2. Deformations of the Structure Constants Generated by DDA

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 basisP0,P1, . . . ,PNfor a given algebra A, takes the structure constantsC_jkⁿ defined by the multiplication table

PjPk^N

n0

Cⁿ_jkPn, j, k0,1, . . . , N, 1.1 and looks for their deformationsC_jkⁿx,wherex x¹, . . . , x^Mis the set of deformation parameters, such that the associativity condition

N m0

C^m_jkxC_mlⁿ x ^N

m0

C^m_klxC_jmⁿ x 1.2

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 ΦjΦkΦl ∂³F/∂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 ∂³F

∂x^j∂x^k∂x^m, 1.3

where the constants areη^lm g^−1lm and g_lm ∂³F/∂x⁰∂x^l∂x^m 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 fieldsΦj.

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 elementspjof the basis and deformation parametersx_jform a certain algebraPoisson, Heisenberg, etc.. A general class of deformations considered in13is characterized by the condition that the idealJ fjk generated by the elementsfjk −pjpk_N

l0C^l_jkxplrepresenting 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 elementsp_jof the basis and deformation parameters x_kdeformation driving algebraDDA, is a Lie algebra. The basic idea is to require that all elements fjk −pjpk _N

l0C^l_jkxpl 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 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 algebraAwithor withoutunit element P0 in the fixed basis composed by the elements P0,P1, . . . ,PN. The multiplication table1 defines the structure constantsC^l_jk. The commutativity of the basis implies thatC^l_jk C^l_kj. In the presence of the unit element one hasC^l_j0δ^l_jwhereδ^l_jis the Kronecker symbol.

Following Gerstenhaber’s suggestion1,2we will treat the structure constantsC^l_jk as the objects to deform and will denote the deformation parameters by x¹, x², . . . , x^M. 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.

Thus, to define deformations one has the following.

1We associate a set of elementsp₀, p₁, . . . , p_N, x¹, x², . . . , x^Mwith the elements of the basisP0,P1, . . . ,PNand deformation parametersx¹, x², . . . , x^M.

2We consider the Lie algebra B of the dimension NMwith the basis elements e1, . . . , e_NMobeying the commutation relations:

eα, eβ

^NM

γ1

Cαβγeγ, α, β1,2, . . . , NM. 2.1 3We identify the elementsp₁, . . . , p_N, x¹, x², . . . , x^Mwith the elementse₁, . . . , e_NM, thus defining the deformation driving algebra DDA. Different identifications define different DDAs. We assume that the elementp0commutes with all elements of DDA and we putp₀1. The commutativity of the basis in the algebra A implies the commutativity betweenp_j, and in this paper we assume the same property for allx^k. So, we will consider the DDAs defined by the commutation relations of the type

p_j, p_k

0, x^j, x^k

0, p_j, x^k

l

α^k_jlx^l

l

β^kl_j p_l, 2.2 whereα^k_jlandβ^kl_j are some constants.

4We consider the elements

f_jk −pjp_k^N

l0

C^l_jkxpl, j, k1, . . . , N 2.3

of the universal enveloping algebra UB of the algebra DDAB. These fjk

“represent” the table1inUB.

5We require that allfjkare left zero divisors and have a common right zero divisor.

In this casefjkgenerate the left idealJ of left zero divisors. We remind that non-zero elementsaandbare called left and right divisors of zero ifab0see e.g.,15.

Definition 2.1. The structure constantsC^l_jkxare said to define deformations of the algebra A generated by given DDA if allfjkare left zero divisors with common right zero divisor.

To justify this definition we first observe that the simplest possible realization of the multiplication table1inUBgiven by the equationsf_jk0, j, k1, . . . , Nis too restrictive in general. Indeed, for instance, for the Heisenberg algrebraB12such equations imply that pl, C^m_jkx ∂C^m_jk/∂x^l0 and, hence, allC^m_jkare constants. So, one should look for a weaker realization of the multiplication table. A condition that allf_jkare left zero divisors is a natural candidate. The condition of compatibility of the corresponding equationsf_jk·Ψjk 0, j, k 1, . . . , NwhereΨjkare right zero divisors requires that the l.h.s. of these equations and, hence, Ψjk should have a common divisorsee, e.g.,15. We restrict ourselves to the case when Ψjk Ψ·Φjk, j, k1, . . . , NwhereΦjkare invertible elements ofUB. In this case one has the set of equations

fjk·Ψ 0, j, k0,1, . . . , N; 2.4 that is, all left zero divisorsf_jkhave common right zero divisorΨ.

These conditions impose constraints onC^m_jkx. To clarify these constraints we will use the associativity ofUB. First we observe that due to the relations2.2one has the identity p01

p_l, C_jk^mx ^N

t0

Δ^mt_jk,lxpt, 2.5

whereΔ^mt_jk,lxare certain functions ofx¹, . . . , x^Monly. Then, taking into account 2.2and associativity ofUB, one obtains

p_jp_k p_l−p_j

p_kp_l ^N

s,t0

K^st_klj·f_st^N

t0

Ω^t_kljx·p_t, j, k, l0,1, . . . , N, 2.6

where

K_klj^st 1 2

δ_k^sδ_l^tδ^t_kδ^s_l pj−1 2

δ_k^sδ^t_jδ^t_kδ^s_j pl1

2

δ_j^sC^t_klδ_j^tC^s_kl

−1 2

δ_l^sC^t_kjδ_l^tC_kj^s

Δ^st_kl,j−Δ^st_kj,l, Ω^t_kljx

s

C^s_jkC^t_ls−

s

C^s_lkC^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_kljx·p_t0, j, k, l0,1, . . . , N. 2.8

Due to the relations2.4,2.8implies that _N

t0

Ω^t_kljx·pt

Ψ 0. 2.9

These equations are satisfied if

Ω^t_kljx

s

C_jk^sC^t_ls−

s

C^s_lkC^t_js

s,n

Δ^sn_kj,l−Δ^sn_kl,j

C^t_sn0, j, k, l, t0,1, . . . , N. 2.10

This system of equations plays a central role in our approach. IfΨhas no left zero divisors linear inpj, the relation2.10is the necessary condition for existence of a common right zero divisor forfjksinceUBhas no zero elements linear inpjsee e.g.,16.

At N ≥ 3 it is also a sufficient condition. Indeed, if C^m_jkx are such that 2.10 is satisfied, then

N s,t0

K_klj^st ·f_st0, j, k, l0,1, . . . , N. 2.11

Generically, it is the system of 1/2N²N − 1 linear equations for NN 1/2 unknownsfstwith noncommuting coefficientsK_klj^st. AtN≥3 for genericnonzeros, nonzero divisorsK_klj^stx, pthe system2.11implies that

α_jkf_jk β_lmf_lm, j, k, l, m1, . . . , N, 2.12 γ_jkf_jk 0, j, k1, . . . , N, 2.13 whereαjk, βlm,andγjkare certain elements ofUB see e.g.,17,18. Thus, allfjkare right zero divisors. They are also left zero divisors. Indeed, due to Ado’s theoremsee e.g.,16 finite-dimensional Lie algebraBand, hence,UBare isomorphic to matrix algebras. For the matrix algebras zero divisorsmatrices with vanishing determinantsare both right and left zero divisors15. Then, under the assumption that allα_jkandβ_lmare not zero divisors, the relations2.12imply that the right divisor of one off_jkis also the right zero divisor for the others.

At N 2 one has only two relations of the type2.12 and a right zero divisor of one off₁₁, f₁₂, f₂₂is 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 bracketspt, C^l_jkxwhich are defined by the relations 2.2for the elements of the basis of DDA. For stationary solutionsΔ^mt_jk,l0the CS2.10is reduced to the associativity conditions1.2.

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, k1, . . . , N, 3.1

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

Δ^mt_jk,lδ^t₀∂C_jk^mx

∂x^l , 3.2

and consequently

Ωⁿ_kljx ∂Cⁿ_jk

∂x^l −∂Cⁿ_kl

∂x^{j N}

m0

C^m_jkC_mlⁿ −C^m_klC_jmⁿ

0, j, k, l, n0,1, . . . , 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_jkidentified with the structure constants C^l_jkΓ^l_jk 12.

In the representation of the Heisenberg algebra3.1by operators acting in a linear spaceHleft divisors of zero are realized by operators with nonempty kernel. The idealJis the left ideal generated by operatorsf_jkwhich have nontrivial common kernel or, equivalently, for which equations

fjk|Ψ0, j, k1,2, . . . , N 3.4

have nontrivial common solutions|Ψ ⊂H. The compatibility condition for3.4is given by the CS3.3. The common kernel of the operatorsf_jkforms a subspaceH_Γin the linear space H. So, in the approach under consideration the multiplication table1is realized only onH_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 subspaceH_Γas 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:

∂Cⁿ_jk

∂x^l −∂Cⁿ_kl

∂x^j 0, j, k, l, n1, . . . , N. 3.5 These equations imply that Cⁿ_jk ∂²Φⁿ/∂x^j∂x^k where Φⁿ are some functions while the associativity condition1.2takes the following form:

N m0

∂²Φ^m

∂x^j∂x^k

∂²Φⁿ

∂x^m∂x^{l N}

m0

∂²Φ^m

∂x^l∂x^k

∂²Φⁿ

∂x^m∂x^j. 3.6

It is the oriented associativity equation introduced in5,20. Under the gradient reduction Φⁿ_N

l0η^nl∂F/∂x^lequation3.7becomes the WDVV equations1.2and1.3.

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 forN2 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_jkδ^l_jkH_j−l^k H_k−l^j , j, k, l0,1,2, . . . 3.7 with

H_k^j 1 Pk

−∂∂logτ

∂x^j , j, k1,2,3, . . . , 3.8 where τ is the famous tau-function for the KP hierarchy and P_k−∂ P_k−∂/∂x¹,−1/2∂/∂x²,−1/3∂/∂x³, . . . where P_kt1, t₂, t₃, . . . are Schur polynomials defined by the generating formula exp_∞

k1λ^ktk _∞

k0λ^kPkt.

Discrete deformations of noncommutative associative algebras are generated by the DDA withMNand commutation relations

p_j, p_k

0, x^j, x^k

0, p_j, x^k

δ^k_jp_j, j, k1, . . . , N. 3.9

In this case

Δ^mt_jk,lδ^t_lTl−1C_jk^mx, j, k, l, m, t0,1,2, . . . , N, 3.10

where for an arbitrary functionϕxthe action ofTj is defined by Tjϕx⁰, . . . , x^j, . . . , x^N ϕx⁰, . . . , x^j1, . . . , x^N.The corresponding CS is of the form

ClTlCj−CjTjCl0, j, l0,1, . . . , N, 3.11 where the matrices Cj are defined as Cj^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-function13.

For coisotropic deformations of commutative algebras10,11 againM N, but the DDA is the Poisson algebra withpjandx^kidentified with the Darboux coordinates, that is,

pj, pk

0, x^j, x^k

0, pj, x^k

−δ^k_j, j, k0,1, . . . , 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 fjk should be functions with common set Γ of zeros. Thus, in the coisotropic case the multiplication table1is realized by the following set of equations10:

fjk 0, j, k0,1,2, . . . , N. 3.13 The compatibility condition for these equations issee e.g.,10

fjk, fnl

|_Γ0, j, k, l, n1,2, . . . , 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 equationsΩⁿ_kljx 0 have the form of associativity conditions1.2:

Ωⁿ_kljx ^N

m0

C^m_jkxC_mlⁿ x−C^m_klxCⁿ_jmx

0. 3.16

Equations 3.15 and 3.16 form the CS for coisotropic deformations 10. In this caseC^l_jkis transformed as the tensor of the type1,2under the general transformations of coordinatesx^j, and the whole CS of3.15and3.16is invariant under these transformations 14. The bracket C, C^m_jklr 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 tensorC^l_jk obeys the algebraic constraint3.16. In7the CS of3.15and3.16has

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 algebraL. The complete list of such algebras contains 9 algebrassee e.g.

16. Denoting the basis elements bye₁, e₂, e₃, one has the following nonequivalent cases:

1abelian algebraL1,

2general algebraL₂:e1, e₂ e₁, e2, e₃ 0, e3, e₁ 0, 3nilpotent algebraL3:e1, e2 0, e2, e3 e1, e3, e1 0,

4–7four nonequivalent solvable algebras:e1, e₂ 0, e2, e₃ αe₁ βe₂, e3, e₁ γe1δe2withαδ−βγ /0,

8-9simple algebrasL₈so3andL₉ so2,1.

In virtue of the one-to-one correspondence between the elements of the basis in DDA and the elementspj,x^kan algebraLshould have an abelian subalgebra and only one of its elements may play the role of the deformation parameterx. For the original algebra A and the algebra B one has two options.

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

2A is a three-dimensional algebra with the unit element and BL₀⊕LwhereL₀is the algebra generated by the unity elementp0.

After the choice of B one should establish a correspondence between p₁, p₂, x and e₁, e₂, e₃ 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

1algebraL₁:p₁e₁, p₂e₂, xe₃; DDA is the commutative algebra with p1, p2

0, p1, x

0, p2, x

0, 4.1

2algebraL2:

caseap1 −e2, p2 e3, xe1; the corresponding DDA is the algebraL2awith the commutation relations:

p1, p2

0, p1, x

x, p2, x

0, 4.2

casebp1e1, p2e3, xe2; the corresponding DDAL2bis defined by p1, p2

0, p1, x

p1, p2, x

0, 4.3

3algebraL₃:p₁e₁, p₂e₂, xe₃; DDAL₃is p₁, p₂

0, p₁, x

0, p₂, x

p₁, 4.4

4solvable algebraL₄withα0, β1, γ−1, δ0 :p₁e₁, p₂e₂, xe₃; DDA L4is

p1, p2

0, p1, x

p1, p2, x

p2, 4.5

5solvable algebraL₅atα1, β0, γ0, δ1 :p₁ e₁, p₂e₂, xe₃; DDAL₅ is

p₁, p₂

0, p₁, x

p₁, p₂, x

−p2. 4.6

For the second choice of the algebra B L₀ ⊕ L mentioned above the table of multiplication1.1consists of the trivial partP0Pj PjP0 Pj, j 0,1,2 and the nontrivial part:

P²₁AP0BP1CP2, P1P2DP0EP1GP2,

P²₂ KP0MP1NP2.

4.7

For the first choiceBKthe multiplication table is given by4.7withADK0.

It is convenient also to arrange the structure constantsA, B, . . . , N into the matrices C₁, C₂defined byCj^l_kC^l_jk. One has

C₁

⎛

⎝0 A D 1 B E 0 C G

⎞

⎠, C₂

⎛

⎝0 D K

0 E M

1 G N

⎞

⎠. 4.8

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

C1C2C2C1. 4.9

Simple algebrasL₈ andL₉ do not contain two commuting elements to be identified withp1andp2, and, hence, they cannot be DDA. Deformations generated by algebrasL6and L₇will 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 functionϕx

p_j, ϕx

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

Ωⁿ_kljx ΔlCⁿ_jk−ΔjCⁿ_kl²

m0

C^m_jkCⁿ_lm−C^m_klCⁿ_jm

0, j, k, l, n0,1,2. 5.2

In terms of the matricesC1andC2defined above this CS has a form of the Lax equation:

x∂C₂

∂x C₂, C₁. 5.3

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

I1 trC2, I2 1

2trC2², I3 1

3trC2³, 5.4

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

C2Φ λΦ, x∂Φ

∂x −C1Φ, 5.5

where Φis the column with three components and λis a spectral parameter. Though the evolution inxdescribed by the second linear problem5.5is too simple, nevertheless the CS 5.2or5.3has the meaning of the isospectral deformations of the matrixC₂that is typical to the class of integrable systemssee e.g.22,23.

CS5.3is the system of six equations for the structure constantsD, E, G, L, M, Nwith freeA, B, C:

DDBKC−AE−DG, KDEKG−AM−DN,

EMC−EG−D, ME²MG−BM−EN−K,

GGBNC−CE−G²A, NGE−CMD,

5.6

whereD x∂D/∂xand so forth. Here we will consider only simple particular cases of the CS5.6. First it corresponds to the constraintA0,B0,C0, that is, to the nilpotentP1. The corresponding solution is

D β

lnx, E−β γ

lnx, G 1

lnx, Kαβ2β²δlnx− βγ lnx, Mαγ3βγμlnx−δlnx²− γ²

lnx, Nαβ− γ lnx,

5.7

whereα, β, γ, δ, μare arbitrary constants. The three integrals for this solution are

I₁α, I₂ 1

2α²3β²2αβμ, I3 1

3

αβ ³−β³

αβ μβ

α2β

−γδ.

5.8

The second example is given by the constraintB 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−6E²4αEβ0, 5.9

and the other structure constants are given by

A2E−α, B0, C1, Dγ−1

2E, G0, K−E²αE1

2β, Mγ1

2E, Nα−N,

5.10

whereα, β, γare arbitrary constants. The corresponding first integrals are

I₁α, I₂ 1 2

βα²

, I₃ 1

3α³γ²1 2αβ−1

4

E ²E³−αE²−1

2βE. 5.11

IntegralI3reproduces 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

C1 B E

C G

, C2

E M G N

5.12

or in components

EMC−EG, ME²MG−BM−EN,

GGBNC−CE−G², NGE−CM.

5.13

In this case there are two independent integrals of motion:

I₁ EN, I₂ 1 2

E²N²2MG

. 5.14

The corresponding spectral problem is given by5.5. Eigenvalues of the matrixC₂, that is,λ1,2 1/2EN±

E−N²4GMare invariant under deformations and detC2 1/2I₁²−I₂. We note also an obvious invariance of5.6and5.13under the rescaling ofx.

The system of5.13contains two arbitrary functionsBandC. In virtue of the possible rescalingP1 → μ1P1,P2 → μ2P2 of the basis for the algebra A with two arbitrary functions μ₁, μ₂, one has four nonequivalent choices1B0,C0,2B1,C0,3B0,C1, and4B1,C1.

In the caseB0,C0nilpotentP1the solution of the system5.13is B0, C0, E β

lnx, G 1

lnx, Mγlnx− β²

lnxαβ, N− β

lnxα, 5.15 whereα,β, γ are arbitrary constants. For this solution the integrals are equal toI₁ α, I₂ γ 1/2α², andλ_1,2 1/2α

α²4γ.

AtB1,C0 the system5.13has the following solution:

B1, C0, E γ

xβ, G x xβ, Mδ

αγβδ−γ² β

1

x γ² β

xβ , N− γ xβ α,

5.16

whereα, β, γ, δare arbitrary constants. The integrals areI₁α, I₂δ 1/2α². 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. AtB0,C1 with additional constraintI₁0 one gets

G2G²G4

G ²2GG0 5.17

while atB1,C1, andI₁0 the system5.13becomes G2G²G4

G ²2GG−G0. 5.18

The second integral for these ODEs is

I₂−1 2G⁴1

2

G ²−2G²G−GG 1

2BG². 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 ofBandCthe CS5.13may be equivalent to the other notable third-order ODEs. It is really the case. Here we will consider only the reductionC1 withI₁NE0. In this case the system5.13is reduced to the following equation:

G2G²G4

G ²2GG−2GΦ−GΦ0, 5.20 whereΦ B 1/2B². The second integral is

I2−1 2G⁴1

2

G ²−2G²G−GG ΦG², 5.21

andλ1,2± I2/2.

Choosing particularBorΦ, 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:

G2G²G2

G ²GG0. 5.22

AtB2G5.20becomes the Chazy VIII equation:

G−6G²G0. 5.23

Choosing the functionΦsuch that 6Φe^1/3G

2G²G

G ²4GG, 5.24

one gets the Chazy III equation:

G−2GG3

G ²0. 5.25

In the above particular cases the integralI₂5.21is reduced to those given in27.

All Chazy equations presented above have the Lax representation5.3withE−N

−1/2GG²GB, M−1/2G3GGG³G²B GB, C1, and the proper choice ofB.

Solutions of all these Chazy equations provide us with the deformations of the structure constants5.12for the two-dimensional algebra A generated by the DDAL_2a.

2Now we pass to the DDAL_2b.The commutation relations4.3imply that p1, ϕx

T−1ϕx·p1,

p2, ϕx

0, 5.26

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

2 m0

Δl1C^m_jkx·Cⁿ_lmx

Δj1 C_kl^mx·Cⁿ_jmx

, j, k, l, n0,1,2, 5.27

whereΔ1 T−1,Δ2 0.In terms of the matricesC₁andC₂, this CS is

C1TC2C2C1. 5.28

For nondegenerated matrixC1one has

TC₂C⁻¹₁ C₂C₁. 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:

I1 trC2, I2 1

2trC2², I3 1

3trC2³, 5.30

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

ΦC2λΦ,

TΦ ΦC1. 5.31

Note that detC₂is 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 matrices5.12it is

BTEETGEBMC, BTMETNE²MG,

CTEGTGBGCN, CTMGTNEGNG,

5.32

whereBandCare arbitrary functions. For nondegenerated matrixC1,that is, atBG−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 DDAL3, in virtue of the defining relations4.5, one has p1, ϕx

0,

p2, ϕx ∂ϕ

∂x ·p1 6.1

or

p_j, ϕx ∂ϕ

∂x ·²

k1

a_jkp_k, 6.2

wherea21 1, a11 a12 a22 0. Using6.2, one gets the following CS:

2 q1

alq

2 m0

Cⁿ_qm∂C^m_jk

∂x −²

q1

ajq

2 m0

Cⁿ_qm∂C^m_kl

∂x ²

m0

C_jk^mC_lmⁿ −C^m_klCⁿ_jm

0, j, k, l, n0,1,2. 6.3

In the matrix form it is

C₁∂C1

∂x C₁, C₂. 6.4

For invertible matrixC1

∂C₁

∂x C⁻¹₁ C1, C₂. 6.5

This system of ODEs has three independent first integrals:

I1 trC1, I2 1

2trC1², I3 1

3trC1³, 6.6

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

C1Φ λΦ, C1∂Φ

∂x C2Φ 0.

6.7

So, as in the previous section the CS6.4describes isospectral deformations of the matrixC₁. This CS governs deformations generated byL₃.

For the two-dimensional algebra A without unit element the CS is given by6.4with the matrices5.12. First integrals in this case areI₁ BG, I₂ 1/2B²G²2CEand detC₁ 1/2I₁²−I2.Since detC₁is a constant on the solutions of the system, then at detC₁/0 one can always introduce the variableydefined byxydetC1such that CS6.5takes the form

BEBGENC−GMC−CE², EGBMGEN−ECM−MG², CBCEBG²MC²−CEG−BNC−GB²,

GCMGCE²−CEN−BGE,

6.8

whereB ∂B/∂yand so forth andM,Nare arbitrary functions. At detC1 BG−CE1 this system becomes

BECEN−GM, EMGEN−GM, CG−BCMC−BN,

G−E−CEN−GM.

6.9

ChoosingMN0, one gets

BE, E0, CG−B, G−E. 6.10

The solution of this system is

Eα, Bαyβ, G−αyγ, C−y²

γ−β yδ, 6.11 whereα, β, γ, δare arbitrary constants subject to the constraintβγ−αδ1. First integrals for this solution areI1βγ, I2 1/2β²γ²2αδ.

With the choiceM0,N1 and under the constraintI₁BG0 the system6.8 takes the form

B 1CE, E−BE, C−2CB. 6.12

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

E ²αE⁴−2E³E²0, 6.13

whereαis an arbitrary constant and

B²−1−αE²2E, 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 DDAL₃.

7. Solvable DDAs

1For the solvable DDAL4the relations of4.5imply that p_j, ϕx

T−1ϕxpj, j1,2, 7.1

whereϕxis an arbitrary function andT is the shift operatorTϕx ϕx1. With the use of7.1one arrives at the following CS:

C₁TC₂C₂TC₁. 7.2

For nondegenerated matrixC₁7.2is equivalent to the equationTC2C⁻¹₁ C⁻¹₁ C₂or

TUC⁻¹₁ UC₁, 7.3

whereUC₂C₁⁻¹. Using this form of the CS, one promptly concludes that the CS7.2has three independent first integrals:

I₁ tr C₂C⁻¹₁

, I₂ 1 2tr

C₂C⁻¹_{1 2}

, I₃ 1 3tr

C₂C₁⁻¹₃

, 7.4

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

ΦC2C⁻¹₁ λΦ,

TΦ ΦC1. 7.5

For the two-dimensional algebra A one has the CS7.2with the matrices5.12. It is the system of four equations for six functions:

BTEETGETBMTC, BTMETNETEMTG,

CTEGTGGTBNTC, CTMGTNGTENTG.

7.6

ChoosingB and Cas free functions and assuming thatBG−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 , TG1M−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 DDAL₅. Since in this case

p1, ϕx

T−1ϕxp1,

p2, ϕx

T⁻¹−1

ϕxp2, 7.8

the CS takes the form

C₁TC₂C₂T⁻¹C₁. 7.9

For nondegeneratedC₂it is equivalent to

TV C₂V C⁻¹₂ , 7.10

whereV T⁻¹C₁·C₂. Similar to the previous case the CS has three first integrals:

I1trC1TC2, I2 1

2trC1TC2², I3 1

3trC1TC2³, 7.11 and it is equivalent to the compatibility condition for the linear system:

T⁻¹C₁

C₂Φ λΦ, TΦ C2Φ.

7.12