 4. Three-Dimensional Lie Algebras as DDA

In the rest of the paper we will study deformations of associative algebras generated by three-dimensional real Lie algebraL. The complete list of such algebras contains 9 algebras (see e.g. [16]). Denoting the basis elements by e1,e2,e3,  one has the following nonequivalent cases:(1)
abelian algebraL1,(2)
general algebraL2: [e1,e2]=e1,[e2,e3]=0,[e3,e1]=0,(3)
nilpotent algebraL3:[e1,e2]=0,[e2,e3]=e1,[e3,e1]=0,(4)–(7)
four nonequivalent solvable algebras:[e1,e2]=0,[e2,e3]=αe1+βe2,[e3,e1]=γe1+δe2 with αδ-βγ≠0,(8)-(9)
simple algebrasL8= so (3) and L9= so (2,1).In virtue of the one-to-one correspondence between the elements of the basis in DDA and the elementspj, xk an algebra L should have an abelian subalgebra and only one of its elements may play the role of the deformation parameter x. For the original algebra Aand the algebra B one has two options.(1)
A is a two-dimensional algebra without unit element and B=L.(2)
A is a three-dimensional algebra with the unit element and B = L0⊕L where L0 is the algebra generated by the unity element p0.After the choice ofB one should establish a correspondence between p1,p2,x and e1,e2,e3 defining DDA. For each algebra Lk there are obviously, in general, six possible identifications if one avoids linear superpositions. Some of them are equivalent. The incomplete list of nonequivalent identifications is as follows(1)
algebraL1:p1=e1,p2=e2,x=e3; DDA is the commutative algebra with
(4.1)[p1,p2]=0,[p1,x]=0,[p2,x]=0,(2)
algebraL2:
case (a)
p1=-e2,p2=e3,x=e1; the corresponding DDA is the algebra L2a with the commutation relations:
(4.2)[p1,p2]=0,[p1,x]=x,[p2,x]=0,case (b)
p1=e1,p2=e3,x=e2; the corresponding DDA L2b is defined by
(4.3)[p1,p2]=0,[p1,x]=p1,[p2,x]=0,(3)
algebraL3: p1=e1,p2=e2,x=e3; DDA L3 is
(4.4)[p1,p2]=0,[p1,x]=0,[p2,x]=p1,(4)
solvable algebraL4 with α=0,β=1,γ=-1,δ=0:p1=e1,p2=e2,x=e3; DDA L4 is
(4.5)[p1,p2]=0,[p1,x]=p1,[p2,x]=p2,(5)
solvable algebraL5 at α=1,β=0,γ=0,δ=1:p1=e1,p2=e2,x=e3; DDA L5 is
(4.6)[p1,p2]=0,[p1,x]=p1,[p2,x]=-p2.For the second choice of the algebraB=L0⊕L mentioned above the table of multiplication  (1.1) consists of the trivial part P0Pj=PjP0=Pj,j=0,1,2  and the nontrivial part:(4.7)P12=AP0+BP1+CP2,P1P2=DP0+EP1+GP2,P22=KP0+MP1+NP2.For the first choiceB=K the multiplication table is given by (4.7) with A=D=K=0.It is convenient also to arrange the structure constantsA,B,…,N into the matrices C1,C2 defined by (Cj)kl=Cjkl. One has(4.8)C1=(0AD1BE0CG),C2=(0DK0EM1GN).In terms of these matrices the associativity conditions (1.2) are written as(4.9)C1C2=C2C1.Simple algebrasL8 and L9 do not contain two commuting elements to be identified with p1 and p2, and, hence, they cannot be DDA. Deformations generated by algebras L6 and L7 will be considered elsewhere.


## 5. Deformations Generated by General DDAs

(1) Commutative DDA (4.1) does not force any deformation of structure constants. So, we begin with the three-dimensional commutative algebra Aand DDA L2a defined by the commutation relations (4.2). These relations imply that for an arbitrary function φ(x)(5.1)[pj,φ(x)]=Δjφ(x),j=1,2,
where Δ1=(x∂/∂x),Δ2=0. Consequently, one has the following CS:(5.2)Ωkljn(x)=ΔlCjkn-ΔjCkln+∑m=02(CjkmClmn-CklmCjmn)=0,j,k,l,n=0,1,2.
In terms of the matrices C1 and C2 defined above this CS has a form of the Lax equation:(5.3)x∂C2∂x=[C2,C1].The CS (5.3) has all remarkable standard properties of the Lax equations (see e.g. [20, 21]): it has three independent first integrals:(5.4)I1=trC2,I2=12tr(C2)2,I3=13tr(C2)3,
and it is equivalent to the compatibility condition of the linear problems:(5.5)C2Φ=λΦ,x∂Φ∂x=-C1Φ,
where Φ is the column with three components and λ is a spectral parameter. Though the evolution in x described by the second linear problem (5.5) is too simple, nevertheless the CS (5.2) or (5.3) has the meaning of the isospectral deformations of the matrix C2 that is typical to the class of integrable systems (see e.g. [22, 23]).CS (5.3) is the system of six equations for the structure constants D,E,G,L,M,N with free A,B,C:(5.6)D′=DB+KC-AE-DG,K′=DE+KG-AM-DN,E′=MC-EG-D,M′=E2+MG-BM-EN-K,G′=GB+NC-CE-G2+A,N′=GE-CM+D,
where D′=x∂D/∂x and so forth. Here we will consider only simple particular cases of the CS (5.6). First it corresponds to the constraint A=0, B=0, C=0, that is, to the nilpotent P1. The corresponding solution is(5.7)D=βlnx,E=-β+γlnx,G=1lnx,K=αβ+2β2+δlnx-βγlnx,M=αγ+3βγ+μlnx-δ(lnx)2-γ2lnx,N=α+β-γlnx,
where α,β,γ,δ,μ are arbitrary constants. The three integrals for this solution are(5.8)I1=α,I2=12α2+3β2+2αβ+μ,I3=13((α+β)3-β3)+(α+β)(μ+β(α+2β))-γδ.The second example is given by the constraintB=0, C=1, G=0 for which the quantum CS (3.3) is equivalent to the Boussinesq equation [12]. Under this constraint the CS (5.6) is reduced to the single equation:(5.9)E′′-6E2+4αE+β=0,
and the other structure constants are given by(5.10)A=2E-α,B=0,C=1,D=γ-12E′,G=0,K=-E2+αE+12β,M=γ+12E′,N=α-N,
where α,β,γ are arbitrary constants. The corresponding first integrals are(5.11)I1=α,I2=12(β+α2),I3=13α3+γ2+12αβ-14(E′)2+E3-αE2-12βE.
Integral I3 reproduces the well-known first integral of (5.9). Solutions of (5.9) are given by elliptic integrals (see e.g., [24]). Any such solution together with the  formulae (5.10) describes deformation of the three-dimensional algebra A driven by DDA L2a.Now we will consider deformations of the two-dimensional algebraAwithout unit element according to the first option mentioned in the previous section. In this case the CS has the form (5.3) with the 2×2 matrices(5.12)C1=(BECG),C2=(EMGN)
or in components(5.13)E′=MC-EG,M′=E2+MG-BM-EN,G′=GB+NC-CE-G2,N′=GE-CM.
In this case there are two independent integrals of motion:(5.14)I1=E+N,I2=12(E2+N2+2MG).The corresponding spectral problem is given by (5.5). Eigenvalues of the matrix C2, that is, λ1,2=(1/2)(E+N±(E-N)2+4GM) are invariant under deformations and detC2=(1/2)I12-I2. We note also an obvious invariance of (5.6) and (5.13) under the rescaling of x.The system of (5.13) contains two arbitrary functions B and C. In virtue of the possible rescaling P1→μ1P1,P2→μ2P2 of the basis for the algebra Awith two arbitrary functions μ1,μ2, one has four nonequivalent choices (1) B=0, C=0, (2) B=1, C=0, (3) B=0, C=1, and (4) B=1, C=1.In the caseB=0, C=0 (nilpotent P1) the solution of the system (5.13) is(5.15)B=0,C=0,E=βlnx,G=1lnx,M=γlnx-β2lnx+αβ,N=-βlnx+α,
where α, β,γ are arbitrary constants. For this solution the integrals are equal to I1=α,I2=γ+(1/2)α2, and λ1,2=(1/2)(α+α2+4γ).AtB=1, C=0 the system (5.13) has the following solution:(5.16)B=1,C=0,E=γx+β,G=xx+β,M=δ+(αγ+βδ-γ2β)1x+γ2β(x+β),N=-γx+β+α,
where α,β,γ,δ are arbitrary constants. The integrals are I1=α,I2=δ+(1/2)α2. The formulae (5.15) and (5.16) provide us with explicit deformations of the structure constants.In the last two cases the CS (5.13) is equivalent to the simple third-order ordinary differential equations. At B=0, C=1 with additional constraint I1=0 one gets(5.17)G′′′+2G2G′+4(G′)2+2GG′′=0
while at B=1, C=1, and I1=0 the system (5.13) becomes(5.18)G′′′+2G2G′+4(G′)2+2GG′′-G′=0.
The second integral for these ODEs is(5.19)I2=-12G4+12(G′)2-2G2G′-GG′′+12BG2.
Equation (5.17) with G′=∂G/∂y is the Chazy V equation from the well-known Chazy-Bureau list of the third-order ODEs having Painlevé property [25, 26]. The integral (5.19) is known too (see e.g. [27]).The appearance of the Chazy V equation among the particular cases of the system (5.13) indicates that for other choices of B and C the CS (5.13) may be equivalent to the other notable third-order ODEs. It is really the case. Here we will consider only the reduction C=1 with I1=N+E=0. In this case the system (5.13) is reduced to the following equation:(5.20)G′′′+2G2G′+4(G′)2+2GG′′-2G′Φ-GΦ′=0,
where Φ=B′+(1/2)B2. The second integral is(5.21)I2=-12G4+12(G′)2-2G2G′-GG′′+ΦG2,
and λ1,2=±I2/2.Choosing particularB or Φ, one gets equations from the Chazy-Bureau list. Indeed, at Φ=0 one has the Chazy V equation (5.17). Choosing Φ=G′, one gets the Chazy VII equation:(5.22)G′′′+2G2G′+2(G′)2+GG′′=0.
At B=2G (5.20) becomes the Chazy VIII equation:(5.23)G′′′-6G2G′=0.Choosing the functionΦ such that(5.24)(6Φe(1/3)G)′=2G2G′+(G′)2+4GG′′,
one gets the Chazy III equation:(5.25)G′′′-2GG′′+3(G′)2=0.
In the above particular cases the integral I2 (5.21) is reduced to those given in [27].All Chazy equations presented above have the Lax representation (5.3) with E=-N=-(1/2)(G′+G2+GB),M=-(1/2)(G′′+3GG′+G3+G2B+(GB)′),C=1, and the proper choice of B.Solutions of all these Chazy equations provide us with the deformations of the structure constants (5.12) for the two-dimensional algebra A generated by the DDA L2a.(2) Now we pass to the DDA L2b. The commutation relations (4.3) imply that(5.26)[p1,φ(x)]=(T-1)φ(x)·p1,[p2,φ(x)]=0,
where φ(x) is an arbitrary function and Tφ(x)=φ(x+1). Using (5.26), one finds the corresponding CS:(5.27)∑m=02((Δl+1)Cjkm(x)·Clmn(x)=(Δj+1)Cklm(x)·Cjmn(x)),j,k,l,n=0,1,2,
where Δ1=T-1,Δ2=0. In terms of the matrices C1 and C2, this CS is(5.28)C1TC2=C2C1.
For nondegenerated matrix C1 one has(5.29)TC2=C1-1C2C1.The CS (5.29) is the discrete version of the Lax equation (5.3) and has similar properties. It has three independent first integrals:(5.30)I1=trC2,I2=12tr(C2)2,I3=13tr(C2)3,
and it represents itself the compatibility condition for the linear problems:(5.31)ΦC2=λΦ,TΦ=ΦC1.
Note that detC2 is the first integral too.The CS (5.28) is the discrete dynamical system in the space of the structure constants. For the two-dimensional algebra A with matrices (5.12) it is(5.32)BTE+ETG=EB+MC,BTM+ETN=E2+MG,CTE+GTG=BG+CN,CTM+GTN=EG+NG,
where B and C are arbitrary functions. For nondegenerated matrix C1, that is, at BG-CE≠0, one has the resolved form (5.29), that is,(5.33)TE=GM-ENBG-CEC,TG=B+BN-CMBG-CEC,TM=GM-ENBG-CEG,TN=E+BN-CMBG-CEG.
This system defines discrete deformations of the structure constants.


## 6. Nilpotent DDA

For the nilpotent DDAL3, in virtue of the defining relations (4.5), one has(6.1)[p1,φ(x)]=0,[p2,φ(x)]=∂φ∂x·p1
or(6.2)[pj,φ(x)]=∂φ∂x·∑k=12ajkpk,
where a21=1,a11=a12=a22=0. Using (6.2), one gets the following CS:(6.3)∑q=12alq∑m=02Cqmn∂Cjkm∂x-∑q=12ajq∑m=02Cqmn∂Cklm∂x+∑m=02(CjkmClmn-CklmCjmn)=0,j,k,l,n=0,1,2.
In the matrix form it is(6.4)C1∂C1∂x=[C1,C2].
For invertible matrix C1(6.5)∂C1∂x=C1-1[C1,C2].
This system of ODEs has three independent first integrals:(6.6)I1=trC1,I2=12tr(C1)2,I3=13tr(C1)3,
and it is equivalent to the compatibility condition for the linear system:(6.7)C1Φ=λΦ,C1∂Φ∂x+C2Φ=0.
So, as in the previous section the CS (6.4) describes isospectral deformations of the matrix C1. This CS governs deformations generated by L3.For the two-dimensional algebraA without unit element the CS is given by (6.4) with the matrices (5.12). First integrals in this case are I1=B+G,I2=(1/2)(B2+G2+2CE) and detC1=(1/2)I12-I2. Since detC1 is a constant on the solutions of the system, then at detC1≠0 one can always introduce the variable y defined by x=ydetC1 such that CS (6.5) takes the form(6.8)B′=EBG+ENC-GMC-CE2,E′=GBM+GEN-ECM-MG2,C′=BCE+BG2+MC2-CEG-BNC-GB2,G′=CMG+CE2-CEN-BGE,
where B′=∂B/∂y and so forth and M, N are arbitrary functions. At detC1=BG-CE=1 this system becomes(6.9)B′=E+C(EN-GM),E′=M+G(EN-GM),C′=G-B+C(MC-BN),G′=-E-C(EN-GM).ChoosingM=N=0, one gets(6.10)B′=E,E′=0,C′=G-B,G′=-E.
The solution of this system is(6.11)E=α,B=αy+β,G=-αy+γ,C=-y2+(γ-β)y+δ,
where α,β,γ,δ are arbitrary constants subject to the constraint βγ-αδ=1. First integrals for this solution are I1=β+γ,I2=(1/2)(β2+γ2+2αδ).With the choiceM=0, N=1 and under the constraint I1=B+G=0 the system (6.8) takes the form(6.12)B′=(1+C)E,E′=-BE,C′=-(2+C)B.
This system can be written as a single equation in the different equivalent forms. One of them is(6.13)(E′)2+αE4-2E3+E2=0,
where α is an arbitrary constant and(6.14)B2=-1-αE2+2E,C=αE-2,G=-B.
The second integral is equal to -1.Solutions of (6.13) can be expressed through the elliptic integrals. Solutions of (6.13) and the formulae (6.14) define deformations of the structure constants driven by DDA L3.


## 7. Solvable DDAs

(1) For the solvable DDA L4 the relations of (4.5) imply that(7.1)[pj,φ(x)]=(T-1)φ(x)pj,j=1,2,
where φ(x) is an arbitrary function and T is the shift operator Tφ(x)=φ(x+1). With the use of (7.1) one arrives at the following CS:(7.2)C1TC2=C2TC1.
For nondegenerated matrix C1 (7.2) is equivalent to the equation T(C2C1-1)=C1-1C2 or(7.3)TU=C1-1UC1,
where U≑C2C1-1. Using this form of the CS, one promptly concludes that the CS (7.2) has three independent first integrals:(7.4)I1=tr(C2C1-1),I2=12tr(C2C1-1)2,I3=13tr(C2C1-1)3,
and it is representable as the commutativity condition for the linear system:(7.5)ΦC2C1-1=λΦ,TΦ=ΦC1.For the two-dimensional algebraA one has the CS (7.2) with the matrices (5.12). It is the system of four equations for six functions:(7.6)BTE+ETG=ETB+MTC,BTM+ETN=ETE+MTG,CTE+GTG=GTB+NTC,CTM+GTN=GTE+NTG.ChoosingB and C as free functions and assuming that BG-CE≠0, one can easily resolve (7.6) with respect to TE,TG,TM,TN. For instance, with B=C=1 one gets the following four-dimensional mapping:(7.7)TE=M-EM-NE-G,TG=1+M-NE-G,TM=N+(N-G)M-NE-G-G(M-NE-G)2,TN=M+(1-E)M-NE-G+(M-NE-G)2.(2) In a similar manner one finds the CS associated with the solvable DDA L5. Since in this case(7.8)[p1,φ(x)]=(T-1)φ(x)p1,[p2,φ(x)]=(T-1-1)φ(x)p2,
the CS takes the form(7.9)C1TC2=C2T-1C1.
For nondegenerated C2 it is equivalent to(7.10)TV=C2VC2-1,
where V≑T-1C1·C2. Similar to the previous case the CS has three first integrals:(7.11)I1=tr(C1TC2),I2=12tr(C1TC2)2,I3=13tr(C1TC2)3,
and it is equivalent to the compatibility condition for the linear system:(7.12)(T-1C1)C2Φ=λΦ,TΦ=C2Φ.
Note that the CS (7.9) is of the form (3.11) with T1=T,T2=T-1. Thus, the deformations generated by L5  can be considered as the reductions of the discrete deformations (3.11) under the constraint T1T2Cjkn=Cjkn.A class of solutions of the CS (7.9) is given by(7.13)Cj=g-1Tjg,
where g is 3×3 matrix and T0=1,T1=T,T2=T-1. Since Cjkn=Ckjn, one has Tjglm=Tlgjm and hence gjm=TjΦm where Φ0,Φ1,Φ2 are arbitrary functions. So, this subclass of deformations are defined by three arbitrary functions.To describe the isoassociative deformations for whichC1(x)C2(x)=C2(x)C1(x) for all x these functions should obey the systems of equations:(7.14)∑l,t=02TjTtΦn·(g-1)lt·TkTmΦl=∑l,t=02TkTtΦn·(g-1)lt·TjTmΦl,j,k,n,m=0,1,2.
It is a version of the discrete oriented associativity equation.



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*Source: 389091-2010-07-14.xml*