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Advances in Mathematical Physics
Volume 2010 (2010), Article ID 389091, 21 pages
On the Deformation Theory of Structure Constants for Associative Algebras
Dipartimento di Fisica, Universita del Salento and INFN, Sezione di Lecce, 73100 Lecce, Italy
Received 5 October 2009; Revised 30 March 2010; Accepted 16 May 2010
Academic Editor: Alexander P. Veselov
Copyright © 2010 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 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.
An idea to study deformations of structure constants for associative algebras goes back to the classical works of Gerstenhaber [1, 2]. As one of the approaches to deformation theory he suggested “to take the point of view that the objects being deformed are not merely algebras, but essentially algebras with a fixed basis” and to treat “the algebraic set of all structure constants as parameter space for deformation theory” .
Thus, following this approach, one chooses the basis for a given algebra , takes the structure constants defined by the multiplication table and looks for their deformations where 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 has been discovered by Witten  and Dijkgraaf et al. . They demonstrated that the function which defines the correlation functions and so forth in the deformed two-dimensional topological field theory obeys the associativity equation (1.2) with the structure constants given by where the constants are and where the variable 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 - dimensional associative algebra of primary fields .
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 . The beautiful and rich theory of Frobenius and F-manifolds has various applications from the singularity theory to quantum cohomology (see, e.g., [6, 8, 9]).
An alternative approach to the deformation theory of the structure constants for commutative associative algebras has been proposed recently in [10–14]. Within this method the deformations of the structure constants are governed by the so-called central system (CS). Its concrete form depends on the class of deformations under consideration and CS contains, as particular reductions, many integrable systems like WDVV equation, oriented associativity equation, and integrable dispersionless, dispersive, and discrete equations (Kadomtsev-Petviashvili equation, etc.). The common feature of the coisotropic, quantum, discrete deformations considered in [10–14] is that for all of them elements of the basis and deformation parameters form a certain algebra (Poisson, Heisenberg, etc.). A general class of deformations considered in  is characterized by the condition that the ideal generated by the elements representing the multiplication table () is the Poisson ideal. It was shown that this class contains a subclass of so-called integrable deformations for which the CS has a simple and nice geometrical meaning.
In the present paper we will discuss a purely algebraic formulation of such integrable deformations. We will consider the case when the algebra generating deformations of the structure constants, that is, the algebra formed by the elements of the basis and deformation parameters (deformation driving algebra (DDA)), is a Lie algebra. The basic idea is to require that all elements are left divisors of zero and that they generate the ideal 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 . Depending on the choice of DDA and identification of , and with the elements of DDA, the corresponding CS takes the form of the system of ordinary differential equations or the system of discrete equations (multidimensional mappings). In the first case the CS contains the third-order ODEs from the Chazy-Bureau list as the particular examples. This approach provides us also with the Lax form of the above equations and their first integrals.
The paper is organized as follows. General formulation of the deformation theory for the structure constants is presented in 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 commutative algebra with (or without) unit element in the fixed basis composed by the elements . The multiplication table () defines the structure constants . The commutativity of the basis implies that . In the presence of the unit element one has where is the Kronecker symbol.
Following Gerstenhaber's suggestion [1, 2] we will treat the structure constants as the objects to deform and will denote the deformation parameters by . For the undeformed structure constants the associativity conditions (1.2) are nothing else than the compatibility conditions for the table of multiplication (1.1). In the construction of deformations we should first specify a “deformed” version of the multiplication table and then require that this realization is self-consistent and meaningful.
Thus, to define deformations one has the following.()We associate a set of elements with the elements of the basis and deformation parameters .()We consider the Lie algebra B of the dimension with the basis elements obeying the commutation relations: ()We identify the elements with the elements , thus defining the deformation driving algebra (DDA). Different identifications define different DDAs. We assume that the element commutes with all elements of DDA and we put . The commutativity of the basis in the algebra A implies the commutativity between , and in this paper we assume the same property for all . So, we will consider the DDAs defined by the commutation relations of the type where and are some constants.()We consider the elements of the universal enveloping algebra of the algebra DDA(B). These “represent” the table () in .()We require that all are left zero divisors and have a common right zero divisor.
In this case generate the left ideal of left zero divisors. We remind that non-zero elements and are called left and right divisors of zero if (see e.g., ).
Definition 2.1. The structure constants are said to define deformations of the algebra A generated by given DDA if all 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 () in given by the equations is too restrictive in general. Indeed, for instance, for the Heisenberg algrebra  such equations imply that and, hence, all are constants. So, one should look for a weaker realization of the multiplication table. A condition that all are left zero divisors is a natural candidate. The condition of compatibility of the corresponding equations where are right zero divisors requires that the l.h.s. of these equations and, hence, should have a common divisor (see, e.g., ). We restrict ourselves to the case when where are invertible elements of . In this case one has the set of equations that is, all left zero divisors have common right zero divisor .
These conditions impose constraints on . To clarify these constraints we will use the associativity of . First we observe that due to the relations (2.2) one has the identity () where are certain functions of only. Then, taking into account (2.2) and associativity of , one obtains where
This system of equations plays a central role in our approach. If has no left zero divisors linear in , the relation (2.10) is the necessary condition for existence of a common right zero divisor for since has no zero elements linear in (see e.g., ).
At it is also a sufficient condition. Indeed, if are such that (2.10) is satisfied, then
Generically, it is the system of linear equations for unknowns with noncommuting coefficients . At for generic (nonzeros, nonzero divisors) the system (2.11) implies that where and are certain elements of (see e.g., [17, 18]). Thus, all are right zero divisors. They are also left zero divisors. Indeed, due to Ado's theorem (see e.g., ) finite-dimensional Lie algebra and, hence, are isomorphic to matrix algebras. For the matrix algebras zero divisors (matrices with vanishing determinants) are both right and left zero divisors . Then, under the assumption that all and are not zero divisors, the relations (2.12) imply that the right divisor of one of is also the right zero divisor for the others.
At one has only two relations of the type (2.12) and a right zero divisor of one of is the right zero divisor of the others. We note that it is not easy to control assumptions mentioned above. Nevertheless, (2.4) and (2.10) certainly are fundamental one for the whole approach.
We will refer to the system (2.10) as the Central System (CS) governing deformations of the structure constants of the algebra A generated by a given DDA. Its concrete form depends strongly on the form of the brackets which are defined by the relations (2.2) for the elements of the basis of DDA. For stationary solutions () the CS (2.10) is reduced to the associativity conditions (1.2).
3. Quantum, Discrete, and Coisotropic Deformations
For the quantum deformations one has and the deformation driving algebra is given by the Heisenberg algebra . 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 where is the real constant (Planck's constant in physics). For the Heisenberg DDA and consequently
Quantum CS (3.3) governs deformations of structure constants for associative algebra driven by the Heisenberg DDA. It has a simple geometrical meaning of vanishing Riemann curvature tensor for torsionless Christoffel symbols identified with the structure constants () .
In the representation of the Heisenberg algebra (3.1) by operators acting in a linear space left divisors of zero are realized by operators with nonempty kernel. The ideal is the left ideal generated by operators which have nontrivial common kernel or, equivalently, for which equations have nontrivial common solutions . The compatibility condition for (3.4) is given by the CS (3.3). The common kernel of the operators forms a subspace in the linear space . So, in the approach under consideration the multiplication table () is realized only on , but not on the whole H. Such type of realization of the constraints is well known in quantum theory as Dirac's recipe for quantization of the first-class constraints . 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 as the “structure constants” subspace. In  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: These equations imply that where are some functions while the associativity condition (1.2) takes the following form: It is the oriented associativity equation introduced in [5, 20]. Under the gradient reduction equation (3.7) becomes the WDVV equations (1.2) and (1.3).
Non-isoassociative deformations for which the condition (3.5) is not valid are of interest too. They are described by some well-known integrable soliton equations . In particular, there are Boussinesq equation among them for and Kadomtsev-Petviashvili (KP) hierarchy for the infinite-dimensional algebra of polynomials in the Faa' de Bruno basis . In the latter case the deformed structure constants are given by with where is the famous tau-function for the KP hierarchy and where are Schur polynomials defined by the generating formula
Discrete deformations of noncommutative associative algebras are generated by the DDA with and commutation relations In this case where for an arbitrary function the action of is defined by The corresponding CS is of the form where the matrices are defined as 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 -function .
For coisotropic deformations of commutative algebras [10, 11] again , but the DDA is the Poisson algebra with and identified with the Darboux coordinates, that is, where is the standard Poisson bracket. The algebra is the commutative ring of functions and divisors of zero are realized by functions with zeros. So, the functions should be functions with common set of zeros. Thus, in the coisotropic case the multiplication table () is realized by the following set of equations : The compatibility condition for these equations is (see e.g., ) The set is the coisotropic submanifold in . The condition (3.14) gives rise to the following system of equations for the structure constants: while the equations have the form of associativity conditions (1.2):
Equations (3.15) and (3.16) form the CS for coisotropic deformations . In this case is transformed as the tensor of the type () under the general transformations of coordinates , and the whole CS of (3.15) and (3.16) is invariant under these transformations . The bracket has appeared for the first time in  where the so-called differential concomitants were studied. It was shown in  that this bracket is a tensor only if the tensor obeys the algebraic constraint (3.16). In  the CS of (3.15) and (3.16) has appeared implicitly as the system of equations which characterizes the structure constants for F-manifolds. In  it has been derived as the CS governing the coisotropic deformations of associative algebras.
The CS of (3.15) and (3.16) contains the oriented associativity equation, the WDVV equation, dispersionless KP hierarchy, and equations from the genus zero universal Whitham hierarchy as the particular cases [10, 11]. Yano manifolds and Yano algebroids associated with the CS of (3.15) and (3.16) are studied in .
We would like to emphasize that for all deformations considered above the stationary solutions of the CSs obey the global associativity condition (1.2).
4. Three-Dimensional Lie Algebras as DDA
In the rest of the paper we will study deformations of associative algebras generated by three-dimensional real Lie algebra . The complete list of such algebras contains 9 algebras (see e.g. ). Denoting the basis elements by , one has the following nonequivalent cases:()abelian algebra ,()general algebra : ,()nilpotent algebra ,()–()four nonequivalent solvable algebras: with ,()-()simple algebras so () and so ().
In virtue of the one-to-one correspondence between the elements of the basis in DDA and the elements , an algebra should have an abelian subalgebra and only one of its elements may play the role of the deformation parameter . For the original algebra A and the algebra B one has two options.(1)A is a two-dimensional algebra without unit element and .(2)A is a three-dimensional algebra with the unit element and B = where is the algebra generated by the unity element .
After the choice of B one should establish a correspondence between and defining DDA. For each algebra there are obviously, in general, six possible identifications if one avoids linear superpositions. Some of them are equivalent. The incomplete list of nonequivalent identifications is as follows(1)algebra DDA is the commutative algebra with (2)algebra : case (a); the corresponding DDA is the algebra with the commutation relations: case (b); the corresponding DDA is defined by (3)algebra : ; DDA is (4)solvable algebra with ; DDA is (5)solvable algebra at ; DDA is
For the second choice of the algebra mentioned above the table of multiplication (1.1) consists of the trivial part and the nontrivial part:
For the first choice the multiplication table is given by (4.7) with .
It is convenient also to arrange the structure constants into the matrices defined by . One has
In terms of these matrices the associativity conditions (1.2) are written as
Simple algebras and do not contain two commuting elements to be identified with and , and, hence, they cannot be DDA. Deformations generated by algebras and will be considered elsewhere.
5. Deformations Generated by General DDAs
() Commutative DDA (4.1) does not force any deformation of structure constants. So, we begin with the three-dimensional commutative algebra A and DDA defined by the commutation relations (4.2). These relations imply that for an arbitrary function where . Consequently, one has the following CS: In terms of the matrices and defined above this CS has a form of the Lax equation:
The CS (5.3) 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 is the column with three components and is a spectral parameter. Though the evolution in described by the second linear problem (5.5) is too simple, nevertheless the CS (5.2) or (5.3) has the meaning of the isospectral deformations of the matrix that is typical to the class of integrable systems (see e.g. [22, 23]).
CS (5.3) is the system of six equations for the structure constants with free : where and so forth. Here we will consider only simple particular cases of the CS (5.6). First it corresponds to the constraint , , , that is, to the nilpotent . The corresponding solution is where are arbitrary constants. The three integrals for this solution are
The second example is given by the constraint , , for which the quantum CS (3.3) is equivalent to the Boussinesq equation . Under this constraint the CS (5.6) is reduced to the single equation: and the other structure constants are given by where are arbitrary constants. The corresponding first integrals are Integral reproduces the well-known first integral of (5.9). Solutions of (5.9) are given by elliptic integrals (see e.g., ). Any such solution together with the formulae (5.10) describes deformation of the three-dimensional algebra A driven by DDA .
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 form (5.3) with the matrices or in components In this case there are two independent integrals of motion:
The corresponding spectral problem is given by (5.5). Eigenvalues of the matrix , that is, are invariant under deformations and . We note also an obvious invariance of (5.6) and (5.13) under the rescaling of .
The system of (5.13) contains two arbitrary functions and . In virtue of the possible rescaling of the basis for the algebra A with two arbitrary functions , one has four nonequivalent choices () , , () , , () , , and () , .
In the case , (nilpotent ) the solution of the system (5.13) is where , are arbitrary constants. For this solution the integrals are equal to , and .
In the last two cases the CS (5.13) is equivalent to the simple third-order ordinary differential equations. At , with additional constraint one gets while at , , and the system (5.13) becomes The second integral for these ODEs is Equation (5.17) with is the Chazy V equation from the well-known Chazy-Bureau list of the third-order ODEs having Painlevé property [25, 26]. The integral (5.19) is known too (see e.g. ).
The appearance of the Chazy V equation among the particular cases of the system (5.13) indicates that for other choices of and the CS (5.13) may be equivalent to the other notable third-order ODEs. It is really the case. Here we will consider only the reduction with . In this case the system (5.13) is reduced to the following equation: where . The second integral is and .
Choosing particular or , one gets equations from the Chazy-Bureau list. Indeed, at one has the Chazy V equation (5.17). Choosing , one gets the Chazy VII equation: At (5.20) becomes the Chazy VIII equation:
All Chazy equations presented above have the Lax representation (5.3) with , and the proper choice of .
Solutions of all these Chazy equations provide us with the deformations of the structure constants (5.12) for the two-dimensional algebra A generated by the DDA .
() Now we pass to the DDA The commutation relations (4.3) imply that where is an arbitrary function and Using (5.26), one finds the corresponding CS: where In terms of the matrices and , this CS is For nondegenerated matrix one has
The CS (5.29) is the discrete version of the Lax equation (5.3) and has similar properties. It has three independent first integrals: and it represents itself the compatibility condition for the linear problems: Note that is the first integral too.
The CS (5.28) is the discrete dynamical system in the space of the structure constants. For the two-dimensional algebra A with matrices (5.12) it is where and are arbitrary functions. For nondegenerated matrix that is, at , one has the resolved form (5.29), that is, This system defines discrete deformations of the structure constants.
6. Nilpotent DDA
For the nilpotent DDA , in virtue of the defining relations (4.5), one has or where . Using (6.2), one gets the following CS: In the matrix form it is For invertible matrix This system of ODEs has three independent first integrals: and it is equivalent to the compatibility condition for the linear system: So, as in the previous section the CS (6.4) describes isospectral deformations of the matrix . This CS governs deformations generated by .
For the two-dimensional algebra A without unit element the CS is given by (6.4) with the matrices (5.12). First integrals in this case are and Since is a constant on the solutions of the system, then at one can always introduce the variable defined by such that CS (6.5) takes the form where and so forth and , are arbitrary functions. At this system becomes
Choosing , one gets The solution of this system is where are arbitrary constants subject to the constraint . First integrals for this solution are
With the choice , and under the constraint the system (6.8) takes the form This system can be written as a single equation in the different equivalent forms. One of them is where is an arbitrary constant and The second integral is equal to .
7. Solvable DDAs
() For the solvable DDA the relations of (4.5) imply that where is an arbitrary function and is the shift operator . With the use of (7.1) one arrives at the following CS: For nondegenerated matrix (7.2) is equivalent to the equation or where . Using this form of the CS, one promptly concludes that the CS (7.2) has three independent first integrals: and it is representable as the commutativity condition for the linear system:
Choosing and as free functions and assuming that , one can easily resolve (7.6) with respect to . For instance, with one gets the following four-dimensional mapping:
() In a similar manner one finds the CS associated with the solvable DDA . Since in this case the CS takes the form For nondegenerated it is equivalent to where . Similar to the previous case the CS has three first integrals: and it is equivalent to the compatibility condition for the linear system: Note that the CS (7.9) is of the form (3.11) with . Thus, the deformations generated by can be considered as the reductions of the discrete deformations (3.11) under the constraint .
A class of solutions of the CS (7.9) is given by where g is matrix and . Since , one has and hence where are arbitrary functions. So, this subclass of deformations are defined by three arbitrary functions.
To describe the isoassociative deformations for which for all these functions should obey the systems of equations: It is a version of the discrete oriented associativity equation.
The author is very grateful to the referees for careful reading of the manuscript and various useful and fruitful remarks.
- M. Gerstenhaber, “On the deformation of rings and algebras,” Annals of Mathematics, vol. 79, pp. 59–103, 1964.
- M. Gerstenhaber, “On the deformation of rings and algebras. II,” Annals of Mathematics, vol. 84, pp. 1–19, 1966.
- E. Witten, “On the structure of the topological phase of two-dimensional gravity,” Nuclear Physics B, vol. 340, no. 2-3, pp. 281–332, 1990.
- R. Dijkgraaf, H. Verlinde, and E. Verlinde, “Topological strings in ,” Nuclear Physics B, vol. 352, no. 1, pp. 59–86, 1991.
- B. Dubrovin, “Integrable systems in topological field theory,” Nuclear Physics B, vol. 379, no. 3, pp. 627–689, 1992.
- B. Dubrovin, “Geometry of 2D topological field theories,” in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), vol. 1620 of Lecture Notes in Mathematics, pp. 120–348, Springer, Berlin, Germany, 1996.
- C. Hertling and Y. I. Manin, “Weak Frobenius manifolds,” International Mathematics Research Notices, no. 6, pp. 277–286, 1999.
- Y. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, vol. 47 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1999.
- C. Hertling and M. Marcolli, Eds., Frobenius Manifolds, Quantum Cohomology and Singularities, vol. E36 of Aspects of Mathematics, Friedrich Vieweg & Sohn, Wiesbaden, Germany, 2004.
- B. G. Konopelchenko and F. Magri, “Coisotropic deformations of associative algebras and dispersionless integrable hierarchies,” Communications in Mathematical Physics, vol. 274, no. 3, pp. 627–658, 2007.
- B. G. Konopelchenko and F. Magri, “Dispersionless integrable equations as coisotropic deformations: generalizations and reductions,” Theoretical and Mathematical Physics, vol. 151, no. 3, pp. 803–819, 2007.
- B. G. Konopelchenko, “Quantum deformations of associative algebras and integrable systems,” Journal of Physics A, vol. 42, no. 9, Article ID 095201, 18 pages, 2009.
- B. G. Konopelchenko, “Discrete integrable systems and deformations of associative algebras,” Journal of Physics A, vol. 42, no. 45, Article ID 454003, 35 pages, 2009.
- B. G. Konopelchenko and F. Magri, “Yano manifolds, F-manifolds and integrable systems,” unpublished.
- B. L. van der Waerden, Algebra, Springer, New York, NY, USA, 1971.
- N. Bourbaki, Groupes et Algebres de Lie, Hermann, Paris, France, 1972.
- J. Dieudonné, “Les déterminants sur un corps non commutatif,” Bulletin de la Société Mathématique de France, vol. 71, pp. 27–45, 1943.
- I. M. Gelfand and V. S. Retakh, “Determinants of matrices over noncommutative rings,” Functional Analysis and Its Applications, vol. 25, no. 2, pp. 91–102, 1991.
- P. A. M. Dirac, Lectures on Quantum Mechanics, Yeshiva University, New York, NY, USA, 1964.
- A. Losev and Y. I. Manin, “Extended modular operads,” in Frobenius Manifolds, Quantum Cohomology and Singularities, C. Hertling and M. Marcoli, Eds., vol. E36 of Aspects Mathematics, pp. 181–211, Vieweg, Wiesbaden, Germany, 2004.
- K. Yano and M. Ako, “On certain operators associated with tensor fields,” Kōdai Mathematical Seminar Reports, vol. 20, pp. 414–436, 1968.
- S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of Solitons, Contemporary Soviet Mathematics, Plenum, New York, NY, USA, 1984.
- M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1981.
- H. F. Baker, Abelian Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 1995.
- J. Chazy, “Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes,” Acta Mathematica, vol. 34, no. 1, pp. 317–385, 1911.
- F. J. Bureau, “Differential equations with fixed critical point. II,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 66, pp. 1–116, 1964.
- C. M. Cosgrove, “Chazy classes IX-XI of third-order differential equations,” Studies in Applied Mathematics, vol. 104, no. 3, pp. 171–228, 2000.