Volume 2010, Article ID 389091,21pages doi:10.1155/2010/389091

*Research Article*

**On the Deformation Theory of Structure Constants** **for Associative Algebras**

**B. G. Konopelchenko**

*Dipartimento di Fisica, Universita del Salento and INFN, Sezione di Lecce, 73100 Lecce, Italy*

Correspondence should be addressed to B. G. Konopelchenko,konopel@le.infn.it Received 5 October 2009; Revised 30 March 2010; Accepted 16 May 2010

Academic Editor: Alexander P. Veselov

Copyrightq2010 B. G. Konopelchenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An algebraic scheme for constructing deformations of structure constants for associative algebras generated by deformation driving algebrasDDAsis 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.

**1. Introduction**

An idea to study deformations of structure constants for associative algebras goes back to the classical works of Gerstenhaber1,2. 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 basis**P**0*,***P**1*, . . . ,***P***N*for a given algebra
*A, takes the structure constantsC*_{jk}* ^{n}* defined by the multiplication table

**P***j***P***k*^{N}

*n0*

*C*^{n}_{jk}**P***n**,* *j, k*0,1, . . . , N, 1.1
and looks for their deformations*C*_{jk}* ^{n}*x,wherex x

^{1}

*, . . . , x*

*is the set of deformation parameters, such that the associativity condition*

^{M}*N*
*m0*

*C*^{m}* _{jk}*xC

_{ml}*x*

^{n}

^{N}*m0*

*C*^{m}* _{kl}*xC

_{jm}*x 1.2*

^{n}or similar equation is satisfied.

A remarkable example of deformations of this type with *M* *N* 1 has been
discovered by Witten 3 and Dijkgraaf et al. 4. They demonstrated that the function *F*
which defines the correlation functions Φ*j*Φ*k*Φ*l* *∂*^{3}*F/∂x*^{j}*∂x*^{k}*∂x** ^{l}* and so forth in the
deformed two-dimensional topological field theory obeys the associativity equation 1.2
with the structure constants given by

*C*^{l}_{jk}^{N}

*m0*

*η*^{lm}*∂*^{3}*F*

*∂x*^{j}*∂x*^{k}*∂x*^{m}*,* 1.3

where the constants are*η** ^{lm}* g

^{−1}

*and*

^{lm}*g*

_{lm}*∂*

^{3}

*F/∂x*

^{0}

*∂x*

^{l}*∂x*

*where the variable*

^{m}*x*

^{0}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 fieldsΦ

*j*.

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 cohomologysee, e.g.,6,8,9.

An alternative approach to the deformation theory of the structure constants for
commutative associative algebras has been proposed recently in10–14. 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 in10–14is that for all of them elements*p**j*of the basis and
deformation parameters*x** _{j}*form a certain algebraPoisson, Heisenberg, etc.. A general class
of deformations considered in13is characterized by the condition that the ideal

*J*f

*jk*generated by the elements

*f*

*jk*−p

*j*

*p*

*k*

_{N}*l0**C*^{l}* _{jk}*xp

*l*representing the multiplication table 1is 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 elements*p** _{j}*of the basis and deformation
parameters

*x*

*deformation driving algebraDDA, is a Lie algebra. The basic idea is to require that all elements*

_{k}*f*

*jk*−p

*j*

*p*

*k*

_{N}*l0**C*^{l}* _{jk}*xp

*l*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 variable*x. Depending*
on the choice of DDA and identification of *p*_{1}*, p*_{2}, 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 equationsmultidimensional 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 Section 2. 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 Sections6and7, respectively.

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

So, we consider a finite-dimensional commutative algebra*A*withor withoutunit element
**P**0 in the fixed basis composed by the elements **P**0*,***P**1*, . . . ,***P***N*. The multiplication table1
defines the structure constants*C*^{l}* _{jk}*. The commutativity of the basis implies that

*C*

^{l}

_{jk}*C*

^{l}*. In the presence of the unit element one has*

_{kj}*C*

^{l}

_{j0}*δ*

^{l}*where*

_{j}*δ*

^{l}*is the Kronecker symbol.*

_{j}Following Gerstenhaber’s suggestion1,2we will treat the structure constants*C*^{l}* _{jk}*
as the objects to deform and will denote the deformation parameters by

*x*

^{1}

*, x*

^{2}

*, . . . , x*

*. For the undeformed structure constants the associativity conditions 1.2are nothing else than the compatibility conditions for the table of multiplication1.1. 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.*

^{M}Thus, to define deformations one has the following.

1We associate a set of elements*p*_{0}*, p*_{1}*, . . . , p*_{N}*, x*^{1}*, x*^{2}*, . . . , x** ^{M}*with the elements of the
basis

**P**0

*,*

**P**1

*, . . . ,*

**P**

*N*and deformation parameters

*x*

^{1}

*, x*

^{2}

*, . . . , x*

*.*

^{M}2*We consider the Lie algebra B of the dimension* *NM*with the basis elements
*e*1*, . . . , e** _{NM}*obeying the commutation relations:

*e**α**, e**β*

^{NM}

*γ1*

*C**αβγ**e**γ**,* *α, β*1,2, . . . , N*M.* 2.1
3We identify the elements*p*_{1}*, . . . , p*_{N}*, x*^{1}*, x*^{2}*, . . . , x** ^{M}*with the elements

*e*

_{1}

*, . . . , e*

*, thus defining the deformation driving algebra DDA. Diﬀerent identifications define diﬀerent DDAs. We assume that the element*

_{NM}*p*0commutes with all elements of DDA and we put

*p*

_{0}

*1. The commutativity of the basis in the algebra A implies*the commutativity between

*p*

*, and in this paper we assume the same property for all*

_{j}*x*

*. So, we will consider the DDAs defined by the commutation relations of the type*

^{k}*p*_{j}*, p*_{k}

0,
*x*^{j}*, x*^{k}

0,
*p*_{j}*, x*^{k}

*l*

*α*^{k}_{jl}*x*^{l}

*l*

*β*^{kl}_{j}*p*_{l}*,* 2.2
where*α*^{k}* _{jl}*and

*β*

^{kl}*are some constants.*

_{j}4We consider the elements

*f** _{jk}* −p

*j*

*p*

_{k}

^{N}*l0*

*C*^{l}* _{jk}*xp

*l*

*,*

*j, k*1, . . . , N 2.3

of the universal enveloping algebra *UB* of the algebra DDAB. These *f**jk*

“represent” the table1in*UB.*

5We require that all*f**jk*are left zero divisors and have a common right zero divisor.

In this case*f**jk*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*0see e.g.,15.

*Definition 2.1. The structure constantsC*^{l}* _{jk}*x

*are said to define deformations of the algebra A*generated by given DDA if all

*f*

*jk*are left zero divisors with common right zero divisor.

To justify this definition we first observe that the simplest possible realization of the
multiplication table1in*UB*given by the equations*f** _{jk}*0, j, k1, . . . , Nis too restrictive
in general. Indeed, for instance, for the Heisenberg algrebra

*B*12such equations imply that p

*l*

*, C*

^{m}*x*

_{jk}*∂C*

^{m}

_{jk}*/∂x*

*0 and, hence, all*

^{l}*C*

^{m}*are constants. So, one should look for a weaker realization of the multiplication table. A condition that all*

_{jk}*f*

*are left zero divisors is a natural candidate. The condition of compatibility of the corresponding equations*

_{jk}*f*

*·Ψ*

_{jk}*jk*0, j, k 1, . . . , NwhereΨ

*jk*are right zero divisors requires that the l.h.s. of these equations and, hence, Ψ

*jk*should have a common divisorsee, e.g.,15. We restrict ourselves to the case when Ψ

*jk*Ψ·Φ

*jk*

*, j, k*1, . . . , NwhereΦ

*jk*are invertible elements of

*UB. In this case one has*the set of equations

*f**jk*·Ψ 0, *j, k*0,1, . . . , N; 2.4
that is, all left zero divisors*f** _{jk}*have common right zero divisorΨ.

These conditions impose constraints on*C*^{m}* _{jk}*x. To clarify these constraints we will use
the associativity of

*UB. First we observe that due to the relations*2.2one has the identity p01

*p*_{l}*, C*_{jk}* ^{m}*x

^{N}*t0*

Δ^{mt}* _{jk,l}*xp

*t*

*,*2.5

whereΔ^{mt}* _{jk,l}*xare certain functions of

*x*

^{1}

*, . . . , x*

*only. Then, taking into account 2.2and associativity of*

^{M}*UB, one obtains*

*p*_{j}*p*_{k}*p** _{l}*−

*p*

_{j}*p*_{k}*p*_{l}^{N}

*s,t0*

*K*^{st}* _{klj}*·

*f*

_{st}

^{N}*t0*

Ω^{t}* _{klj}*x·

*p*

_{t}*,*

*j, k, l*0,1, . . . , N, 2.6

where

*K*_{klj}* ^{st}* 1
2

*δ*_{k}^{s}*δ*_{l}^{t}*δ*^{t}_{k}*δ*^{s}_{l}*p**j*−1
2

*δ*_{k}^{s}*δ*^{t}_{j}*δ*^{t}_{k}*δ*^{s}_{j}*p**l*1

2

*δ*_{j}^{s}*C*^{t}_{kl}*δ*_{j}^{t}*C*^{s}_{kl}

−1 2

*δ*_{l}^{s}*C*^{t}_{kj}*δ*_{l}^{t}*C*_{kj}^{s}

Δ^{st}* _{kl,j}*−Δ

^{st}

_{kj,l}*,*Ω

^{t}*x*

_{klj}*s*

*C*^{s}_{jk}*C*^{t}* _{ls}*−

*s*

*C*^{s}_{lk}*C*^{t}_{js}

*s,n*

Δ^{sn}* _{kj,l}*−Δ

^{sn}

_{kl,j}*C*

^{t}

_{sn}*.*

2.7

Thus, the identity2.6gives
*N*

*s,t0*

*K*_{klj}* ^{st}* ·

*f*

_{st}

^{N}*t0*

Ω^{t}* _{klj}*x·

*p*

*0,*

_{t}*j, k, l*0,1, . . . , N. 2.8

Due to the relations2.4,2.8implies that
_{N}

*t0*

Ω^{t}* _{klj}*x·

*p*

*t*

Ψ 0. 2.9

These equations are satisfied if

Ω^{t}* _{klj}*x

*s*

*C*_{jk}^{s}*C*^{t}* _{ls}*−

*s*

*C*^{s}_{lk}*C*^{t}_{js}

*s,n*

Δ^{sn}* _{kj,l}*−Δ

^{sn}

_{kl,j}*C*^{t}* _{sn}*0,

*j, k, l, t*0,1, . . . , N. 2.10

This system of equations plays a central role in our approach. IfΨhas no left zero
divisors linear in*p**j*, the relation2.10is the necessary condition for existence of a common
right zero divisor for*f**jk*since*UB*has no zero elements linear in*p**j*see e.g.,16.

At *N* ≥ 3 it is also a suﬃcient condition. Indeed, if *C*^{m}* _{jk}*x are such that 2.10 is
satisfied, then

*N*
*s,t0*

*K*_{klj}* ^{st}* ·

*f*

*0,*

_{st}*j, k, l*0,1, . . . , N. 2.11

Generically, it is the system of 1/2N^{2}N − 1 linear equations for *NN* 1/2
unknowns*f**st*with noncommuting coeﬃcients*K*_{klj}* ^{st}*. At

*N*≥3 for genericnonzeros, nonzero divisors

*K*

_{klj}*x, pthe system2.11implies that*

^{st}*α*_{jk}*f*_{jk}*β*_{lm}*f*_{lm}*,* *j, k, l, m*1, . . . , N, 2.12
*γ*_{jk}*f** _{jk}* 0,

*j, k*1, . . . , N, 2.13 where

*α*

*jk*

*, β*

*lm*

*,*and

*γ*

*jk*are certain elements of

*UB see e.g.,*17,18. Thus, all

*f*

*jk*are right zero divisors. They are also left zero divisors. Indeed, due to Ado’s theoremsee e.g.,16 finite-dimensional Lie algebra

*B*and, hence,

*UB*are isomorphic to matrix algebras. For the matrix algebras zero divisorsmatrices with vanishing determinantsare both right and left zero divisors15. Then, under the assumption that all

*α*

*and*

_{jk}*β*

*are not zero divisors, the relations2.12imply that the right divisor of one of*

_{lm}*f*

*is also the right zero divisor for the others.*

_{jk}At *N* 2 one has only two relations of the type2.12 and a right zero divisor of
one of*f*_{11}*, f*_{12}*, f*_{22}is the right zero divisor of the others. We note that it is not easy to control
assumptions mentioned above. Nevertheless,2.4and2.10certainly are fundamental one
for the whole approach.

We will refer to the system2.10as the Central SystemCSgoverning deformations
*of the structure constants of the algebra A generated by a given DDA. Its concrete form*
depends strongly on the form of the bracketsp*t**, C*^{l}* _{jk}*xwhich are defined by the relations
2.2for the elements of the basis of DDA. For stationary solutionsΔ

^{mt}*0the CS2.10is reduced to the associativity conditions1.2.*

_{jk,l}**3. Quantum, Discrete, and Coisotropic Deformations**

Coisotropic, quantum, and discrete deformations of associative algebras considered in10–

14represent particular realizations of the above general scheme associated with diﬀerent DDAs.

*For the quantum deformations one hasM* *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

*p*_{j}*, p*_{k}

0,
*x*^{j}*, x*^{k}

0,
*p*_{j}*, x*^{k}

δ_{j}^{k}*,* *j, k*1, . . . , N, 3.1

whereis the real constantPlanck’s constant in physics. For the Heisenberg DDA

Δ^{mt}* _{jk,l}*δ

^{t}_{0}

*∂C*

_{jk}*x*

^{m}*∂x*^{l}*,* 3.2

and consequently

Ω^{n}* _{klj}*x

*∂C*

^{n}

_{jk}*∂x** ^{l}* −

*∂C*

^{n}

_{kl}*∂x*^{j}^{N}

*m0*

*C*^{m}_{jk}*C*_{ml}* ^{n}* −

*C*

^{m}

_{kl}*C*

_{jm}

^{n}0, *j, k, l, n*0,1, . . . , N. 3.3

Quantum CS3.3governs 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 symbolsΓ^{l}* _{jk}*identified with the structure constants
C

^{l}*Γ*

_{jk}

^{l}*12.*

_{jk}In the representation of the Heisenberg algebra3.1by 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*f** _{jk}*which have nontrivial common kernel or, equivalently,
for which equations

*f**jk*|Ψ0, *j, k*1,2, . . . , N 3.4

have nontrivial common solutions|Ψ ⊂*H. The compatibility condition for*3.4is given by
the CS3.3. The common kernel of the operators*f** _{jk}*forms a subspace

*H*

_{Γ}in the linear space

*H. So, in the approach under consideration the multiplication table*1is realized only on

*H*

_{Γ},

*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 constraints19. 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*H*_{Γ}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:

*∂C*^{n}_{jk}

*∂x** ^{l}* −

*∂C*

^{n}

_{kl}*∂x** ^{j}* 0,

*j, k, l, n*1, . . . , N. 3.5 These equations imply that

*C*

^{n}

_{jk}*∂*

^{2}Φ

^{n}*/∂x*

^{j}*∂x*

*where Φ*

^{k}*are some functions while the associativity condition1.2takes the following form:*

^{n}*N*
*m0*

*∂*^{2}Φ^{m}

*∂x*^{j}*∂x*^{k}

*∂*^{2}Φ^{n}

*∂x*^{m}*∂x*^{l}^{N}

*m0*

*∂*^{2}Φ^{m}

*∂x*^{l}*∂x*^{k}

*∂*^{2}Φ^{n}

*∂x*^{m}*∂x*^{j}*.* 3.6

It is the oriented associativity equation introduced in5,20. Under the gradient reduction
Φ^{n}_{N}

*l0**η** ^{nl}*∂F/∂x

*equation3.7becomes the WDVV equations1.2and1.3.*

^{l}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
KPhierarchy 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

*C*^{l}_{jk}*δ*^{l}_{jk}*H*_{j−l}^{k}*H*_{k−l}^{j}*,* *j, k, l*0,1,2, . . . 3.7
with

*H*_{k}* ^{j}* 1

*P*

*k*

−*∂∂*log*τ*

*∂x*^{j}*,* *j, k*1,2,3, . . . , 3.8
where *τ* is the famous tau-function for the KP hierarchy and *P** _{k}*−

*∂*

*P*

*−∂/∂x*

_{k}^{1},−1/2∂/∂x

^{2},−1/3∂/∂x

^{3}, . . . where

*P*

*t1*

_{k}*, t*

_{2}

*, t*

_{3}

*, . . .*are Schur polynomials defined by the generating formula exp

_{∞}

*k1**λ*^{k}*t**k* _{∞}

*k0**λ*^{k}*P**k*t.

Discrete deformations of noncommutative associative algebras are generated by the
DDA with*MN*and commutation relations

*p*_{j}*, p*_{k}

0,
*x*^{j}*, x*^{k}

0,
*p*_{j}*, x*^{k}

*δ*^{k}_{j}*p*_{j}*,* *j, k*1, . . . , N. 3.9

In this case

Δ^{mt}_{jk,l}*δ*^{t}* _{l}*T

*l*−1C

_{jk}*x,*

^{m}*j, k, l, m, t*0,1,2, . . . , N, 3.10

where for an arbitrary function*ϕx*the action of*T**j* is defined by *T**j**ϕx*^{0}*, . . . , x*^{j}*, . . . , x*^{N}*ϕx*^{0}*, . . . , x** ^{j}*1, . . . , x

*.The corresponding CS is of the form*

^{N}*C**l**T**l**C**j*−*C**j**T**j**C**l*0, *j, l*0,1, . . . , N, 3.11
where the matrices *C**j* are defined as C*j*^{l}_{k}*C*^{l}_{jk}*, 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.

*For coisotropic deformations of commutative algebras*10,11 again*M* *N, but the*
DDA is the Poisson algebra with*p**j*and*x** ^{k}*identified with the Darboux coordinates, that is,

*p**j**, p**k*

0,
*x*^{j}*, x*^{k}

0,
*p**j**, x*^{k}

−δ^{k}_{j}*,* *j, k*0,1, . . . , N, 3.12

where {,} is the standard Poisson bracket. The algebra *UB* is the commutative ring
of functions and divisors of zero are realized by functions with zeros. So, the functions
*f**jk* should be functions with common set Γ of zeros. Thus, in the coisotropic case the
multiplication table1is realized by the following set of equations10:

*f**jk* 0, *j, k*0,1,2, . . . , N. 3.13
The compatibility condition for these equations issee e.g.,10

*f**jk**, f**nl*

|_{Γ}0, *j, k, l, n*1,2, . . . , N. 3.14

The set Γ is the coisotropic submanifold in *R*^{2N1}. The condition 3.14 gives rise to the
following system of equations for the structure constants:

C, C^{m}_{jklr}^{N}

*s0*

*C*^{m}_{sj}*∂C*^{s}_{lr}

*∂x*^{k}*C*^{m}_{sk}*∂C*_{lr}^{s}

*∂x** ^{j}* −

*C*

_{sr}

^{m}*∂C*

^{s}

_{jk}*∂x** ^{l}* −

*C*

^{m}

_{sl}*∂C*

^{s}

_{jk}*∂x*^{r}*C*^{s}_{lr}*∂C*^{m}_{jk}

*∂x** ^{s}* −

*C*

^{s}

_{jk}*∂C*

^{m}

_{lr}*∂x*^{s}

0 3.15

while the equationsΩ^{n}* _{klj}*x 0 have the form of associativity conditions1.2:

Ω^{n}* _{klj}*x

^{N}*m0*

*C*^{m}* _{jk}*xC

_{ml}*x−*

^{n}*C*

^{m}*xC*

_{kl}

^{n}_{jm}x

0. 3.16

Equations 3.15 and 3.16 form the CS for coisotropic deformations 10. In this
case*C*^{l}* _{jk}*is transformed as the tensor of the type1,2under the general transformations of
coordinates

*x*

*, and the whole CS of3.15and3.16is invariant under these transformations 14. The bracket C, C*

^{j}

^{m}*has appeared for the first time in 21 where the so-called diﬀerential concomitants were studied. It was shown in16that this bracket is a tensor only if the tensor*

_{jklr}*C*

^{l}*obeys the algebraic constraint3.16. In7the CS of3.15and3.16has*

_{jk}appeared implicitly as the system of equations which characterizes the structure constants for F-manifolds. In10it has been derived as the CS governing the coisotropic deformations of associative algebras.

The CS of3.15 and3.16contains the oriented associativity equation, the WDVV equation, dispersionless KP hierarchy, and equations from the genus zero universal Whitham hierarchy as the particular cases10,11. Yano manifolds and Yano algebroids associated with the CS of3.15and3.16are studied in14.

We would like to emphasize that for all deformations considered above the stationary solutions of the CSs obey the global associativity condition1.2.

**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 algebra*L. The complete list of such algebras contains 9 algebras*see e.g.

16. Denoting the basis elements by*e*_{1}*, e*_{2}*, e*_{3}, one has the following nonequivalent cases:

1abelian algebra*L*1,

2general algebra*L*_{2}:e1*, e*_{2} *e*_{1}*,* e2*, e*_{3} 0, e3*, e*_{1} 0,
3nilpotent algebra*L*3:e1*, e*2 0, e2*, e*3 *e*1*,* e3*, e*1 0,

4–7four nonequivalent solvable algebras:e1*, e*_{2} 0, e2*, e*_{3} *αe*_{1} *βe*_{2}*,* e3*, e*_{1}
*γe*1*δe*2with*αδ*−*βγ /*0,

8-9simple algebras*L*_{8}so3and*L*_{9} so2,1.

In virtue of the one-to-one correspondence between the elements of the basis in DDA
and the elements*p**j*,*x** ^{k}*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 A and*

*the algebra B one has two options.*

1*A is a two-dimensional algebra without unit element andBL.*

2*A is a three-dimensional algebra with the unit element and BL*_{0}⊕*L*where*L*_{0}is
the algebra generated by the unity element*p*0.

*After the choice of B one should establish a correspondence between* *p*_{1}*, p*_{2}*, x* and
*e*_{1}*, e*_{2}*, e*_{3} defining DDA. For each algebra *L** _{k}* 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

1algebra*L*_{1}:*p*_{1}*e*_{1}*, p*_{2}*e*_{2}*, xe*_{3}; DDA is the commutative algebra with
*p*1*, p*2

0,
*p*1*, x*

0,
*p*2*, x*

0, 4.1

2algebra*L*2:

casea*p*1 −e2*, p*2 *e*3*, xe*1; the corresponding DDA is the algebra*L*2awith the
commutation relations:

*p*1*, p*2

0,
*p*1*, x*

*x,*
*p*2*, x*

0, 4.2

caseb*p*1*e*1*, p*2*e*3*, xe*2; the corresponding DDA*L*2bis defined by
*p*1*, p*2

0,
*p*1*, x*

*p*1*,*
*p*2*, x*

0, 4.3

3algebra*L*_{3}:*p*_{1}*e*_{1}*, p*_{2}*e*_{2}*, xe*_{3}; DDA*L*_{3}is
*p*_{1}*, p*_{2}

0,
*p*_{1}*, x*

0,
*p*_{2}*, x*

*p*_{1}*,* 4.4

4solvable algebra*L*_{4}with*α*0, β1, γ−1, δ0 :*p*_{1}*e*_{1}*, p*_{2}*e*_{2}*, xe*_{3}; DDA
*L*4is

*p*1*, p*2

0,
*p*1*, x*

*p*1*,*
*p*2*, x*

*p*2*,* 4.5

5solvable algebra*L*_{5}at*α*1, β0, γ0, δ1 :*p*_{1} *e*_{1}*, p*_{2}*e*_{2}*, xe*_{3}; DDA*L*_{5}
is

*p*_{1}*, p*_{2}

0,
*p*_{1}*, x*

*p*_{1}*,*
*p*_{2}*, x*

−p2*.* 4.6

For the second choice of the algebra *B* *L*_{0} ⊕ *L* mentioned above the table of
multiplication1.1consists of the trivial part**P**0**P***j* **P***j***P**0 **P***j**, j* 0,1,2 and the nontrivial
part:

**P**^{2}_{1}*AP*0*BP*1*CP*2*,*
**P**1**P**2*DP*0*EP*1*GP*2*,*

**P**^{2}_{2} *KP*0*MP*1*NP*2*.*

4.7

For the first choice*BK*the multiplication table is given by4.7with*ADK*0.

It is convenient also to arrange the structure constants*A, B, . . . , N* into the matrices
*C*_{1}*, C*_{2}defined byC*j*^{l}_{k}*C*^{l}* _{jk}*. One has

*C*_{1}

⎛

⎝0 *A D*
1 *B E*
0 *C G*

⎞

⎠*,* *C*_{2}

⎛

⎝0 *D* K

0 *E M*

1 *G N*

⎞

⎠*.* 4.8

In terms of these matrices the associativity conditions1.2are written as

*C*1*C*2*C*2*C*1*.* 4.9

Simple algebras*L*_{8} and*L*_{9} do not contain two commuting elements to be identified
with*p*1and*p*2, and, hence, they cannot be DDA. Deformations generated by algebras*L*6and
*L*_{7}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 A and DDAL*_{2a} defined by the
commutation relations4.2. These relations imply that for an arbitrary function*ϕx*

*p*_{j}*, ϕx*

Δ*j**ϕx,* *j* 1,2, 5.1
where Δ1 x∂/∂x,Δ20. Consequently, one has the following CS:

Ω^{n}* _{klj}*x Δ

*l*

*C*

^{n}*−Δ*

_{jk}*j*

*C*

^{n}

_{kl}^{2}

*m0*

*C*^{m}_{jk}*C*^{n}* _{lm}*−

*C*

^{m}

_{kl}*C*

^{n}

_{jm}0, *j, k, l, n*0,1,2. 5.2

In terms of the matrices*C*1and*C*2defined above this CS has a form of the Lax equation:

*x∂C*_{2}

*∂x* *C*_{2}*, C*_{1}. 5.3

The CS5.3has all remarkable standard properties of the Lax equationssee e.g.20, 21: it has three independent first integrals:

*I*1 tr*C*2*,* *I*2 1

2trC2^{2}*,* *I*3 1

3trC2^{3}*,* 5.4

and it is equivalent to the compatibility condition of the linear problems:

*C*2Φ *λΦ,*
*x∂Φ*

*∂x* −C1Φ, 5.5

where Φis the column with three components and *λ*is a spectral parameter. Though the
evolution in*x*described by the second linear problem5.5is too simple, nevertheless the CS
5.2or5.3has the meaning of the isospectral deformations of the matrix*C*_{2}that is typical
to the class of integrable systemssee e.g.22,23.

CS5.3is the system of six equations for the structure constants*D, E, G, L, M, N*with
free*A, B, C:*

*D*^{}*DBKC*−*AE*−*DG,*
*K*^{}*DEKG*−*AM*−*DN,*

*E*^{}*MC*−*EG*−*D,*
*M*^{}*E*^{2}*MG*−*BM*−*EN*−*K,*

*G*^{}*GBNC*−*CE*−*G*^{2}*A,*
*N*^{}*GE*−*CMD,*

5.6

where*D*^{} *x∂D/∂x*and so forth. Here we will consider only simple particular cases of the
CS5.6. First it corresponds to the constraint*A*0,*B*0,*C*0, that is, to the nilpotent**P**1.
The corresponding solution is

*D* *β*

ln*x,* *E*−β *γ*

ln*x,* *G* 1

ln*x,* *Kαβ*2β^{2}*δ*ln*x*− *βγ*
ln*x,*
*Mαγ*3βγ*μ*ln*x*−*δlnx*^{2}− *γ*^{2}

ln*x,* *Nαβ*− *γ*
ln*x,*

5.7

where*α, β, γ, δ, μ*are arbitrary constants. The three integrals for this solution are

*I*_{1}*α,* *I*_{2} 1

2*α*^{2}3β^{2}2αβ*μ,*
*I*3 1

3

*αβ* ^{3}−*β*^{3}

*αβ* *μβ*

*α*2β

−*γδ.*

5.8

The second example is given by the constraint*B* 0, *C* 1, *G* 0 for which the
quantum CS3.3is equivalent to the Boussinesq equation12. Under this constraint the CS
5.6is reduced to the single equation:

*E*^{}−6E^{2}4αE*β*0, 5.9

and the other structure constants are given by

*A*2E−*α,* *B*0, *C*1, *Dγ*−1

2*E*^{}*,* *G*0,
*K*−E^{2}*αE*1

2*β,* *Mγ*1

2*E*^{}*,* *Nα*−*N,*

5.10

where*α, β, γ*are arbitrary constants. The corresponding first integrals are

*I*_{1}*α,* *I*_{2} 1
2

*βα*^{2}

*,* *I*_{3} 1

3*α*^{3}*γ*^{2}1
2*αβ*−1

4

*E*^{} ^{2}*E*^{3}−*αE*^{2}−1

2*βE.* 5.11

Integral*I*3reproduces the well-known first integral of5.9. Solutions of5.9are given by
elliptic integralssee e.g.,24. Any such solution together with the formulae5.10describes
*deformation of the three-dimensional algebra A driven by DDAL*_{2a}.

*Now we will consider deformations of the two-dimensional algebra A without unit*
element according to the first option mentioned in the previous section. In this case the CS
has the form5.3with the 2×2 matrices

*C*1
*B E*

*C G*

*,* *C*2

*E M*
*G N*

5.12

or in components

*E*^{}*MC*−*EG,*
*M*^{}*E*^{2}*MG*−*BM*−*EN,*

*G*^{}*GBNC*−*CE*−*G*^{2}*,*
*N*^{}*GE*−*CM.*

5.13

In this case there are two independent integrals of motion:

*I*_{1} *EN,* *I*_{2} 1
2

*E*^{2}*N*^{2}2MG

*.* 5.14

The corresponding spectral problem is given by5.5. Eigenvalues of the matrix*C*_{2},
that is,*λ*1,2 1/2EN±

E−*N*^{2}4GMare invariant under deformations and det*C*2
1/2I_{1}^{2}−*I*_{2}. We note also an obvious invariance of5.6and5.13under the rescaling of*x.*

The system of5.13contains two arbitrary functions*B*and*C. In virtue of the possible*
rescaling**P**1 → *μ*1**P**1*,***P**2 → *μ*2**P**2 *of the basis for the algebra A with two arbitrary functions*
*μ*_{1}*, μ*_{2}, one has four nonequivalent choices1*B*0,*C*0,2*B*1,*C*0,3*B*0,*C*1,
and4*B*1,*C*1.

In the case*B*0,*C*0nilpotent**P**1the solution of the system5.13is
*B*0, *C*0, *E* *β*

ln*x,* *G* 1

ln*x,* *Mγ*ln*x*− *β*^{2}

ln*xαβ,* *N*− *β*

ln*xα,* 5.15
where*α,β, γ* are arbitrary constants. For this solution the integrals are equal to*I*_{1} *α, I*_{2}
*γ* 1/2α^{2}, and*λ*_{1,2} 1/2α

*α*^{2}4γ.

At*B*1,*C*0 the system5.13has the following solution:

*B*1, *C*0, *E* *γ*

*xβ,* *G* *x*
*xβ,*
*Mδ*

*αγβδ*−*γ*^{2}
*β*

1

*x* *γ*^{2}
*β*

*xβ* *,* *N*− *γ*
*xβ* *α,*

5.16

where*α, β, γ, δ*are arbitrary constants. The integrals are*I*_{1}*α, I*_{2}*δ* 1/2α^{2}. The formulae
5.15and5.16provide us with explicit deformations of the structure constants.

In the last two cases the CS5.13 is equivalent to the simple third-order ordinary
diﬀerential equations. At*B*0,*C*1 with additional constraint*I*_{1}0 one gets

*G*^{}2G^{2}*G*^{}4

*G*^{} ^{2}2GG^{}0 5.17

while at*B*1,*C*1, and*I*_{1}0 the system5.13becomes
*G*^{}2G^{2}*G*^{}4

*G*^{} ^{2}2GG^{}−*G*^{}0. 5.18

The second integral for these ODEs is

*I*_{2}−1
2*G*^{4}1

2

*G*^{} ^{2}−2G^{2}*G*^{}−*GG*^{} 1

2*BG*^{2}*.* 5.19

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 Painlev´e property 25,26. The integral 5.19
is known toosee e.g.27.

The appearance of the Chazy V equation among the particular cases of the system
5.13indicates that for other choices of*B*and*C*the CS5.13may be equivalent to the other
notable third-order ODEs. It is really the case. Here we will consider only the reduction*C*1
with*I*_{1}*NE*0. In this case the system5.13is reduced to the following equation:

*G*^{}2G^{2}*G*^{}4

*G*^{} ^{2}2GG^{}−2G^{}Φ−*GΦ*^{}0, 5.20
whereΦ *B*^{} 1/2B^{2}. The second integral is

*I*2−1
2*G*^{4}1

2

*G*^{} ^{2}−2G^{2}*G*^{}−*GG*^{} ΦG^{2}*,* 5.21

and*λ*1,2±
*I*2*/2.*

Choosing particular*B*orΦ, 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:

*G*^{}2G^{2}G^{}2

*G*^{} ^{2}*GG*^{}0. 5.22

At*B*2G5.20becomes the Chazy VIII equation:

*G*^{}−6G^{2}*G*^{}0. 5.23

Choosing the functionΦsuch that
6Φe^{1/3G}_{}

2G^{2}*G*^{}

*G*^{} ^{2}4GG^{}*,* 5.24

one gets the Chazy III equation:

*G*^{}−2GG^{}3

*G*^{} ^{2}0. 5.25

In the above particular cases the integral*I*_{2}5.21is reduced to those given in27.

All Chazy equations presented above have the Lax representation5.3with*E*−N

−1/2G^{}*G*^{2}*GB, M*−1/2G^{}3GG^{}*G*^{3}*G*^{2}*B* GB^{}, C1, and the proper choice
of*B.*

Solutions of all these Chazy equations provide us with the deformations of the
structure constants5.12*for the two-dimensional algebra A generated by the DDAL*_{2a}.

2Now we pass to the DDA*L*_{2b}*.*The commutation relations4.3imply that
*p*1*, ϕx*

T−1ϕx·*p*1*,*

*p*2*, ϕx*

0, 5.26

where *ϕx* is an arbitrary function and *Tϕx * *ϕx* 1. Using 5.26, one finds the
corresponding CS:

2
*m0*

Δ*l*1C^{m}* _{jk}*x·

*C*

^{n}*x*

_{lm}Δ*j*1 *C*_{kl}* ^{m}*x·

*C*

^{n}*x*

_{jm}*,* *j, k, l, n*0,1,2, 5.27

whereΔ1 *T*−1,Δ2 0.In terms of the matrices*C*_{1}and*C*_{2}, this CS is

*C*1*TC*2*C*2*C*1*.* 5.28

For nondegenerated matrix*C*1one has

*TC*_{2}*C*^{−1}_{1} *C*_{2}*C*_{1}*.* 5.29

The CS 5.29 is the discrete version of the Lax equation 5.3 and has similar properties. It has three independent first integrals:

*I*1 tr*C*2*,* *I*2 1

2trC2^{2}*,* *I*3 1

3trC2^{3}*,* 5.30

and it represents itself the compatibility condition for the linear problems:

ΦC2*λΦ,*

*TΦ ΦC*1*.* 5.31

Note that det*C*_{2}is the first integral too.

The CS5.28is the discrete dynamical system in the space of the structure constants.

*For the two-dimensional algebra A with matrices*5.12it is

*BTEETGEBMC,*
*BTMETNE*^{2}*MG,*

*CTEGTGBGCN,*
*CTMGTNEGNG,*

5.32

where*B*and*C*are arbitrary functions. For nondegenerated matrix*C*1*,*that is, at*BG*−*CE /*0,
one has the resolved form5.29, that is,

*TE* *GM*−*EN*

*BG*−*CE* *C,* *TGBBN*−*CM*
*BG*−*CE* *C,*
*TM* *GM*−*EN*

*BG*−*CE* *G,* *TNEBN*−*CM*
*BG*−*CE* *G.*

5.33

This system defines discrete deformations of the structure constants.

**6. Nilpotent DDA**

For the nilpotent DDA*L*3, in virtue of the defining relations4.5, one has
*p*1*, ϕx*

0,

*p*2*, ϕx*
*∂ϕ*

*∂x* ·*p*1 6.1

or

*p*_{j}*, ϕx*
*∂ϕ*

*∂x* ·^{2}

*k1*

*a*_{jk}*p*_{k}*,* 6.2

where*a*21 1, a11 *a*12 *a*22 0. Using6.2, one gets the following CS:

2
*q1*

*a**lq*

2
*m0*

*C*^{n}_{qm}*∂C*^{m}_{jk}

*∂x* −^{2}

*q1*

*a**jq*

2
*m0*

*C*^{n}_{qm}*∂C*^{m}_{kl}

*∂x* ^{2}

*m0*

*C*_{jk}^{m}*C*_{lm}* ^{n}* −

*C*

^{m}

_{kl}*C*

^{n}

_{jm}0, *j, k, l, n*0,1,2. 6.3

In the matrix form it is

*C*_{1}*∂C*1

*∂x* *C*_{1}*, C*_{2}. 6.4

For invertible matrix*C*1

*∂C*_{1}

*∂x* *C*^{−1}_{1} C1*, C*_{2}. 6.5

This system of ODEs has three independent first integrals:

*I*1 tr*C*1*,* *I*2 1

2trC1^{2}*,* *I*3 1

3trC1^{3}*,* 6.6

and it is equivalent to the compatibility condition for the linear system:

*C*1Φ *λΦ,*
*C*1*∂Φ*

*∂x* *C*2Φ 0.

6.7

So, as in the previous section the CS6.4describes isospectral deformations of the matrix*C*_{1}.
This CS governs deformations generated by*L*_{3}.

*For the two-dimensional algebra A without unit element the CS is given by*6.4with
the matrices5.12. First integrals in this case are*I*_{1} *BG, I*_{2} 1/2B^{2}*G*^{2}2CEand
det*C*_{1} 1/2I_{1}^{2}−I2*.*Since det*C*_{1}is a constant on the solutions of the system, then at det*C*_{1}*/*0
one can always introduce the variable*y*defined by*xy*det*C*1such that CS6.5takes the
form

*B*^{}*EBGENC*−*GMC*−*CE*^{2}*,*
*E*^{}*GBMGEN*−*ECM*−*MG*^{2}*,*
*C*^{}*BCEBG*^{2}*MC*^{2}−*CEG*−*BNC*−*GB*^{2}*,*

*G*^{}*CMGCE*^{2}−*CEN*−*BGE,*

6.8

where*B*^{} *∂B/∂y*and so forth and*M,N*are arbitrary functions. At det*C*1 *BG*−*CE*1
this system becomes

*B*^{}*ECEN*−*GM,*
*E*^{}*MGEN*−*GM,*
*C*^{}*G*−*BCMC*−*BN,*

*G*^{}−E−*CEN*−*GM.*

6.9

Choosing*MN*0, one gets

*B*^{}*E,* *E*^{}0, *C*^{}*G*−*B,* *G*^{}−E. 6.10

The solution of this system is

*Eα,* *Bαyβ,* *G*−αy*γ,* *C*−y^{2}

*γ*−*β* *yδ,* 6.11
where*α, β, γ, δ*are arbitrary constants subject to the constraint*βγ*−*αδ*1. First integrals for
this solution are*I*1*βγ, I*2 1/2β^{2}*γ*^{2}2αδ.

With the choice*M*0,*N*1 and under the constraint*I*_{1}*BG*0 the system6.8
takes the form

*B*^{} 1*CE,* *E*^{}−BE, *C*^{}−2*CB.* 6.12

This system can be written as a single equation in the diﬀerent equivalent forms. One of them is

*E*^{} ^{2}*αE*^{4}−2E^{3}*E*^{2}0, 6.13

where*α*is an arbitrary constant and

*B*^{2}−1−*αE*^{2}2E, *CαE*−2, *G*−B. 6.14

The second integral is equal to−1.

Solutions of6.13can be expressed through the elliptic integrals. Solutions of6.13
and the formulae6.14define deformations of the structure constants driven by DDA*L*_{3}.

**7. Solvable DDAs**

1For the solvable DDA*L*4the relations of4.5imply that
*p*_{j}*, ϕx*

*T*−1ϕxp*j**,* *j*1,2, 7.1

where*ϕx*is an arbitrary function and*T* is the shift operator*Tϕx ϕx*1. With the use
of7.1one arrives at the following CS:

*C*_{1}*TC*_{2}*C*_{2}*TC*_{1}*.* 7.2

For nondegenerated matrix*C*_{1}7.2is equivalent to the equation*TC*2*C*^{−1}_{1} *C*^{−1}_{1} *C*_{2}or

*TUC*^{−1}_{1} *UC*_{1}*,* 7.3

where*UC*_{2}*C*_{1}^{−1}. Using this form of the CS, one promptly concludes that the CS7.2has
three independent first integrals:

*I*_{1} tr
*C*_{2}*C*^{−1}_{1}

*,* *I*_{2} 1
2tr

*C*_{2}*C*^{−1}_{1} _{2}

*,* *I*_{3} 1
3tr

*C*_{2}*C*_{1}^{−1}_{3}

*,* 7.4

and it is representable as the commutativity condition for the linear system:

ΦC2*C*^{−1}_{1} *λΦ,*

*TΦ ΦC*1*.* 7.5

*For the two-dimensional algebra A one has the CS*7.2with the matrices5.12. It is
the system of four equations for six functions:

*BTEETGETBMTC,*
*BTMETNETEMTG,*

*CTEGTGGTBNTC,*
*CTMGTNGTENTG.*

7.6

Choosing*B* and *C*as free functions and assuming that*BG*−*CE /*0, one can easily
resolve 7.6with respect to *TE, TG, TM, TN. For instance, withB* *C* 1 one gets the
following four-dimensional mapping:

*TEM*−*EM*−*N*

*E*−*G* *,* *TG*1*M*−*N*
*E*−*G* *,*
*TMN* *N*−*GM*−*N*

*E*−*G* −*G*

*M*−*N*
*E*−*G*

_{2}
*,*

*TNM* 1−*EM*−*N*
*E*−*G*

*M*−*N*
*E*−*G*

_{2}
*.*

7.7

2In a similar manner one finds the CS associated with the solvable DDA*L*_{5}. Since in
this case

*p*1*, ϕx*

T−1ϕxp1*,*

*p*2*, ϕx*

*T*^{−1}−1

*ϕxp*2*,* 7.8

the CS takes the form

*C*_{1}*TC*_{2}*C*_{2}*T*^{−1}*C*_{1}*.* 7.9

For nondegenerated*C*_{2}it is equivalent to

*TV* *C*_{2}*V C*^{−1}_{2} *,* 7.10

where*V* *T*^{−1}*C*_{1}·*C*_{2}. Similar to the previous case the CS has three first integrals:

*I*1trC1*TC*2, *I*2 1

2trC1*TC*2^{2}*,* *I*3 1

3trC1*TC*2^{3}*,* 7.11
and it is equivalent to the compatibility condition for the linear system:

*T*^{−1}*C*_{1}

*C*_{2}Φ *λΦ,*
*TΦ C*2Φ.

7.12