Formal Pseudodifferential Operators in One and Several Variables , Central Extensions , and Integrable Systems

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.


Introduction
This paper is on some cohomological aspects of Lie algebras of formal pseudodifferential operators in one and several independent variables, motivated by previous works on algebras of importance for integrable systems and symplectic geometry such as the Lie algebra of vector fields on the circle and their deformations [1,2] or the Lie algebra of differential operators; see, for example, [3][4][5][6].
A very well-known infinite-dimensional Lie algebra of interest for physics is the Virasoro algebra [7].This algebra is a one-dimensional central extension of the Lie algebra of vector fields on the circle (also called centerless Virasoro algebra) which also appears naturally in applications.We mention, as a recent example, that the centerless Virasoro algebra can be realized as an algebra of nonlocal symmetries for the Camassa-Holm and Hunter-Saxton equations [8,9].Now, the Lie algebra of vector fields on the circle is included naturally in the Lie algebra of differential operators on the circle.This algebra, in turn, is included in the Lie algebra of formal pseudodifferential operators on the circle which has been studied very carefully, for example, by Dickey [5] and Adler [10], in connection with the algebraic and geometric theory of the famous Korteweg-de Vries (KdV) equation and other integrable systems.Moreover, there exist nontrivial twisted versions of these works related to "twisted" and "quantum" analogs of classical integrable systems; see, for instance, [11][12][13][14].It is certainly reasonable to consider formal pseudodifferential operators as a general arena for integrable systems.
Our aim in this work is to study central extensions of Lie algebras of formal pseudodifferential operators in a general algebraic setting and to apply this study to the construction of hierarchies of centrally extended Lie algebras, Manin triples [15], and nonlinear integrable equations in one and several independent variables.
We mention four examples of relevant central extensions.Centerless Virasoro has a unique central extension; see [3].Also, a 2-cocycle of the algebra of differential operators on the circle was constructed in [6], and a 2-cocycle of the algebra of pseudodifferential operators on the circle was constructed by Kravchenko and Khesin using logarithms; see [16,17].Finally, a 2-cocycle for a quantum analog of the algebra of pseudodifferential operators was considered in [11].
In this paper we consider algebraic versions of these results, and we present classifications of central extensions.In particular, we show that many of the constructions of cocycles appearing in the literature (see, for instance, [17] or [4]) are valid well beyond their original framework.
We have divided our work in three main sections.

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In Section 2 we introduce the main objects considered in this work.We define formal pseudodifferential operators (or, "pseudodifferential symbols") in one variable on an arbitrary associative and commutative algebra, and we construct the Kravchenko-Khesin logarithmic 2-cocycle in this general context.We also construct a hierarchy of centrally extended Lie algebras of differential operators via the logarithmic 2cocycle.This construction generalizes a theorem by Khesin [18] on hierarchies of Lie algebras of differential operators on the circle.Finally, motivated by [11,19,20] and the later paper [12], we consider twisted pseudodifferential symbols on arbitrary associative and commutative algebras and, in analogy with the untwisted case, we construct central extensions and hierarchies of twisted centrally extended algebras.
In Section 3 we consider pseudodifferential symbols in several independent variables on an arbitrary associative and commutative algebra.Our main motivation for considering the several variables case in detail is the relative absence of examples of integrable equations in this context.Indeed, besides the equations of the standard KP hierarchy [5] and their cousins, there are not very many general constructions of integrable equations in several independent variables.Important exceptions are the equations introduced by Tenenblat and her coworkers (see [21] and references therein) and the hierarchies considered by Parshin, [22].We construct central extensions in the several variables case using logarithmic cocycles, and we also exhibit hierarchies of Lie algebras of differential operators in several independent variables admitting central extensions.We then consider the work [23] by Dzhumadil' daev, in which he presents a classification of central extensions.The paper [23] is quite technical and it requires a very careful and critical reading, and so we have decided to explain how to prove the main result of [23] using an inductive argument.We present a full inductive proof of Dzhumadil' daev's theorem using some technical homological tools elsewhere; see [24].
Finally, in Section 4 we introduce Manin triples, we define double extensions for the algebras of (twisted) pseudodifferential symbols in one and several independent variables and, using a general algebraic theorem [25], we construct Manin triples for these algebras, thereby putting [17,[26][27][28][29] in a very general framework.We also apply our algebraic results to the construction of integrable systems in one and several independent variables, roughly following the techniques of [4,11,22].

Basic Definitions and Preliminary Results.
Let A be an associative and commutative algebra, and let  : A → A be a derivation on A; that is,  is a linear map such that () = () + () for all ,  ∈ A. The algebra of formal differential symbols DO is generated by A and a symbol  with the relation for all  ∈ A. The algebra A is a subalgebra of DO and we can prove inductively that for all  ∈ A and  ≥ 0. We extend the algebra DO to obtain the algebra of pseudodifferential symbols ΨDO by introducing differentiations with negative exponents.A general element of ΨDO is a formal series  of the form We set and so (2) generalizes to for all  ∈ Z, in which the binomial coefficient is defined by for ,  ∈ Z.
The Lie algebra structure on ΨDO is given by the usual commutator [, ] =  ∘  −  ∘ , so that, for instance, Lemma 1.For any nonnegative integer  and ,  ∈ A, we have Proof.Proof is by induction.
We now proceed by induction on .

Proposition 3.
Let A be an algebra,  a derivation on A, and  a -invariant trace.Then the linear map res : Ψ → C defined by res is a trace on Ψ; that is, res is linear and it satisfies res() = res() for all ,  ∈ Ψ.This is the Adler-Manin noncommutative residue introduced in [10,30].
Proof.We use the elementary identity for  < 0 and  ≥ 0. Since res is linear, it is sufficient to show that for any ,  ∈ A and ,  ∈ Z We consider several cases.
Corollary 5.The bilinear form ⟨, ⟩ = res( ∘ ) is invariant; that is, it satisfies ⟨[, ], ⟩ = ⟨, [, ]⟩.Now, if  =     +  −1  −1 + ⋅ ⋅ ⋅ is a pseudodifferential symbol such that   ̸ = 0 and   = 0 for all  > , we say that  is the order or degree of .The following observation has resulted to be fundamental for the theory; see, for instance, [5,10,17,31].The algebra ΨDO can be decomposed as a (vector space) direct sum ΨDO = DO ⊕ INT, where Proposition 6.The subalgebras  (of differential operators) and  (of pseudodifferential symbols of order ≤ −1) are isotropic subspaces of Ψ with respect to the bilinear form defined in Corollary 5; that is, the restrictions of this form to both  and  vanish.

On Cohomology of Lie Algebras.
Having reviewed the elementary properties of ΨDO, we now summarize some basic facts on the cohomology of Lie algebras in order to fix our notation.We will study the cohomology of ΨDO in Section 2.3.

Algebraic Interpretations of Cohomology.
A derivation  of the Lie algebra g is a linear map  : where  ∈ g is a fixed element.Outer derivations are by definition elements of the quotient space of all derivations module the subspace of inner derivations.The proof of the following proposition is in [32], Chapter 1, Section 4. Proposition 7.  1 (g, g) can be interpreted as the space of outer derivations of the algebra g.Definition 8.A central extension of a Lie algebra g by a vector space n is a Lie algebra g whose underlying vector space g = g ⊕ n is equipped with the following Lie bracket: for some bilinear map  : g × g → n.
Note that  depends only on  and  but not on  and V.This implies that n is the center of the Lie algebra g.

Proposition 9.
There is a one-to-one correspondence between equivalence classes of central extensions of g by n and elements of  2 (g, n).
The referee has pointed out that Proposition 9 combined with Proposition 10 below allows us to effectively calculate central extensions.
Proposition 10.Let g be a Lie algebra.The space of onedimensional central extensions  2 (g, C) is isomorphic to a subspace of the first cohomology space  1 (g, g * ).Specifically, if we denote by  1 (g) the subspace of  1 (g, g * ) generated by cohomology classes of cocycles  : g → g * such that This general proposition is due to Dzhumadil' daev, who used it in [33] (in the case of one-dimensional central extensions induced by fields of characteristic  > 0) and in [34] (in the zero characteristic case) for the study of central extensions of Lie algebras of Cartan type; see, for example, [33][34][35].Proposition 10 has the following corollary, also pointed out by the referee (see also [23,33]).

Corollary 11.
If the Lie algebra g admits an invariant, symmetric, and nondegenerate bilinear form ⟨ , ⟩, then the space of onedimensional central extensions  2 (g, C) is isomorphic to the space of outer derivations  : g → g such that ⟨(), ⟩ = 0 for all  ∈ g.
Proof.The existence of ⟨ , ⟩ allows us to identify  1 (g, g) with  1 (g, g * ).Now, if we take  so that the cohomology class of  is in  1 (g), we obtain a set of derivations of the form for some  ∈ g.Then, invariance of ⟨ , ⟩ implies that  ∈  1 (g) if and only if ⟨  (), ⟩ = 0 for all  ∈ g.

Outer Derivations and Central Extensions of Ψ𝐷𝑂.
We go back to the algebra ΨDO considered in Section 2.1.Following [16], we write formally the identity   =   log  .This implies that Hence, setting  =  in ( 8) and differentiating at  = 0 using It follows that if  ∈ ΨDO, then [log , ] is also an element of ΨDO, even though log  itself is not.
In the next proposition we will make use of the following combinatorial identity (see, for instance, [12]).Lemma 12. Let  ≥ 1 and  ≥ 0 be integers and  ∈ Z. Then Proposition 13 and Theorem 14 below were proved by Kravchenko and Khesin [16] in the case A = Diff( 1 ).

Proposition 13. [log 𝜉, ⋅] defines a (resp., an outer) derivation of the associative (resp., Lie) algebra Ψ𝐷𝑂.
Proof.We note that the proposition does not follow from the fact that for any associative algebra the map   → [, ⋅] determines a derivation, since in our case log  is not an element of ΨDO.
First of all, it is not difficult to see using ( 27) that [log , ] belongs to ΨDO for any  ∈ ΨDO.Now, assuming that [log , ⋅] is a derivation, it is trivial to prove that it is outer derivation of the Lie algebra ΨDO: if [log , ⋅] = [, ⋅] for some  ∈ ΨDO, then log  −  belongs to the center of ΨDO, and so log  ∈ ΨDO, a contradiction.
We show that [log , ⋅] is a derivation.It is sufficient to prove that, for any ,  ∈ , ,  ∈ Z, Indeed, for the left side of ( 29) we have On the other hand, for the right side of (29) we have Now, for any integer  ⩾ 1, the coefficient of  +− in (30) is where the summation is over all integers  ⩾ 1,  ⩾ 0 such that  +  = .Using (9) we have where both summations are over all integers  ⩾ 1,  ⩾ 0 such that  +  = .On the other hand, for  ⩾ 1, the coefficient where the sum is over all integers  ⩾ 1,  ⩾ 0 such that  +  = .Therefore, ( 33) and ( 34) are equal if for fixed integers ,  as above we have where the sum is over all integers  ⩾ 1,  ⩾ 0, and  ⩾ 0 such that  =  − ,  +  = .This amounts to showing that and this is consequence of (28).
Proof.It is easy to see that res([log ,   ]) = 0 for  ≤ −1, while for  ≥ 0 we have res and so res([log , ]) = 0 for all  ∈ ΨDO.It follows that and so  is skew-symmetric.It remains to prove the cocycle identity (24).This a direct calculation using Corollary 5 and the fact that [log , ⋅] is a Lie algebra derivation.

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Remark 15.In the case of the Lie algebra of pseudodifferential symbols on  1 , the restriction of the cocycle (37) to the subalgebra of vector fields on  1 is the Gelfand-Fuchs cocycle see, for instance, [17,Prop. 4.12].This cocycle is nontrivial; see [3], and therefore cocycle (37) is nontrivial [16].It is also known (see, for instance, [17]) that the restriction of 2-cocycle (37) to the subalgebra of differential operator DO( 1 ) ⊂ ΨDO( 1 ) is a multiple of the Kac-Peterson cocycle [6]: Interestingly, the Lie algebra DO( 1 ) has exactly one central extension [36] but ΨDO( 1 ) has two independent central extensions [17,37]: in addition to (37), the following expression defines a nontrivial cocycle: We will reprove this result as a corollary of our study of central extensions of Lie algebras of formal pseudodifferential operators in several variables; see Lemma 33 in Section 3.4.
Remark 16.Let Σ be a compact Riemann surface and let M be the space of meromorphic functions on Σ. Fix a meromorphic vector field V on Σ and denote by  V the operator of Lie derivative  V along the field V: locally, if V() = ()/, and  ∈ M, then  V () =  V (()) = ()  ().The associative algebra of meromorphic pseudodifferential symbols is (see [4]) with multiplication defined as in (7).We consider the Lie algebra structure of ΨDO and the residue map V is understood as a meromorphic differential on Σ.We further define the trace associated to the point  ∈ Σ by Tr = res  (res  ()).Then as a consequence of Theorem 14 we have a nontrivial 2-cocycle on ΨDO given by  V (, ) = Tr([log  V , ]).This cocycle first appeared in [4].

A Hierarchy of Centrally Extended Lie Algebras. ΨDO
is not unique in admitting nontrivial central extensions.In fact, a whole hierarchy of Lie algebras does.This fact was first observed by Khesin, [18], in the case A = Diff( 1 ).
For any positive integer , we let ΨDO  be the subalgebra of ΨDO consisting of differential operators of the form ∑  =     for some nonnegative integer .
We note that in this case we cannot use Corollary 11 to prove that the 2-cocycle  determines a nontrivial central extension because [log , ⋅] is not a derivation on ΨDO  .

The Algebra of Twisted Pseudodifferential Symbols in One
Variable.We consider the algebra of twisted pseudodifferential symbols and its corresponding logarithmic cocycle following [12].Particular examples have appeared much earlier; see, for instance, [11,19,20].Definition 18.Let  : A → A be an automorphism of fixed algebra A, and let ,  ∈ A.
Given a triplet (A, , ) as above, the algebra of twisted formal pseudodifferential symbols ΨDO  is the set of all formal Laurent series in  with coefficients in A: equipped with a multiplication determined by the rules For example, for each  ≥ 0 we have where  , (, ) is a noncommutative polynomial in  and  with (   ) terms of total degree  such that the degree of  is .If  = 2, for instance, we get  2  =  2 () + (() + ()) +  2 () 2 .We extend (50) for  < 0. We obtain and it follows that if where  = ( 1 , . . .,   ) is an -tuple of integers and || =  1 + ⋅ ⋅ ⋅ +   .The next proposition and theorem are proved in [12].

Proposition 19.
Let A be algebra,  an automorphism of A,  a -trace on A, and  a -derivation on A. If ∘ = 0, then for any ,  ∈ A and any -tuple  = ( 1 , . . .,   ) of nonnegative integers, we have Theorem 20.Let A be algebra,  an automorphism of A,  a -trace on A, and  a -derivation on A. If  ∘  = 0, then the linear functional res : Ψ  → C defined by res is a trace on Ψ  .
As pointed out in [12], if  ∘  =  ∘ , formulae (50) and (51) simplify to For example, the twisted pseudodifferential operators considered in [11] satisfy (55).We introduce a twisted logarithmic cocycle assuming that  and  commute.Let   be a 1parameter group of automorphisms of A with  1 = .We formally replace the integer  by  ∈ R in (55) and obtain Taking derivatives with respect to  at  = 0, as in Section 2.3 we obtain the commutation relation We note that [log , ] = 0.The following two results are also proven in [12].Now we go beyond [12].The algebra Ψ  has a direct sum decomposition as a vector space, where The proof now follows along the lines of the demonstration of Theorem 17.
Let A be an algebra on C and let   , with  = 1, . . ., , be (commuting) derivations on A. The algebra of formal differential symbols in several variables DO  is, by definition, the algebra generated by A and symbols   with the relations for all  ∈ A and  = 1, . . ., .Elements of DO  are of the form  = ∑ ∈Γ +      .Using (64), we can prove that We extend the algebra DO  to the algebra Ψ  DO of formal pseudodifferential operators by introducing differentiations with negative exponents via and we define a structure of Lie algebra on Ψ  DO by the usual commutator where   /  and   are determined by linearity and the rules (68)
(71)  ( As in Section 2.1, we define the bilinear form ⟨, ⟩ = res() for ,  ∈  DO.We can prove that it is symmetric and invariant, and we will assume hereafter that it is nondegenerate.Examples of nondegenerate bilinear forms as above appear in [22,23] for special choices of algebras A. Proof.Skew-symmetry is proven as in Theorem 14.Also, as before, a straightforward computation using that [log   , ⋅] is a Lie algebra derivation yields   ([, ], ) +   ([, ], ) +   ([, ], ) = 0.

Hierarchies of Centrally Extended Algebras of Pseudodifferential Symbols in Several
Variables.We present two examples of hierarchies of subalgebras of Ψ  DO and we prove that they admit nontrivial central extensions.
For our first example, we set   =  > 0 for fixed 1 ≤  ≤ , and The coefficient of  − verifies that on the th position.On the other hand, we have that The coefficient of  − satisfies that on the th position.Comparing (78) with (80) for arbitrary ,  we have that   =   , and this is a contradiction, because   ≤ −  − 1 and   ≥ 0.
For our second example, let us say that  ≥  if   ≥   for all .Fix  ∈ Γ +  , and define Ψ  DO  as the subalgebra of Ψ  DO with elements of the form ∑ ∈Γ      such that  ≥ .

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Theorem 29.The cocycle generated by log   for each  is a nontrivial cocycle in Ψ    .Hence, it defines a nontrivial central extension of Ψ    .
Proof.The proof is similar to the proof of Theorem 28.We see that if the cocycle generated by log   were trivial, then (78) and (80) would be true for all  components, and we would have   =   for all , which is impossible.

The Dzhumadil' daev Classification
Theorem.We sketch a new proof of the principal theorem of [23].It is a new proof, in the sense that we argue by induction and we use tools from homological algebra to perform the inductive step.It is a sketch, in the sense that full homological details are left for the paper [24].We continue using the notation introduced in Section 3.1.
Let  +  = {∑      |  ∈ Γ +  } be the algebra of polynomials in variables  1 , . . .,   , and let   be the algebra of Laurent power series of the form ∑ ∈Γ      such that the number of  ∈ Γ +  's with nonzero   is finite.The action of the derivation   on this algebra is determined by   (  ) =    −  , with  ∈ Γ  .
Let U be another algebra with derivations δ1 , . . ., δ .The tensor product U ⊗   becomes an associative algebra if we endow it with the multiplication rule Clearly this algebra contains U ⊗  +  as a subalgebra and we have the following.

Proposition 30.
There is an isomorphism between the associative algebra of formal pseudodifferential symbols Ψ   and U ⊗   determined by the correspondence This isomorphism determines a Lie algebra isomorphism between Ψ   and U ⊗   .Now, following [23], we let H  be the Lie algebra associated to U⊗  with U =   .Then, identifying   +   − ≃   ⊗  , we see that an element of H  is of the form and that the Lie bracket on H  is given by where  + ,  − are derivations acting on H  as and Definition 31.We define a linear function  0 on It is not hard to check that  0 is an outer derivation of the Lie algebra H  .The following result is the main theorem of this section.
Proof.We sketch an inductive proof of Theorem 32.We consider first the one variable case; that is, the basic elements of the Lie algebra are  + =  +1 and  − =  −1 .We write H instead of H 1 .Proposition 25 implies that [log  + , ⋅] and [log  − , ⋅] are outer derivations, and it is easy to check that the derivations [log  + , ⋅], [log  − , ⋅], and  0 are linearly independent.Now we have the following.

Lemma 33.
Let  be a derivation on H with ( ± ) ̸ = 0.Then, there is  ∈ H such that for some , .
Lemmas 33 and 34 imply that Theorem 32 holds in the case  = 1.In order to perform the inductive step we use an appropriate version of the Künneth formula.Again we refer to [24].
We use Theorem 32 to classify central extensions of H  : Theorem 1 of [23] tells us that H  can be equipped with the bilinear form ⟨, V⟩ = res(V) = (V) − and that this form is symmetric, nondegenerate, and invariant.Then, reasoning as in Theorem 27 we obtain that   = res([log  ± , ]) and  0 = res([ 0 , ]) are Lie algebra cocycles of H  .Now we use the key Corollary 11: we have that ⟨ 0 (), ⟩ ̸ = 0 for  =  − and therefore  0 does not determine a central extension.On the other hand, in [23, page 135] the author shows that ⟨[log  ± , ], ⟩ = 0 for all .We conclude that the space of central extensions of H  is of dimension 2.The case  = 1 is discussed from a geometric point of view in [17,Remark 4.16].

Manin Triples and Double
Extensions of (Twisted) Pseudodifferential Symbols.Manin triples are ubiquitous in integrable systems; see, for instance, [17].They were introduced by Drinfel' d in his seminal paper [15] on Hopf algebras and the quantum inverse scattering method.Definition 35.Three Lie algebras g, g − , and g + form a Manin triple if the following conditions are satisfied.
(1) The Lie algebras g − and g + are Lie subalgebras of g such that g = g − ⊕ g − as vector spaces.
(2) There exists a nondegenerate invariant bilinear form on g such that g + and g − are isotropic subspaces; that is, the restrictions of this form to both g + and g − vanish.
For example, if we assume that the bilinear form ⟨, ⟩ = res() on ΨDO is nondegenerate, then Proposition 6 tells us that the algebras ΨDO, DO, and INT form a Manin triple with respect to the bilinear form ⟨, ⟩.In the same way, Proposition 23 implies that the algebras ΨDO  , DO  , and INT  form a Manin triple with respect to the twisted bilinear form ⟨, ⟩  .
Let us discuss the several variables case in some detail.We say that the order of the pseudodifferential operator  = ∑ ∈Γ      is , if there is  = ( 1 , . . .,   ) with   =  such that    ̸ = 0 and for all  = ( 1 , . . .,   ) such that   >  we have   = 0.This definition is analogous to the notion of order used by Parshin in [22].
The Lie algebra Ψ  DO has two natural subalgebras: and clearly, as a vector space, Ψ  DO is a direct sum of these algebras: Proposition 36.If the bilinear form ⟨, ⟩ = res() is nondegenerate, the algebras (Ψ  ,   ,   ) form a Manin triple with respect to ⟨ , ⟩.
Proof.We reason as in the proof of Proposition 6.
Theorem 38.Let (A, ) be a metrized algebra over a field K, and let  be a Lie algebra over K. Suppose that there is a Lie homomorphism  :  → Der  (A), where Der  (A) denotes the space of all -antisymmetric derivations of A (i.e., the derivations  of A for which (, ã) + (, ã) = 0 for all , ã ∈ A).
Let  * denote the dual space of .We define  : A × A → B * as the bilinear antisymmetric map The pair (A  ,   ) is a metrized algebra over K called the double extension of A by (, ).
We sketch the proof of Theorem 38 in the case of interest for us.We assume that A is a Lie algebra and that (, ) = (, ) is a bilineal symmetric form on A which is invariant and nondegenerate.Then, it is in fact easy to check that the bilineal form (102) is symmetric, nondegenerate and invariant.On the other hand, that (101) defines a Lie bracket on A  follows from a straightforward computation using the identities and invariant symmetric linear form This extension is certainly known; see [17].In an analogous way we obtain a double extension ΨDO  of the Lie algebra of twisted pseudodifferential operators.Now, we note that the Lie algebra ΨDO (and also ΨDO  ) admits the (vector space) direct sum decomposition ΨDO = DO ⊕ ĨNT (resp., ΨDO  = DO  ⊕ ĨNT  ), in which and analogous definitions for the twisted case.We have the following.Theorem 38 also allows us to define a double extension of Ψ DO by considering the -central extension R ⋅  1 ⊕ ⋅ ⋅ ⋅ R ⋅   and the  symbols log   .We have the vector space decomposition

Hierarchies of Differential Equations.
In this last section we apply our work to the construction of hierarchies of partial differential equations.We begin with some standard facts on hamiltonian systems modelled on (dual spaces of) Lie algebras (see, for instance, [17] or the recent paper [38]).
The operator  is called a classical -matrix if the bracket [ , ]  satisfies the Jacobi identity.
an abstract form of the ubiquitous Korteweg-de Vries equation.
Definition 41.If g is a Lie algebra and g * is its dual space, the functional derivative of a function  : g * → R at  ∈ g * is the unique element / of g determined by * , in which ⟨ , ⟩ denotes a natural paring between g and g * .* → R and  ∈ g * . = [ () , ] + [,  ()] ,  ∈ g.