On the deformation theory of structure constants for associative algebras

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a 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.


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 algebra 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 constants C n jk defined by the multiplication table C n jk P n , j, k = 0, 1, ..., N and look for their deformations C n jk (x), where (x) = (x 1 , ..., x M ) is the set of deformation parameters, such that the associativity condition 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-Verlinde-Verlinde [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 etc in the deformed two-dimensional topological field theory obeys the associativity equation (2) with the structure constants given by where constants η lm = (g −1 ) lm and g lm = ∂ 3 F ∂x 0 ∂x l ∂x m where the variable x 0 is associated with the unite element.Each solution of the WDVV equation ( 2), (3) describes a deformation of the structure constants of the N+1-dimensional associative algebra of primary fields Φ j .
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].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][11][12][13][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, integrable dispersionless, dispersive and discrete equations (Kadomtsev-Petviashvili equation etc).The common feature of the coisotropic, quantum, discrete deformations considered in [10][11][12][13][14] is that for all of them elements p j of the basis and deformation parameters x j form a certain algebra ( Poisson, Heisenberg etc).A general class of deformations considered in [13] is characterized by the condition that the ideal J =< f jk > generated by the elements f jk = −p j p k + N l=0 C l jk (x)p l representing the multiplication table (1) is closed.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 pure algebraic formulation of such integrable deformations.We will consider the case when the algebra generating deformations of the structure constants, i.e. the algebra formed by the elements p j of the basis and deformation parameters x k ( deformation driving algebra (DDA)), is a Lie algebra.The basic idea is to require that all elements f jk = −p j p k + N l=0 C l jk (x)p l are left divisors of zero and that they generate the ideal J =< f jk > of left divisors of zero.This requirement gives rise to the central system which governs deformations generated by DDA.
Here we will study the deformations of the structure constants for the threedimensional algebra in the case when the DDA is given by one of the threedimensional 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 equations (multi-dimensional 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 sections 6 and 7, respectively.
2 Deformations of the structure constants generated by DDA So, we consider a finite-dimensional noncommutative algebra A with ( or without ) unite element P 0 .We will restrict overself to a class of algebras which possess a basis composed by pairwise commuting elements P 0 , P 1 , ..., P N .The table of multiplication (1) defines the structure constants C l jk .The commutativity of the basis implies that C l jk = C l kj .In the presence of the unite element one has C l j0 = δ l j where δ l j is the Kroneker symbol.Following the Gerstenhaber's suggestion [1,2] we will treat the structure constants C l jk in a given basis as the objects to deform and will denote the deformation parameters by x 1 , x 2 , ..., x M .For the undeformed structure constants the associativity conditions (2) are nothing else than the compatibility conditions for the table of multiplication (1).In the construction of deformations we should first to specify a "deformed " version of the multiplication table and then to require that this realization is selfconsistence and meaningful.
Thus, to define deformations we 1) 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) consider the Lie algebra B of the dimension N+M+1 with the basis elements e 1 , ..., e N +M+1 obeying the commutation relations 3) identify the elements p 0 , p 1 , ..., p N , x 1 , x 2 , ..., x M with the elements e 1 , ..., e N +M+1 thus defining the deformation driving algebra (DDA).Different identifications define different DDAs.We will assume that the element p 0 is always a central element of DDA.The commutativity of the basis in the algebra A implies the commutativity between p j and in this paper we assume the same property for all x k .So, we will consider the DDAs defined by the commutation relations of the type ) where α k jl and β kl j are some constants, 4) consider the elements of the universal enveloping algebra U(B ) of the algebra DDA(B ).These f jk "represent" the table (1) in U(B ).Note that f j0 = f 0j = 0. 5) require the all f jk are non-zero left divisors of zero and have a common right zero divisor.
In this case f jk generate the left ideal J =< f jk > of left divisors of zero.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.The structure constants C 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 table (1) in U(B ) given by the equations f jk = 0, j, k = 0, 1, ..., N is too restrictive in general.Indeed, for instance, for the Heisenberg algrebra B [12] such equations imply that [p l , C m jk (x)] = 0 and , hence, no deformations are allowed.So, one should look for a weaker realization of the multiplication table .A condition that all f jk are just non-zero divisors of zero is a natural candidate.Then, the condition of compatibility of the corresponding equations f jk • Ψ jk = 0, j, k = 1, ..., N where Ψ jk are right zero divisors requires that the l.h.s. of these equations and, hence, Ψ jk should have a common divisor (see e.g.[15] ).We restrict ourself to the case when Ψ jk = Ψ •Φ jk , j, k = 1, ..., N where Φ jk are invertible elements of U(B).In this case one has the compatible set of equations that is all left zero divisors f jk have common right zero divisor Ψ.These conditions impose constraints on C m jk (x).To clarify these constraints we will use the basic property of the algebra A, i.e. its associativity.First we observe that due to the relations (4) one has the identity where ∆ mt jk,l (x) are certain functions of x 1 , ..., x M only.Then, taking into account (4), one obtains the identity where and The identity (6) implies that for an associative algebra N s,t=0 Due to the relations ( 5) equations ( 7) imply that These equations are satisfied if (8) This system of equations plays a central role in our approach.If Ψ has no left zero divisors linear in p j and U(B) has no zero elements linear in p j then the relation ( 8) is the necessary condition for existence of a common right zero divisor for f jk .
At N≥ 3 it is also a sufficient condition.Indeed, if C m jk (x) are such that equations ( 8) are satisfied then N s,t=0 Generically, it is the system of 1 2 N 2 (N − 1) linear equations for N (N +1) 2 unknowns f st with noncommuting coefficients K st klj .At N ≥ 3 for generic (non zeros, non zero divisors) K st klj (x, p) the system (9) implies that and where α jk , β lm , γ jk are certain elements of U(B) ( see e.g.[16,17] ).Thus, all f jk are right zero divisors.They are also left zero divisors.Indeed, due to Ado's theorem ( see e.g.[18] ) 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 α jk and β lm are not zero divisors, the relations (10) imply that the right divisor of one of f jk is also the right zero divisor for the others.At N=2 one has only two relations of the type (10) 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 isn't that easy to control assumptions mentioned above.Nevertheless, the equations ( 5) and ( 8) certainly are fundamental one for the whole approach.
We shall refer to the system (8) as the Central System (CS) governing deformations of the structure constants of the algebra A generated by a given DDA.Its concrete form depends strongly on the form of the brackets p t , C l jk (x) which are defined by the relations (4) for the elements of the basis of DDA.For stationary solutions (∆ t jk,l = 0) the CS ( 8) is reduced to the associativity conditions (2).
For the quantum deformations of noncommutative algebra 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 deformations parameters are identified with the elements of the Heisenberg algebra in such a way that where is the (Planck's) constant.For the Heisenberg DDA and consequently Quantum CS ( 14) 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 Γ l jk identified with the structure constants (C l jk = Γ l jk ) [12].In the representation of the Heisenberg algebra (12) 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 f jk which have nontrivial common kernel or, equivalently, for which equations have nontrivial common solutions |Ψ ⊂ H .The compatibility condition for equations (15) is given by the CS (14).The common kernel of the operators f jk form a subspace H Γ in the linear space H. So, in the approach under consideration the multiplication table (1) is 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 the Dirac's recipe for quantization of the first-class constraints [19].
In quantum theory context equations (15) serve 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 (15) was the starting point for construction of the quantum deformations.Quantum CS ( 14) contains various classes of solutions which describe different classes of deformations.An important subclass is given by iso-associative deformations, i.e. by deformations for which the associativity condition ( 2) is valid for all values of deformation parameters.For such quantum deformations the structure constants should obey the equations These equations imply that C n jk = ∂ 2 Φ n ∂x j ∂x k where Φ n are some functions while the associativity condition (2) takes the form It is the oriented associativity equation introduced in [20,5].Under the gradient reduction Φ n = N l=0 η nl ∂F ∂x l equation ( 18) becomes the WDVV equation ( 2), (3).
Non iso-associative deformations for which the condition ( 16) is not valid are of interest too.They are described by some well-known integrable soliton equations [12].In particular, there are the Boussinesq equation among them for N=2 and the 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 with where τ is the famous tau-function for the KP hierarchy and .. where P k (t 1 , t 2 , t 3 , ...) are Schur polynomials defined by the generating formula exp Discrete deformations of noncommutative associative algebras are generated by the DDA with M = N 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.
In this case where for an arbitrary function ϕ(x) the action of T j is defined by T j ϕ(x 0 , ..., x j , ..., x N ) = ϕ(x 0 , ..., x j + 1, ...., x N ).The corresponding CS is of the form where the matrices C j are defined as (C j ) l k = C l jk , j, k, l = 0, 1, ..., N. The discrete CS (22) governs discrete deformations of associative algebras.The CS (22) contains, as particular cases, the discrete versions of the oriented associativity equation, WDVV equation, Boussinesq equation, discrete KP hierarchy and Hirota-Miwa bilinear equations for KP τ -function.
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 f jk should be functions with common set Γ of zeros.Thus, in the coisotropic case the multiplication table ( 1) is realized by the set of equations [10] f jk = 0, j, k = 0, 1, 2, ..., N.
Well-known compatibility conditon for these equations is The set Γ is the coisotropic submanifold in R 2(N +1) .The condition (25) gives rise to the following system of equations for the structure constants Equations ( 26) and ( 27) form the CS for coisotropic deformations [10].In this case C l jk is transformed as the tensor of the type (1,2) under the general tranformations of coordinates x j and the whole CS ( 26), ( 27) is invariant under these tranformations [14].The bracket [C, C] m jklr has appeared for the first time in the paper [21] where the co-called differential concomitants were studied.It was shown in [18] that this bracket is a tensor only if the tensor C l jk obeys the algebraic constraint (27).In the paper [7] the CS ( 26), (27) 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.
We would like to emphasize that for all deformations considered above the stationary solutions of the CSs obey the global associativity condition (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.[18]).Denoting the basis elements by e 1 , e 2 , e 3 , one has the following nonequivalent cases: 1) abelian algebra L 1 , 2) general algebra 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 has an abelian subalgebra and only one its element may play a 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 unite element and B =L 2) A is a three-dimensional algebra with the unite element and B = L 0 ⊕ L where L 0 is the algebra generated by the unite 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: 1) algebra L 1 : p 1 = e 1 , p 2 = e 2 , x = e 3 ; DDA is the commutative algebra with 2) algebra L 2 : case a) p 1 = −e 2 , p 2 = e 3 , x = e 1 ; the corresponding DDA is the algebra L 2a with the commutation relations 3) algebra L 3 : 5) solvable algebra L 5 at α = 1, β = 0, γ = 0, δ = 1 : For the second choice of the algebra B = L 0 ⊕L mentioned above the table of multiplication (1) consists from the trivial part P 0 P j = P j P 0 = P j , j = 0, 1, 2 and the nontrivial part For the first choice B =L the multiplication table is given by (34) with A=D=L=0.
It is convenient also to arrange the structure constants A,B,...,N into the matrices C 1 , C 2 defined by (C j ) l k = C l jk .One has In terms of these matrices the associativity conditions (2) are written as 5 Deformations generated by general DDAs Commutative DDA (28) obviously does not generate any deformation.So, we begin with the three-dimensional commutative algebra A and DDA L 2a defined by the commutation relations (29).These relations imply that for an arbitrary function ϕ(x) where (38) In terms of the matrices C 1 and C 2 defined above this CS has a form of the Lax equation The CS (39) has all remarkable standard properties of the Lax equations (see e.g.[20,21]): it has three independent first integrals and it is equivalent to the compatibility condition of the linear problems 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 (41) is too simple, nevertheless the CS (38) or (39) have the meaning of the iso-spectral deformations of the matrix C 2 that is typical to the class of integrable systems (see e.g.[22,23]).CS (39) is the system of six equations for the structure constants D,E,G,L,M,N with free A,B,C: where D ′ = x ∂D ∂x etc.Here we will consider only simple particular cases of the CS (42).First corresponds to the constraint A=0, B=0, C=0, i.e. to the nilpotent P 1 .The corresponding solution is where α, β, γ, δ, µ are arbitrary constants.The three integrals for this solution are (44) The second example is given by the constraint B=0, C=1, G=0 for which the quantum CS ( 14) is equivalent to the Boussinesq equation [12].Under this constraint the CS (42) is reduced to the single equation and the other structure constants are given by where α, β, γ are arbitrary constants.The corresponding first integrals are Integral I 3 reproduces the well-known first integral of equation ( 45).Solutions of equation ( 45) are given by elliptic integrals (see e.g.[24]).Any such solution together with the formulae (46) describes deformation of the three-dimensional algebra A driven by DDA L 2a .Now we will consider deformations of the two-dimensional algebra A without unite element according to the first option mentioned in the previous section.In this case the CS has the form (39) with the 2 × 2 matrices or in components In this case there are two independent integrals of motion The corresponding spectral problem is given by (41).Eigenvalues of the matrix C 2 , i.e. λ 1,2 = 1 2 (E + N ± (E − N ) 2 + 4GM ) are invariant under deformations and det C 2 = 1 2 I 2 1 − I 2 .We note also an obviously invariance of equations ( 42) and (49) under the rescaling of x.
In the last two cases the CS (49) is equivalent to the simple third order ordinary differential equations.At B=0, C=1 with additional constraint I 1 = 0 one gets while at B=1,C=1 and I 1 = 0 the system (49) becomes The second integral for these ODEs is Equation ( 53) with G ′ = ∂G ∂y is the Chazy V equation from the well-known Chazy-Bureau list of the third order ODEs having Painleve property [25,26].The integral (55) is known too (see e.g.[27]).
The appearance of the Chazy V equation among the particular cases of the system (49) indicates that for other choices of B and C the CS (49) 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 I 1 = N + E = 0.In this case the system (49) is reduced to the following equation where Φ = B ′ + 1 2 B 2 .The second integral is and λ 1,2 = ± I2 2 .
Choosing particular B or Φ, one gets equations from the Chazy-Bureau list.Indeed, at Φ = 0 one has the Chazy V equation (53).Choosing Φ = G ′ , one gets the Chazy VII equation At B=2G equation (56) becomes the Chazy VIII equation Choosing the function Φ such that 6Φe one gets the Chazy III equation In the above particular cases the integral I 2 (57) is reduced to those given in [27].All Chazy equations presented above have the Lax representation (39) with Solutions of all these Chazy equations provide us with the deformations of the structure constants (48) for the two-dimensional algebra A generated by the DDA L 2a .
2.Now we pass to the DDA L 2b .The commutation relations (30) imply that where ϕ(x) is an arbitrary function and T ϕ(x) = ϕ(x + 1).Using (62), one finds the corresponding CS where ∆ 1 = T − 1, ∆ 2 = 0.In terms of the matrices C 1 and C 2 this CS is For nondegenerate matrix C 1 one has The CS (65) is the discrete version of the Lax equation (39) and has similar properties.It has three independent first integrals and represents itself the compatibility condition for the linear problems Note that det C 2 is the first integral too.
The CS (64) is the discrete dynamical system in the space of the structure constants.For the two-dimensional algebra A with matrices (48) it is where B and C are arbitrary functions.For nondegenerate matrix C 1 , i.e. at BG − CE = 0 , one has the resolved form (65), i.e.
This system defines discrete deformations of the structure constants.

Nilpotent DDA
For the nilpotent DDA L 3 , in virtue of the defining relations (32), one has (72) In the matrix form it is For invertible matrix C 1 This system of ODEs has three independent first integrals and equivalent to the compatibility condition for the linear system So, as in the previous section the CS (73) describes iso-spectral deformations of the matrix C 1 .This CS governs deformations generated by L 3 .
For the two-dimensional algebra A without unite element the CS is given by equation (73) with the matrices (48).First integrals in this case are I 1 = B + G, I 2 = 1 2 (B 2 + G 2 + 2CE) and det C 1 = 1 2 I 2 1 − I 2 .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 x = y det C 1 such that CS (74) takes the form (89) Chosing B and C as free functions and assuming that BG-CE = 0, one can easily resolve (89) with respect to TE,TG,TM,TN.For instance, with B=C=1 one gets the following four-dimensional mapping 2. In a similar manner one finds the CS associated with the solvable DDA L 5 .Since in this case [p 1 , ϕ(x)] = (T − 1)ϕ(x)p 1 , [p 2 , ϕ(x)] = (T −1 − 1)ϕ(x)p 2 (91) the CS takes the form For nondegenerate C 2 it is equivalent to