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H_T Vertex Algebras and the Infinite Toda Lattice

Let H_T=C[T,T^{-1}] be the Hopf algebra of symmetries of a lattice of rank 1, or equivalently, H_T is the group algebra of a free Abelian group with one generator T. We construct conformal algebras, vertex Poisson algebras and vertex algebras with H_T as symmetry. For example, the Hamiltonian structure for the infinite Toda lattice gives rise to an H_T-vertex Poisson structure on a free difference algebra. Examples of H_T-vertex algebras are constructed from representations of a class of infinite dimensional Lie algebras related to H_T in the same way loop algebras are related to the Hopf algebra H_D=C[D] of infinitesimal translations used in the usual vertex algebras.


Introduction 1. Classical Vertex Algebras and the Hopf Algebra H D
In this section we recall what a vertex algebra in the usual sense is and indicate how it is related to a Hopf algebra H D . See e.g., Frenkel-Lepowsky-Meurman ( [FLM88]) or Kac ([Kac98] for more details on classical vertex algebras and Borcherds ([Bor98]) on the idea of emphasizing the role of the Hopf algebra, cf., also Snydal ([Sny]).
Briefly, a vertex algebra structure on a vector space V is a map Y : V ⊗ V → V ((z)) associating to a pair of vectors a, b in V a formal Laurent series Y (a ⊗ b, z) with values in V . As usual we write Y (a ⊗ b, z) = Y (a, z)b to obtain the vertex operator Y (a, z) = ∑ a (n) z −n−1 with the a (n) linear maps V → V . Also an essential ingredient is the infinitesimal translation operator D : V → V , such that [D,Y (a, z)] = ∂ z Y (a, z). These ingredients satisfy a number of axioms that we will not recall at this point. One of the properties of a vertex algebra is that two vertex operators satisfy a commutator formula of the form where δ (z, w) = ∑ k∈Z z k w −k−1 is the formal Dirac delta function, and for any linear operator D we write D (n) = D n n! . The simplest nontrivial example is given by the vertex algebra of the Heisenberg algebra, the Lie algebra spanned by a central element c and elements b (n) , n ∈ Z, with relations ] of formal power series in a variable t, the functions on the formal disk. Note that the map Y in a vertex algebra structure takes values in formal Laurent series, which are the singular functions on the formal disk. In the same spirit note that the commutator is singular, in the sense that we cannot put z = w in this formula. The delta function can be obtained from the difference of two expansions of the basic singularity 1 z−w . The occurence of H D and the singularities in its dual is no coincidence but the tip of an iceberg. Borcherds, [Bor98] has a general construction of what he calls G-vertex algebras. The construction is based on the notion of a vertex group G, which is, roughly speaking, of a free rank one Abelian group A = T generated by an element T . The dual algebra is the algebra C Z of functions on the integers, and we have found a natural way to introduce singularities for C Z by localizing to get an algebra K. This leads to a class of what we call H T -vertex algebras. These occurs naturally in the theory of the infinite Toda Lattice. To explain this we first recall some facts about Gelfand-Dickey Hamiltonian structures, and how they are related to classical vertex algebras.

Gelfand-Dickey Structures and the Hopf Algebra H D
In the simplest case one wants to describe an infinite hierarchy of differential equations for a single unknown v depending on spatial variable x and infinitely many time variables t i , i > 0. For instance, the Modified Korteweg-de Vries (mKdV) hierarchy is a system of differential equations of the form where f i is a differential polynomial, i.e., an element of the polynomial algebra V = C[v (n) ] n≥0 . This algebra is a module over H D via Dv (n) = v (n+1) . V is a commutative algebra, and its (maximal) spectrum Specm(V ) = Hom C−alg (V, C) can be identified with C , so we will from now on allow s ∈ C((t)). The reader will be excited to note that here the same space of singular functions on the formal disk appears as in the previous section in the context of vertex algebras. But there is more . . . Define the Classical Field C( f , z) of f ∈ V as the generating series of functionals A Gelfand-Dickey Hamiltonian structure on V is then a prescription to define the Poisson brackets of functionals f (n) , or, equivalently, for the corresponding classical fields. For a precise definition (of the Poisson bracket { f (0) , g (0) } of the residues of the classical fields) see Dickey's book ( [Dic03]). The simplest nontrivial example is, perhaps, the Hamiltonian structure for the modified Korteweg-de Vries hierarchy. In this case the Poisson bracket for two functionals is given by where the variational derivative of a functional is given by Then one checks that the Poisson bracket of the basic classical field C(v, z) is given by {C(v, z),C(v, w)} = ∂ w δ (z, w). More generally, the Poisson bracket of two classical fields will be a linear combination of derivatives of delta functions, with classical fields as coefficients. Of course, this is very similar to the commutator formula (1.2) for free bosons.
The basic reason H D appears in the theory of equations of the form (2.1) is that if v(x) solves this system, also the infinitesimal translate v(x +δ ) will solve it. In the theory of Toda lattices one studies equations of the following form (or the obvious multicomponent generalization) where x = (x n (t)) n∈Z is a sequence (of functions of the times t j , j > 0) and f j is a polynomial in the entries of the sequence x. In this case we get as basic symmetry invariance under shifts: if x(t) solves (2.3) then also the shifted sequence T x := (x n+1 (t)) n∈Z is a solution. In this case the symmetry algebra is the Hopf algebra H T of (1.3). First we will develop some of the machinery of the Hopf algebra H T in the next chapter. Then we use this to define in chapter 3 the notion of a Singular Hamiltonian structure and the associated Poisson brackets. As an example we discuss the Hamiltonian structure for the infinite Toda lattice in chapter 4. Then we define a generalization of singular Hamiltonian structures called H T -conformal algebras in 5. Finally we define a class of vertex algebras associated to H T in chapter 6 and discuss some simple examples.

CHAPTER 2
The Hopf Algebra H T and Sequences

Introduction
The goal of this chapter is to indicate how one can construct for the Hopf algebra H T analogs of well known structures related to formal series used in classical vertex algebras. Recall that the dual of the symmetry algebra • The expansion maps (in regions |x| > |y|, resp. |y| > |x|) • The formal delta function, the difference of the two expansions: δ (x, y) = e −y∂ x 1 x + e −x∂ y 1 y .
For H T = C[T, T −1 ] the dual Hopf algebra is the algebra C Z of (arbitrary) functions from the integers to C as will be explained below. In other words the elements of C Z are sequences s = (s n ) n∈Z , with s n ∈ C. Let δ n be the Kronecker sequence with value 1 at n and zero elsewhere, so that s = ∑ n∈Z s n δ n . Note that Kronecker sequences are zero divisors in C Z , as are all non invertible elements of C Z .. Because of the zero divisors in C Z localizing is a somewhat delicate matter. In general, we want to choose a multiplicative set M ⊂ C Z of elements that can be inverted, but we must be careful not to have f , g ∈ M with f .g = 0 , because in that case 1/ f = g/ f .g. In particular inverting more than one of the Kronecker sequences δ n would not be a useful strategy. It turns out that we need to find appropriate analogs in C Z of the powers of t occuring in the dual C [[t]] of H D . These are the sequences τ(ℓ) introduced in section 4.

The Hopf Algebra H T
Let H T = C[T, T −1 ], as before, so that H T is a commutative and cocommutative Hopf algebra and we have the following structures on H T : • The unit i : C → H T , 1 → T 0 and multiplication map m : H T ⊗ H T → H T , given by T m ⊗ T n → T m+n , m, n ∈ Z. • The coproduct π : H T → H T ⊗ H T , given by T m → T m ⊗ T m and the counit ε : H T → C, T m → 1. • The antipode S : H T → H T , T m → T −m . These maps satisfy a large number of compatibility conditions that we will not recall here, see e.g., [CP95].
The powers of the translation T form a basis {T m } m∈Z for H T . Another basis will also be useful for us. Instead of translations we can use differences as basis: let ∆ = T − 1 and∆ = 1 − T −1 . The (divided) powers of ∆ and∆ form the difference operator basis {∆[m]} m∈Z , where Note that∆ = −S∆. Then we get for the antipode acting on this basis: The coproduct of these basis elements becomes

The Dual of H T and Sequences
H T is the group algebra (over the complex numbers) of the free Abelian group of rank 1, i.e., of the additive group Z. Therefore the linear dual H * T is the space of complex-valued maps on Z. So let C Z be the vector space of two sided infinite sequences s = (s n ) n∈Z , s n ∈ C. Then the sequence s defines a function on Z with value s n at n ∈ Z. Let δ n be the Kronecker sequence with a 1 on the nth position and for the rest zeroes, so that any sequence s has an expansion s = ∑ n∈Z s n δ n .
Define an action of H T on C Z by T δ k = δ k−1 so that if s,ŝ are sequences withŝ = T s thenŝ n = s n+1 . Define a pairing (3.1) , so that we have T k , δ l = δ k,l . Now {T m } and {δ m } are dual 'bases' of H T and C Z respectively (we need infinite sums of the δ m in C Z so they don't form a basis in the algebraic sense). We use this to identify C Z with the dual H * T of H T . The pairing satisfies The coproduct on H T induces via the pairing (3.1) a multiplication on C Z : we have π(T m ), δ n ⊗ δ p = T m , δ n · δ p = δ m,n δ m,p , so that the multiplication of basis elements of C Z is δ n · δ p = δ n δ n,p . The multiplication is therefore just the usual termwise multiplication of sequences, The antipode S of H T induces an antipode on C Z via The product on H T induces a map In particular We see that the image of δ n under m * does not belong to C Z ⊗ C Z . In a sense C Z is too large.

The Hopf Dual of H T and Difference Equations
Recall that the Hopf dual of a Hopf algebra is the subspace H • of the full linear dual H * consisting of φ such that One proves that in fact in this case and that H • is alternatively characterized as the subspace of elements φ such that the H-submodule generated by φ is finite dimensional. H • is then a Hopf algebra in its own right, see [Abe80] for details. In Solutions of these equations are easy to describe explicitly. First note that without loss of generality we can assume that The space of solutions of (4.1) is then N dimensional. Now we can factor P(T ) = ∏ n i (T − λ i ) d i and solving (4.1) reduces to solving (T − λ ) d s = 0. Define for λ = 0 an exponential sequence (4.2) For any sequence f we have To find such f we introduce polynomial sequences τ(ℓ) ∈ H • T , ℓ ≥ 0. First put τ(0) = ∑ n∈Z δ n , the sequence with 1 in all positions; this is the identity 1 C Z in the C-algebra C Z . Then define recursively sequences τ(ℓ), as the unique solutions of the difference equations (4.3) ∆τ(ℓ) = ℓτ(ℓ − 1), satisfying the initial conditions τ(ℓ) 0 = 0. We have an explicit formula for the τ(ℓ). First of all We abbreviate τ = τ(1). Then where (n) ℓ is the Jordan factorial: (n) ℓ = n(n − 1) . . . (n − ℓ + 1), for n, ℓ ∈ Z, ℓ ≥ 0. In particular τ(ℓ) is a sequence with 0, 1, . . ., ℓ − 1 as its only zeroes. The formula (4.4) will follow from (4.7) below.
, from C to Z, the sequence τ(1) = τ corresponding to the polynomial x.
Note that the sequences τ(ℓ) are analogs for ], satisfying the differential recursion Of course, the multiplicative properties of the τ(ℓ)'s are more complicated. We have and hence we can calculate products in C pol Z : We see that the product of two basis elements of C pol Z is a finite (integral) linear combination of basis elements and C pol Z is a subalgebra of H • T . To derive (4.7) we use the Leibniz rule for the difference of a product: by the coproduct formula (2.3) We have ∆(τT −1 τ) = ∆(τ)τ + τ(T −1 ∆τ) = 2τ, (τT −1 τ) 0 = 0 so that by uniqueness τ(2) = τT −1 τ. Assuming that (4.7) is true for all ℓ, m such that ℓ + m = n then by induction and so by uniqueness of solutions of (4.3) (4.7) is true for ℓ + 1, m. A similar calculation shows the same for ℓ, m + 1.

The Dual Difference Basis
The difference operator basis {∆[n]} n∈Z of (2.1) has as dual the set of sequences ∆ * [n] defined by ∆[n], ∆ * [m] = δ nm . We refer to {∆ * [n]} n∈Z as the dual difference basis. Introduce notation PROOF. For any φ ∈ C Z we have φ = ∑ n T n , φ δ n . Now T = 1 + ∆, By the binomial formula we then obtain Comparing (6.2) with (5.3) we see that for m = 0 ∆ * [m] is a projection of τ[m]: where Π + is the restriction of functions on Z to Z ≥0 and Π − is the restriction to Z <0 .
We can conversely express the polynomials τ[k] in terms of the dual difference basis. We have by calculating τ for certain integers c ℓs , see section 22 for an explicit formula. In particular

Localization for C pol Z
Let M ⊂ C pol Z be the multiplicative set generated by 1 and the translates T k τ of τ, k ∈ Z. Elements of M correspond to polynomial functions on C with only zeroes at the integers. Define REMARK 7.1. We could also consider the localization This does not seem to give an essential different theory, so we will restrict ourselves to considering K.
Since M is stable under the action of the group generated by T we can extend the action of H T on C pol Z to K via Using the Leibniz rule (4.9) we find the quotient rule It follows from this that Then we can summarize (7.2), using the notations (6.1), (5.2), Recall that elements of C pol Z correspond to polynomial functions on C, via the algebra homomorphism Also recall that by the division algorithm in C[x] we can write, uniquely, for where in case r = 0 we have deg r < deg g. We can and will assume that r and g have no common zeroes. Furthermore we have a partial fraction expansion for r g : if g has zeroes {n i } i∈I with n i ∈ Z and with multiplicities d i then r g Translating this result from C[x] to C pol Z , using the map Φ of (7.4), we see that any y ∈ K can be uniquely written as Here the n i are integers (as y − is supposed to come, via Φ, from a fraction r g where r, g ∈ C[x] and g has only integral zeroes). Hence We have then a projection (7.8) Sing : K → K Sing .

8.Ĥ T andĤ *
T From (7.7) it is clear that K Sing is freely generated by 1 τ as a module over the Hopf algebraĤ where ∂ τ commutes with ∆, ∆, has antipode S(∂ τ ) = −∂ τ , coproduct π(∂ τ ) = ∂ τ ⊗ 1 + 1 ⊗ ∂ τ and counit ε(∂ τ ) = 0. We have an action ofĤ T on C pol Z by putting and this gives K anĤ T -module structure. A basis forĤ T is given by ] of formal power series in a variable σ with coefficients in C Z , the dual of H T . We define an action ofĤ T on C Z [[σ ]] by extending the action of H T on C Z by putting: has an expansion and we extend the counit of C Z to a map ε : We use this to define a pairing σ .e n,ℓ = e n,ℓ−1 ℓ > 0 0

Action of C Z [[σ ]] on K Sing
Let K Sing = K /C pol Z . As a vector space andĤ T -module this is the same as K Sing but K Sing is a C pol Z -module, which K Sing is not.
We have anĤ T -module isomorphism Domain and range of α are C pol Z -modules.
(c) Hence the map α of (10.2) is also an isomorphism of C pol Z -modules.
PROOF. Part (a) follows from Lemma (9.1) and (10.1). Part (b) follows from the observation that in anyĤ T -and C pol Z -module we have recalling the notation (6.1). Then part (c) follows from the other parts.
We use α to extend the C pol Z action on K Sing to an action of C Z [[σ ]] on K Sing . In particular the product e * n,k .F is well defined for all F ∈ K Sing , and for instance we have, by (9.2), , ℓ, m ≥ 0.
Here we use that K is the localization M −1 C pol Z (with M the multiplicative set of translates of τ), and that K has a decomposition in singular and holomorphic elements. Now we can localize also the full dual C Z = H * T : define Now, maybe, one expects that similarly the C Z -action on K Sing can be written in the form (10.4), or equivalently that the C Z -action on K Sing comes from a projection of the natural action of C Z on K Z . However, this is not the case. Indeed, let f = δ k ∈ C Z , the Kronecker sequence with support at k > 0, for example. Then we have δ k τ(k + 1) = 0, see (4.4), so that for all g h ∈ K (or K Z ) we have On the other hand we have by (9.1) In fact, inside K Z there is no distinction between holomorphic and singular elements. To see this note that we have an exact sequence Then we see that also where we identify M with its image in C tf Z , as M ∩ C pol Z = / 0. Now if m ∈ M then m has at most a finite number of zeroes (as function on the integers), so modulo C tor Z m is invertible. Hence as subset of C tf Z M consists of units and so we find K Z ≃ C tf Z . There are no singular elements in C tf Z .

Trace for K
NowĤ T , being a Hopf algebra, has a counit, a multiplicative map ε :Ĥ T → C given on the basis (8.2) by The counit induces, via the isomorphism α, a map K Sing → C, which we extend (by zero) to all of K to get a map, called the Trace where Sing is the projection (7.8)

then
Tr(hF) = ε(h) Tr(F). Recall that an element F ∈ K corresponds to a rational function F(x) on C with at most a finite number of finite order poles at the integers. Let F(x)dx be the corresponding rational 1-form on C. Then the trace of F is the sum of residues of this 1-form.
LEMMA 11.2. If F ∈ K then (11.3) Note that if f ∈ C pol Z the value of f at 0 can be expressed as a trace The terminology Trace for the map (11.1) is used to indicate the analogy with the trace of finite matrices. The Hamiltonians of the finite Toda lattice are traces of Lax matrices. Similarly the Hamiltonians for the infinite Toda lattice are traces in the sense of (11.1). This is discussed in more detail at the end of section 2.
REMARK 11.3. The trace on K can not reasonably extended to a trace on the localization K Z introduced in Remark 10.2. As explained there, in K Z there is no distinction between singular and holomorphic elements.
However, what we can and will do is define a pairing, by abuse of notation also called trace, between C Z and K Sing : We warn the reader that in general, for g h ∈ K Sing unless f ∈ C pol Z , see Lemma 10.1, part c.

Orthogonality Relations
We introduce notation: for ℓ ≥ 0 we put . PROOF. We consider the left hand identity first. In case m, n ≥ 0 the trace is zero because τ(m)τ(n) is nonsingular. In case m, n ≤ −1 the one form 1 f dx corresponding to 1 τ(−m)τ(−n) is regular at ∞ and has only poles at the integers, so we can, using (11.3), replace the trace by the sum over the residues of 1 f dx at all points of P 1 , which is zero by the residue theorem. So assume m ≥ 0 and n = −k − 1 for k ≥ 0. Then we get using the quotient rule (7.2), the partial summation rule (11.2), (4.3), and (4.7) and (11.4) The rest of the proof is similar.
Recall the action of C Z [[σ ]] on K Sing described by combining Lemma 9.1 with the isomorphism α.

Expansions in K
A basis for K is given by So any F ∈ K has a (finite) expansion (13.2) The coefficients of F are traces: using the action of C Z [[σ ]] on K Sing , see section 10.

(Twisted) Coproduct and (Twisted) Exponential Operators
Let H be a commutative and cocommutative Hopf algebra, for simplicity, and let H * be its dual. Fix a basis {e i } i∈I for H and let e * j be linear functions on H such that e * j (e i ) = δ i j . Any φ ∈ H * can then be expanded as Dual to the twisted multiplication we get a map PROOF. If, for instance, B : H → H is a linear map, then the dual is given by B * φ (e) = φ (Be) for φ ∈ H * , e ∈ H. Then we have By a similar calculation we find the expression of the A ⊗ B twisted coproduct in terms of R A⊗B .
Because of Example 14.2 below we call L A⊗B , R A⊗B twisted exponential operators. In case A = B = 1 we write L 1⊗1 = L , R 1⊗1 = R and refer to L , R as untwisted exponential operators. We will mainly interested in the case A = 1, B = S, and we will write for brevity . Here and below we will write t 1 ,t 2 for t ⊗ 1, 1 ⊗ t. We have two expansions: In this case the untwisted exponential operators are The 1 ⊗ S-twisted comultiplication is EXAMPLE 14.3. Now take H = H T , with as basis the difference operators {e n = ∆[n]} of (2.1) and as dual basis the sequences {e * n = ∆ * [n]} introduced in section 6. If φ ∈ C Z then φ is a function on the integers. The coproduct m * (φ ) and the twisted coproduct m * 1⊗S (φ ) are functions on the product Z × Z given by The untwisted and twisted exponential operators for H T using the difference operator basis are

H-Covariance of (Twisted) Exponentials
The domain H * and the range (H ⊗ H) * of the twisted comultiplications admit actions of H and H ⊗ H, and one can expect that for suitable A, B the twisted exponentials relate these actions.
We will write h 1 = h ⊗ 1 and h 2 = 1 ⊗ h and similar for other operators acting on tensor products.
PROOF. We have by the commutativity of H . On the other hand, introducing constants h j i ∈ C describing the action of h ∈ H on H * and H: we see that The proof of part (b) is the same. Similarly we find On the other hand Part (d) is proved similarly. In this situation Lemma 15.1 will still be true, with the same proof. We will frequently omit the subscript M on the exponentials in case the module they are acting on should be clear.

Multiplicativity of (Twisted) Exponentials
The domain H * and range (H ⊗H) * of twisted exponentials are algebras so one can expect, for suitable twistings A, B, that these algebra structures are related. We will need the following cases.
PROOF. Introduce structure constants C k i j ∈ C for the multiplication in H * and the comultiplication in H: Then, for instance, Similar computations gives the multiplicativity of the other exponentials.

H T -Leibniz Algebras and Exponentials
If H is a Hopf algebra, then an H-Leibniz algebra is a associative algebra in the category of H-modules. So V is both an H-module and an algebra, such that the unital algebra structure maps In case H = H D = C[D] a (commutative) H D -Leibniz algebra is just a (commutative) differential algebra, and the last equation of (17.1) is the Leibniz rule of differentiation of a product. For any cocommutative Hopf algebra the dual H * is a commutative H-Leibniz algebra. We will assume cocommutativity from now on, being interested in the case REMARK 17.1. We noted in Remark 15.2 that (twisted) exponentials act on any H-module and that they will have then the H-covariance properties of Lemma 15.1. In case V is a H T -Leibnitz algebra (twisted) exponentials will act on V , and in this situation furthermore the multiplicativity properties of Lemma 16.1 will still hold, with the same proof.

Inverses of exponentials.
The twisted exponential operators L S , R S are, in a sense, inverses of the untwisted ones. Indeed, recall that for φ ∈ C Z , i.e., if φ is a function on the integers, then the copoducts m * (φ ) and m * 1⊗S are functions on Z × Z given by (14.2). We can define functions on Z × Z × Z by m * (m * 1⊗S (φ ))(p, q, r) = φ (p − q + r), and restricting to the diagonal q = r we get φ back. This holds not only in C Z but is an identity for exponential operators acting on any H T -module. More precisely, we have the following Lemma.

LEMMA 18.1. Let V be an H T -module. Then
is the coproduct on H • dual to the product on H. Hence where µ S⊗1 is the twisted multiplication on H • , so using the connection i * (φ )1 = ∑ S(φ ′ )φ ′′ between counit, antipode and coproduct on a Hopf algebra, hence Similarly one proves the other parts.

Adjoint action of exponentials
Define for h ∈ H and X ∈ End(V ), for some H-module V , the adjoint action of h by The twisted coproduct π 1⊗S : h → h ′ ⊗ S(h ′′ ) is dual to the twisted multiplication µ 1⊗S on H * . Extend the adjoint action termwise to exponential operators: gives End(V ) the structure of H-Leibniz algebra. In particular

Multivariable expansions
Let H be a Hopf algebra, commutative and cocommutative for simplicity. If M is an H-module, and m ∈ M, then we get a distribution R M (m) on H and the twisted version R S M (m). We think of these distributions as expandable in a basis e * i of H * , and to emphasize this we write R M (e * )(m), R S M (e * )(m). We will also consider distributions on . In this section we discuss how the invertibility properties of exponential operators, see section 18, leads to identities between distributions in several variables. First we need some notation. If e * ∈ H * we write e * 12 = m * 1⊗S (e * ) = ∑ e * ′ ⊗ e * ′′ , e * 13 = ∑ e * ′ ⊗ 1 ⊗ e * ′′ and e * 23 = 1 ⊗ e * ,′ ⊗ e * ,′′ . We can consider exponential operators in these variables, for instance Then, by section 18 and Lemma 20.1, we get identities like .
PROOF. We have

Extension of Exponential Distributions
Let W, H be vector spaces. A W -valued distribution on H is just a linear map H → W . In this section we describe the following phenomenon. Assume that H is an algebra and let H ⊂ H ′ be an inclusion of H in a bigger algebra H ′ . Let {e i } be a basis for H, with e * i ∈ H * dual functions. Let M be an H ′ -module and consider for m ∈ M the action by the exponential for H on m: Now it can happen that the infinite series (21.1) makes sense, for all m ∈ M, as a distribution on the bigger algebra H ′ , in such a way that In this case we say that the distributions R H (m) (on H) can be extended to H ′ . This means that we can replace on M the exponential operator R H ′ by the simpler R H . The Then, for all m ∈ M, we have

PROOF. We have an inclusion of
so that c i,ℓ are the same coefficients that appear in (21.3). Then the proof of the lemma is a simple computation. For instance, . The rest is proved similarly.
More generally, on C pol Z we have an action ofĤ T , so we have a twisted exponential forĤ T on C pol Z : and we calculate the twisted coproduct on C pol Z by thinking of it as a subspace of the dual ofĤ T . For instance, in this way we get It is no longer true, as on C pol Z , that ∆ acts locally nilpotent on K Sing . We would like to write (21.5) but the infinite sum on the right does not make sense in K Sing . However, we can think of K Sing as consisting of distributions on C pol Z , via the embedding

Now we have for any distribution on
and (21.5) makes sense in the dual of C pol Z and similarly we can expand

LOCALIZATION OF (TWISTED) COPRODUCTS AND DIRAC DISTRIBUTIONS 33
as a power series in ∆. This is sufficient to check (21.3) and the Lemma 21.1 applies to K Sing . Now K = C pol Z ⊕ K Sing , so by combining with Example 21.2 we see that we can replace on K the exponential operators for H T or H T by the simpler ones for C[∆].
EXAMPLE 21.4. We can generalize Example 21.2 as follows. Let M be any H T -module that is locally finite, i.e., M decomposes into generalized eigenspaces for T and T −1 :

Localization of (Twisted) Coproducts and Dirac Distributions
We defined the twisted coproduct of φ ∈ H * as the distribution on H ⊗H given by . We are interested in localizations K of H * , and we would like to extend the definition of m * 1⊗S to K. We can not use (22.1), as φ ∈ K is in general a singular function on H, and not defined on all h ∈ H. Now we expressed the twisted coproduct m * 1⊗S on H * in terms of twisted exponentials L S , R S . These expressions make sense for F ∈ K, K being an H-module, but one finds that the two calculations give in this case in general a different result. So define for F ∈ K the obstruction of extending m * 1⊗S to F as the Dirac Distribution associated to F by In case F ∈ H * we have δ (F) = 0, of course. EXAMPLE 22.1. We continue the example 14.2. We consider the local- This is the usual formal delta distribution, usually denoted by δ (t 1 − t 2 ) or δ (t 1 ,t 2 ), see [Kac98].
EXAMPLE 22.2. We continue with Example 14.3 and consider the lo- .
and so we get 3 forms of the Dirac distribution δ ( 1

Difference Operators
Let H be a Hopf algebra and let V be vector space. Denote by V H the vector space of V -valued H-operators: elements of V H are finite sums of the form (23.1) In case H = H T we will call elements of V H T difference operators with values in V . In case V is an H T -Leibniz algebra, see section 17, V H T will be an algebra in the obvious way, but we will not need this at this point.
The following operations make sense for any space V H. First we have the adjoint map of V H given, for P = ∑ p i e i , by In case V happens to be an H-Leibniz algebra this is an anti-involution. Secondly there is the antipodal operator (23.3) Thirdly we will need later the map

Expansions of Distributions
Let W be a vector space. We denote by W n the space of all W -valued distributions on K ⊗n . For simplicity consider first D ∈ W 1 . It has a decomposition Then we have for all F ∈ K To make sense of this formula, let Then the trace extends toK. Now D(τ)F is an infinite sum of terms in W ⊗K, of which at most a finite number have a second component with nonvanishing trace. The trace then also extends then to such infinite sums.
One also sees that we can let F ∈K, and D(F) is well defined in that case. Now note that in the expression (24.1) for D Hol theĤ T form of the distribution ρ( 1 τ ) appears, see Example 22.2. So we can write D Hol in the form adapted to C[∆]: where we think of D Hol as a distribution on C[∆] ≃ S(C[∆]) 1 τ that can be extended to a distribution on all ofĤ T ≃ K Sing , as discussed in section 21. This means that we have a convenient uniform expression for the kernel of any distribution in one variable: using the notation (12.1). If F ∈ K it defines a rational distribution D F with F as kernel: This means that any F ∈ K (or rather its image in K * ) has a formal expansion For example we have the following infinite expansion of the rational function 1 τ 2 ∈ K Sing : 1 if we think of 1 τ 2 as a distribution (on C pol Z ). In the same way we have for D ∈ W 2 a kernel (24.6) with for all F, G ∈ K D(F ⊗ G) = Tr τ 1 Tr τ 2 (D(τ 1 , τ 2 )F(τ 1 )G(τ 2 )).
We will frequently identify a distribution with its kernel. We will also write and more generally we will use the notation a(τ i , τ j ) for a distribution de- Similarly any distribution D ∈ W n has an expansion in τ i , i = 1, . . ., n. We will write for this reason

Some Properties of Distributions
Distributions on K ⊗n can be multiplied by difference operators, via the action of difference operators on K.
using the adjoint operation on V H T , see (23.2). Here we extend the Cvalued distribution D on K ⊗ K to one on V ⊗ K ⊗ K or K ⊗V ⊗ K by linearity in the obvious way, to obtain a V -valued distribution. We can not multiply distributions except in special cases. Any distribution D ∈ W n can be multiplied by a rational distribution in n ways. For example, for n = 2, F, L, M ∈ K and D F the rational distribution (24.4) we have In general we would like to define the product of a distribution D in one variable with a two variable distribution E by However, for general D and E the products of kernels doesn't make sense. We have the special two variable distributions ρ( 1 τ ), λ ( 1 τ ) and δ ( 1 τ ). The claim now is that in case E = ρ, λ or δ the definition (25.1) does make sense for any D. This can be easily checked.
To investigate the properties of such products we need the notion of the trace of a distribution in several variable with respect to one of the variables.
For instance, let D(τ 1 , τ 2 ) be a W -valued distribution on K ⊗ K. We can produce distributions in a single variable from D by
We use this to give explicit formulas for the product of an arbitrary distribution D with ρ, λ , δ .
PROOF. Part (a) follows from the explicit forms for ρ, λ in Example 22.2, combined with the expansions of F and G (see (24.5))) and with the orthogonality relations of section 12. The part (b) follows from this. For 26. DISTRIBUTIONS OF THE FORM P 1 τ 39 part (c) we calculate for instance since by Lemma 25.1 we have identifying G with the rational distribution of which it is the kernel. The other identities follow similarly.

Distributions of the form P 1 τ
We will need, if P ∈ V H T , expressions of the form in case of opposite sign m > 0 > n or m < 0 < n. In case of the same sign (m, n ≥ 0 or m, n ≤ 0) we have

THE HOPF ALGEBRA H T AND SEQUENCES
We use this to note that for n = 0 we have Recall, see section 24, the notion of the singular part of a distribution D, We write Sing(D) = D Sing .
Then, using the coproduct (2.3), the factorization (5.5) and (26.4) we find In the same way one calculates (26.5) for P − = P n ∆[n] for n < 0. The proof of (26.6) is similar.
where W is the H T -module generated by 1 τ 1 PROOF. This is (26.5) in case P = T −a 1 1 τ T b .

Rationality of Distributions on K
Recall the notion of a rational distribution, see (24.4). We say that a distribution D on K has rational singularities in case the singular part D Sing is rational, i.e., there is K PROOF. If D has rational singularities with singular kernel K for all G ∈ C pol Z , since the argument of Tr is here nonsingular. Conversely, let such that F.D Sing = 0 (as a distribution on C pol Z ), for Now the action of C pol Z on singular distributions is given by τ(m + ℓ).
(27.1) By using the last equation of (27.1) we can write F.D Sing = 0 as a collection of recursion equations for the coefficients of D Sing : This leaves d 0 , d 1 , . . ., d N−1 arbitrary, and determines uniquely d p+N , for p ≥ 0. So the subspace of distributions that solve F.D = 0 is n dimensional. Now consider the rational element containing N = deg(F) arbitrary constants a (n,i) . In the summation (n, d) runs over the same set of pairs that occur in F. Then F.A = 0 mod C pol Z . We can expand A (or rather its image in K * ) as and the first N coefficients are determined by a system of equations of the form

Rationality of Multivariable Distributions
We define PROOF. Let X be the left hand side in (28.2). We have which is zero unless s = 0.
Recall the formal expansion of f ∈ K, see (24.5). Similarly we will often consider distributions that have a formal expansion The coefficients of such an expansion are traces by Lemma 28.1: .
Recall from section 8 that the singular part K Sing is generated by 1 τ over then, by Lemma 15.1, Recall also the notion of an element σ ∈ V [[τ ±1 ]] having rational singularities, see section 13. Similarly we say that a distribution a(τ 1 , The proof of this Lemma is similar to that of Lemma 27.1. Note that the expansion (28.3) of a rational a(τ 1 , τ 2 ) will in general be infinite.  In section 3 we will define classical fields as generating series of functions on V (L), for the case H = H T . The rest of this chapter develops then the theory of (singular) Poisson brackets between classical fields.

Spectrum and Singular Points
Let V be a commutative H-Leibniz algebra. The H-action on V induces an H-action on V * and by duality we obtain a coaction in an infinite sum of elements of V ⊗ H * T . As we discussed in section 14 we can calculate m * V using the exponential operator R introduced there: introducing a basis {e i } for H and functions e * i such that e * i (e j ) = δ i j we have m *

Then m *
V is a multiplicative map (see Lemma 16.1) and satisfies H-covariance (see Lemma 15.1). Since we assume V to be a commutative algebra we can define its spectrum Specm(V ): a point σ ∈ Specm(V ) is a C-algebra homomorphism σ : V → C. Given such σ we construct, using the exponential Indeed, this is an algebra homomorphism: Similarly, using part (a) of Lemma 15.1, we see thatσ is an H-module morphism: Hence as sets we find an isomorphism and we refer to V (H * ) as the set of nonsingular points of the (variety associated to the) H-Leibniz algebra V .

Classical Fields
From now on we restrict ourselves for convenience to the case H = H T and consider the localization K = M −1 C pol Z described in section 7. Let V be a commutative H T -Leibniz algebra and let V (K) be the set of H T -Leibniz algebra homomorphismsσ : V → K. (By taking the non singular part ofσ we get a mapσ Hol : V → H * T , i.e.,σ Hol corresponds to a point of Specm(V ).) For f ∈ V we we define the classical field C( f ) as the distribution on K with values in the space of functions V (K) → C, defined tautologically by: For eachσ we get a rational distribution C( f )(σ) on K (with values in C). Rational distributions can be multiplied, see Section 25. This allows us to define the (commutative, associative) product C( f ).C(g) of classical fields associated to f , g ∈ V by C( f ).C(g)(σ) = C( f )(σ)C(g)(σ).
Also we have an H T action on distributions and this turns the space V (τ) of classical fields (spanned by C( f ), f ∈ V ) into a commutative H T -Leibniz algebra.

Expansions of Classical Fields
Any distribution has an expansion (24.3). In the case of classical fields this becomes: .
We will write f (τ i ) for the expansion of C( f ) in the variable τ i .
. This mean that we can identify the holomorphic part of the classical field C( f ) with the distribution In particular the constant term of a classical fields C( f ) = f (τ) can be identified with the element f ∈ V :

Classical Fields and Affinization
If M is an H T -module, its affinization is the H T -module LM = M ⊗ K,where the tensor product is given the H T -module structure by using the coproduct: for h ∈ H T , m ∈ M and k ∈ K, Recall the counit ε : H T → C. This is an algebra homomorphism, so that m = ε −1 (0) ⊂ H T is an ideal (called the augmentation ideal ); it is the ideal generated by ∆, or equivalently the ideal generated by ∆. Define for any H T -module M the module of coinvariants So M m is the quotient by total differences. Note that M m is a trivial H Tmodule. Therefore we will denote the canonical projection by Often we will suppress the subscript M and write just Tr. Denote by L M the quotient LM/mLM. Elements of L M will be written as We have an action ofĤ T on L M: if h ∈Ĥ T we put

In this section we consider
.
Suppose that f ⊗ p ∈ mLV , so that Then, for anyσ ∈ V (K), Hence we see that in fact any representative in LV of the element f {p} ∈ L V will define the same function on V (K). Also we note that f {τ(n)} defines the same function on V (K) as the coefficient f (n) of the classical field C( f ), see (4.1). More generally f {p} defines the same function as the value C( f )(p) of the classical field, where we think of the classical filed as a distribution on K with values in functions on V (K). We will from now on identify the coefficients of C( f ) with elements of L (V ) so that classical fields will be generating sequences for elements of L V .

Derivations and Poisson Structures
Let There is an A-module Ω 1 (A) together with a universal derivation d : A → Ω 1 (A), such that for any X ∈ Der(A, M) there exists a unique A-module morphism γ X : Ω 1 (A) → M making the following diagram commute: The module of differentials Ω 1 (A) is generated by elements da, a ∈ A, subject to the relations d(ab) = adb + bda, for a, b ∈ A. There is a pairing In this case we call the bracket of Γ the Poisson bracket; (A, {}) is then a Lie algebra. The bracket factors through the differentials: In case A is a polynomial algebra we get simple explicit formulae for the Poisson bracket in terms of generators: so that the Poisson bracket is uniquely determined by the Poisson bracket between the generators.

Evolutionary Derivations and Singular Poisson Brackets
Let V be an H T -Leibniz algebra. The module of differentials Ω 1 (V ) for any algebra V a V -module, but it is in this case in fact a V H T -module, via We call, for M an V H T -module, a derivation X : In this case the V -module morphism γ X : Ω 1 (V ) → M of (7.1) is in fact a morphism of V H T -modules. Now we would like to generalize the notion of a Hamiltonian map to the case of evolutionary derivations. An obstacle is that the space Der ev (V,V ) of evolutionary derivations (from V to itself) is not (in an obvious way) a module over V or V H T : if X is evolutionary then as h ∈ H, f ∈ V do not commute in V H T , in general. In case V is free as H T -module there is a V -module structure on Der(V,V ), using a basis for V , see section (10.1) below.
Therefore we consider instead the space D 2 = Der ev (V,V (τ 1 , τ 2 )) of evolutionary derivations from V to the module V (τ 1 , τ 2 ) of two variable classical fields introduced in section 6. So an element X ∈ D 2 acts on f ∈ V by and satisfies In other words, we use the 1 ⊗V H T -module structure on the target to define the words evolutionary and derivation. Now note that we can define a V H T module structure on D 2 by using the other V H T module structure on the target: if X is an evolutionary derivation then define for P ∈ V H T P.
The two actions of V H T on V (τ 1 , τ 2 ) commute: from which we see that P.X is again evolutionary: Any X ∈ D 2 factors through Ω 1 (V ): is a morphism of V H T -modules (for the 1 ⊗ V H T structure on the target). Instead of considering derivations with target V (τ 1 , τ 2 ) we could use distributions in any pair of variables as target. In this case we will write We can extend any X ∈ Der ev (V,V (τ i , τ k )) to a map Then X i jk is an evolutionary derivation, for the (V H T ) k -module structures on V (τ j , τ k ) and V (τ i , τ j , τ k ). Now we will define the evolutionary analog of the Hamiltonian maps introduced in section 7. Consider a map Γ : V → Der ev (V,V (τ 1 , τ 2 )), DEFINITION 8.1. We call Γ a singular Hamiltonian map in case (a) Γ is an evolutionary derivation, Here we extend in (c) the derivation { f (τ 1 ), ·} = Γ( f ) from a map on V to a map Γ( f ) 123 : V (τ 2 , τ 3 ) → V (τ 1 , τ 2 , τ 3 ), as discussed above, and similar for the other terms.
We call, in case Γ is a singular Hamiltonian map for a H T -Leibniz algebra V , the pair (V, Γ) an H T -Poisson algebra.
The singular Poisson bracket { f (τ 1 ), g(τ 2 )} is an evolutionary derivation in both f and g and so factors through Ω 1 (V ) ⊗2 . We can write τ 2 ) is the canonical pairing, and γ Γ is the unique V H T -module morphism Ω 1 (V ) → D 2 making the diagram 9. Projection from V (τ 1 , τ 2 ) to V Let F(τ 1 , τ 2 ) ∈ V (τ 1 , τ 2 ) be a two variable classical field, see section 6. It can be can be uniquely expanded as

see (28.3). Define a map
see Lemma 28.1 and Section 4.
Here we write for instance P i for the classical field valued difference operator P i = ∑ n∈Z p n (τ i )∆ i [n], for i = 1, 2.

. It suffices to check part (a) in case P = p∆[m], Q = q∆[n], and in this case we have
Then part (b) follows from (a): we have LEMMA 9.2. Let X be an evolutionary derivation X : V → V (τ 1 , τ 2 ). Then the composition is an evolutionary derivation of V .

PROOF.
We have for f , g ∈ V by definition of evolutionary derivation with values in V (τ 1 τ 2 ), see (8.1), using Part a of Lemma 9.1 with P = f , Q = 1. In case h ∈ H T similarly with P = h and Q = 1 So we obtain a map (9.4) Π ev : Der ev (V,V (τ 1 , τ 2 )) → Der ev (V,V ), X → X 0 . Now let Γ : V → Der ev (V,V (τ 1 , τ 2 ) be a singular Hamiltonian structure, see section 8. Recall that if P ∈ V H T then Γ(P. f )(g) = P 1 Γ( f )(g). We define a map by composition with Π ev : Recall that Γ factors through the differentials: if f ∈ V then by (9.1). This means that Γ 0 factors through, Ω 1 (V ) m , the module of coinvariants. Then we obtain the following commutative diagram For V a free H T -Leibniz algebra B : Ω 1 (V ) m → Der ev (V,V ) is a matrix of difference operators, i.e., an matrix with coefficients in V H T , as we will explain in the next section.

Free Difference Algebras
Let V be the free difference algebra generated by v α , α ∈ A = {1, 2, . . ., k}. So V is the polynomial algebra where v (0) α = v α and the action of H T is determined by together with linearity and the Leibniz rule for H T . In this case Der ev (V,V ) and Ω 1 (V ) m are free V -modules of rank k = |A |, which simplifies the discription of singular Hamiltonian structures on V .
Any X ∈ Der ev (V,V ) is uniquely determined by its values X α = X (v α ) on the generators of V . Introduce notation Define an action of V on evolutionary derivations by Then Der ev (V,V ) is a free V -module: In particular for f ∈ V we can express the differential in terms of difference operators: where the d α f are difference operators Inside We have a pairing (10.1) It is known (see [Kup85], chapter 2) that Recall the module of coinvariants Ω 1 (V ) m = Ω 1 (V )/mΩ 1 (V ). We have for all P ∈ V H T Pdv α ≡ P V dv α mod mΩ 1 (V ), (where P V is defined in (23.4)). This implies, combined with (10.2), that So we have a splitting of the exact sequence where the splitting map s : In particular we obtain a variational differential δ given by where the variational derivative is The variational derivative δ f is the unique element of Ω 1 (V ) 0 such that for all X ∈ Der ev X ( f ) ≡ X , δ f mod mV, using the pairing (10.1). Indeed, if δ = ∑ δ α dv α ∈ Ω 1 (V ) 0 also satisfies this then we have for all X ∈ Der ev X , δ f − δ ≡ 0 mod mV , or for each α but it is known that this forces δ f δ v α = δ α , see [Kup85], Lemma 17. The module of coinvariants is a free V -module: We can therefore represent an element ω = ∑ α ω α dv α of Ω 1 (V ) m as a column vector ω = In our case B is a map between free rank k V -modules, and we will now see that B is given by an k × k matrix of difference operators. .
by Lemma 9.1 and (10.3). So so that B is the matrix of the map B.

The Finite Toda Lattice
Recall, see [Tod89], that the finite non periodic Toda lattice with N particles is a Hamiltonian system on a phase space with global coordinates p i , q i for i = 1, . . ., N and Hamiltonian

So the equations of motion arė
Introduce new variables B i = p i , i = 1, . . . , N and C j = exp(q j−1 − q j ), j = 2, . . ., N. Then we obtain the following system in the new variables: Then the system (1.1) is equivalent to the Lax equation where L − is the strictly lower triangular part of L. We also have for the Hamiltonian Now the system (1.1) make still sense if we let i, j run over all integers, and we can introduce a doubly infinite Lax matrix such that this infinite system is stil given by (1.3). Now, however, the meaning of (1.4) becomes less clear (but see [Ike94]). Kuperschmidt, for instance, advocates in [Kup85] an other way of dealing with infinite systems, where one replaces the infinite Lax matrix by a difference operator. We follow in this chapter Kuperschmidt's approach, expressed in the formalism of singular Hamiltonian maps, and also indicating the connection with the formalism of r-matrices.

Lax Operator and r-Matrix
Consider the difference operator ("Lax Operator") where V is the H T -Leibniz algebra freely generated by A, B,C. So V is the polynomial algebra We will also write the Lax operator as Any evolutionary derivation X of V defines a time-flow on the Lax operator byL Conversely we will use L to define singular Hamiltonian structures on V .
We have a map, if P = ∑ α p α T α , For all P, Q ∈ V H T we have Indeed, it suffices to check this for P = pT α , Q = qT −α , for α ∈ Z, p, q ∈ V . But in that case Recall the pairing (10.1). In our situation we have encoded the evolutionary derivation X in the difference operator X L . Similarly we can encode an element ω = ∑ ω α dv α ∈ Ω 1 (V ) 0 in a difference operator ω L = ∑ T −α ω α . Then we have Note that the difference operator δ L f is not uniquely determined by this condition: we are free to add to δ L f any difference operator of the form δ ≥2 + δ ≤−2 , where δ ≥2 = ∑ α≥2 T α δ α and δ ≤−2 = ∑ α≤−2 T α δ α , since δ ≥2 and δ ≤−2 are orthogonal to X L with respect to Sp. We have projections and similar for the other projections. We use Sp also to define the dual of a map σ : Then, for instance, we have Then ρ is not skew-adjoint, but we have ρ = S + A, where A = Π >0 − Π <0 is skew and S = Π 0 is symmetric, so that ρ * = S − A.
Besides the Lax operator L (with coefficients in V ) we also have the Lax operator L(τ i ) with coefficients classical fields depending on τ i : i . Define distributions associated to the r-matrix ρ and its adjoint ρ * : We define commutators of the (field-valued) Lax operator with these distributions: If h ∈ H T , x ∈ V , we write h x for the action of h on x, and hx for the multiplication in V H T . For instance, T k x = T k x T k .
We will write all expressions in the form ∑ α,β ℓ α,β T α 1 T β 2 , i.e., with the powers of T to the right. Consider the coefficient of by the property (6.1) of the action of V H T ⊗V H T on two variable classical fields. In the same manner one checks that the coefficients of for k > 0 all vanish. There remain only 2 nonzero coefficients. For T 0 1 T 1 2 the coefficient is Given a singular Hamiltonian structure on V we define the Poisson bracket of the field Lax operator L(τ) = ∑ α v α (τ)T α by Conversely we define on V a singular Hamiltonian structure by putting By Lemma 2.1 this means that we have specified the Poisson brackets of the generators of V to be and all other Poisson brackets are zero, in particular {A(τ 1 ), F(τ 2 )} = 0, for all F ∈ V . Then one checks that this can be extended consistently to a Poisson brackets on all of V , essentially since The projection (9.4) of Γ( f ) ∈ Der ev (V,V (τ 1 , τ 2 ) gives an evolutionary derivation X ( f ) ∈ Der ev (V,V ) which reads in column form In terms of the Lax operator we have We can express this in terms of the difference operator δ L f and the classical r-matrix.
(2.6) by (2.3). Therefore we can write (2.6) as a sum of two terms (involving r and r * respectively). The first term is (suppressing the summation over the repeated indices here and below) (2.7) Now recall that the projection Π from section 9 satisfies for all difference operators P, Q the relation Π (P 1 Q 2 δ (τ 1 , τ 2 )) = Q(P V ) and that we have . Therefore, by applying Π to the coefficients of (2.7) we get as first contribution to Next consider Taking the coefficient of T β 1 of the last expression we see that the second term of (2.6) is . Now the r-matrix satisfies ρ(T k 1 )δ (τ 1 , τ 2 ) = ρ * (T −k 2 )δ (τ 1 , τ 2 ), so that we can rewrite (2.8) as Applying Π to the coefficients we get as second contribution to X f (L) (again writing T for T 2 ) Recall that in (2.2) the difference operator δ L f was determined up to addition of terms δ ≥2 and δ ≤−2 . We have the same freedom in the vector field X f (L): we have so that X f (L) is independent of addition of these terms.
Next consider the Hamiltonians We have for any X ∈ Der ev (V,V ) so that we can represent the differential δ L H n of H n by the difference operator L n−1 /2. Therefore the Hamiltonian vector field of H n is given by the Lax-equation We want now clarify the motivation for the terminology Tr for the maps K → C, (11.1) and LV → L V , see section 5. In the finite Toda lattice the variables q i , p i or B i ,C i give local functions on the phase space (attached to a site i of the finite lattice, roughly speaking), and the Hamiltonians, global functions on phase space, are obtained as traces over the powers of the Lax matrix (1.2). Similarly, for the infinite Toda lattice the elements of V , difference polynomials in B,C, define densities of functions on the phase space V (K), and to obtain honest functions one has to take the trace.

Introduction
In chapter 3 we introduced the notion of a singular Hamiltonian structure on an H T -Leibniz algebra V ; this was given by an evolutionary derivation Γ : V → Der ev (V,V (τ 1 , τ 2 )) satisfying the conditions of Definition 8.1. By projecting using the map Π : V (τ 1 , τ 2 ) → V , see section 9, we obtain for all f ∈ V an evolutionary derivation Π In this chapter we will see that X f is just one of an infinite collection of products labeled by h * ∈ H * T : In this notation X f = f {1} , where 1 = 1 H * T , the identity of the algebra H * T = C Z , see section 3. The collection of all conformal products f {h * } g for all h * ∈ H * T will give V the structure of what we will call an H T -Conformal Algebra. These are analogs of the conformal algebras from the usual theory of vertex algebras, see [Kac98], cf., also [Li] where they are called vertex Lie algebras.
We will define for any H T -conformal algebra C a singular vertex operator Y Sing ( f , τ) for all f ∈ C, see Section 4 below. This is a generating series for the conformal products f {h * } and we will formulate the axioms of an H T -conformal algebra both in terms of the components f {h * } and also in terms of the singular vertex operator.
The singular vertex operators will be generalized in the next Chapter 6 to the vertex operators of an H T -vertex algebra.

H T -Conformal Algebras and H T -Vertex Poisson Algebras.
If Γ is a singular Hamiltonian map for an H T -Leibniz algebra V we obtain for f , g ∈ V a distribution Γ( f )(g) ∈ V (τ 1 , τ 2 ) that we can uniquely write as where The properties of the Poisson brackets { f (τ 1 ), g(τ 2 )} = Γ( f )g of a singular Hamiltonian structure can be expressed in terms of the difference operators D( f ⊗ g). The fact that Γ is evolutionary implies that and the skew symmetry translates to Making the substitution q + r = n, r = p we obtain so that by the Lie property of Γ we find We now observe that the operations f n (which are derivations of the commutative algebra structure of V ) with the properties (a)-(d) of Lemma 2.1 make sense also if V is just an H T -module, not an H T -Leibniz algebra. So we define:

DEFINITION 2.2. An H T -Conformal algebra is an H T -module C with operations
n : C ⊗C → C, n ∈ Z, satisfying (a) (Finiteness) For all f , g ∈ C f n g = 0 for all but a finite number of n ∈ Z.
As a first result we have the Leibniz rule for conformal products.
LEMMA 2.3. Let C be an H T -conformal algebra, and f , g ∈ C. If h ∈ H T has coproduct ∑ h ′ ⊗ h ′′ then for all n ∈ Z we have PROOF. By H T -covariance we have f n g = (T n f ) 0 g so it suffices to check (2.6) in case n = 0. Also, it suffices to check for h = T p , p ∈ Z, in which case the coproduct is just T p ⊗ T p . Then using skew-symmetry and H T -covariance repeatedly.
EXAMPLE 2.4. Let CToda be the free H T -module generated by two elements B and C. We define products x n y for x, y ∈ CToda, n ∈ Z by putting on generators (2.7) B n B = C n C = 0, n ∈ Z, and all other products B n C and C n B are zero. Since the fundamental Poisson brackets (2.4) define a singular Hamiltonian structure on V = C[B (n) ,C (n) ] the products (2.7), (2.8) and (2.9) define an H T -conformal structure on CToda, called the first Toda conformal structure on CToda, cf., [Kup85]. We plan to discuss higher Toda conformal structures in a future paper.
Note that CToda is a finite rank H T -conformal subalgebra of V and in a sense V is the symmetric H T -conformal algebra of CToda. Also note that CToda is a non trivial extension: if we let CC, CB be the rank 1 free Abelian H T -conformal algebras generated C, resp B then we have an exact sequence 0 → CC → CToda → CB → 0. EXAMPLE 2.5. Let g be a Lie algebra, and let Cg be the free H T -module generated by g. Then define, for x, y ∈ g and p, q, n ∈ Z products Then one checks that this gives Cg the structure of H T -conformal algebra, the commutator axiom follows from the Jacobi identity for g. We call Cg the affine H T -conformal algebra of the Lie algebra g.
The notion of a singular Hamiltonian structure on an H T -Leibniz algebra gives rise to a special class of H T -conformal algebras.  (τ 1 , τ 2 )) by One then checks that Γ is in fact a singular Hamiltonian map.

The Lie Algebra of a Conformal Algebra
To obtain from a singular Hamiltonian structure Γ on V a conformal algebra structure we used the expansion of the difference operators D( f ⊗g) in the basis {T n } of H T . The operations n : V ⊗ V → V related to the difference basis, see (2.4), satisfy properties similar to those of n, but they are more complicated. We will not need explicitly the operations n in the sequel, except for the zeroeth one, which is given by In the notation of (2.5) we have Recall the augmentation ideal m ⊂ H T and the submodule mM of total differences of an H T -module M, see Section 5, LEMMA 3.1. In any H T -conformal algebra V the operation 0 given by (3.1) satisfies PROOF. We have by H T covariance By skew-symmetry and H T -covariance we get Using the commutator we find Recall the trace map Tr : M → M/mM, see section 5, and writef for the image Tr( f ) in C m of f ∈ C, see (5.2). Then the lemma shows that in case C is an H T -conformal algebra the bilinear map is well defined, and defines a Lie algebra structure on C m .

Singular Vertex operators
In this section we reformulate the axioms of an H T -conformal algebra C, see Definition 2.2, in terms of the generating series of the conformal products of f ∈ C. Define the singular vertex operator of f by LEMMA 4.1. Let C be an H T -conformal algebra. The singular vertex operator for f ∈ C satisfies (a) (Finiteness) Y Sing ( f , τ)g belongs to CH T 1 τ for all g ∈ C.

PROOF. Finiteness is clear. For H T -covariance note that
For skew-symmetry we use (2.3) to calculate: using Lemma 26.1, where where P a,b = T −a 1 τ 1 T b . Then by Corollary 26.2 we have Of course, we can reverse our procedure, and define an H T -conformal algebra as an H T -module C together with for all f ∈ C a singular vertex operator Y Sing ( f , τ), satisfying the properties of Lemma 4.1. Expanding the singular vertex operator we find that the components f n satisfy the properties of Definition 2.2.

Holomorphic H T -vertex algebras
Let V be a commutative H T -Leibniz algebra. For f ∈ V define the holomorphic vertex operator by where R is the untwisted exponential operator for H T introduced in section 14, see (14.3), so that Y Hol is just a suggestive notation for the coaction map m * V : V → V⊗H * T introduced in section 2. In particular Y Hol is multiplicative, see section 16. We define By Lemma 18.1 we have It then follows that we have Skew-symmetry: The H T -covariance of exponential operators, see section 15, implies that

. A holomorphic H T -vertex algebra is an H T -module, together with a distinguished element 1 V ∈ V and a map
It is clear that any commutative H T -Leibniz algebra V is a holomorphic H T -vertex algebra, via the choice (5.1) of holomorphic vertex operator. Conversely, if V is an holomorphic H T -vertex algebra, then we define a product on V by evaluating at zero: Then one checks that this gives V the structure of a commutative H T -Leibniz algebra.

Extension of H T -conformal Structure to the Affinization
Recall the notion of the affinization LM = M ⊗ K of an H T -module M and the canonical trace map Tr: LM → L M = LM/mLM, see Section 5. In this section we take M = C, where C is an H T -conformal algebra, and show that the affinization LC inherits an H T -conformal structure from the H T -conformal structure of C and the holomorphic vertex algebra structure of K. In particular this gives L C a Lie algebra structure, by Section 3. In case C = V is an H T -vertex Poisson algebra, see Section 2, the Lie bracket on L V is the Poisson bracket of the components of the classical fields of Chapter 3, see Example 6.4.
For f ∈ C we have the singular vertex operator For p ∈ K we have the holomorphic vertex operator Y Hol (p), see (5.1), and by combining Y Sing on C with Y Hol on K we define a singular vertex operator on LC: Here⊗ is a combination of tensor product and multiplication: we have explicitly Now by (26.2) we have a vanishing contribution to (6.1), and hence to Y LC Sing , from the terms in which m, n have opposite sign. In case 0 ≤ m < n is nonsingular, so these terms will also not contribute to Y LC Sing . Finally, by (26.3) and the factorization (4.7) we have, if m = n + s, n, s ≥ 0 LEMMA 6.1. Y LC Sing gives LC the structure of H T -conformal algebra. PROOF. We need to check the properties of Lemma 4.1. Finiteness is manifest from (6.2), as in any H T -conformal algebra f m g = 0 for all but a finite number of m ∈ Z.
For h ∈ H T we have, if π(h) = ∑ h ′ ⊗ h ′′ is the coproduct, using the H T -covariance of Y Sing and Y Hol ,

This proves H T -covariance for Y LC
Sing . For skew-symmetry we need the following simple fact about the projection Sing, cf., [Li]: let X ,Y be vector spaces and let A ∈ X ⊗ K and B ∈ Y ⊗ H T , then Then we have τ)g⊗Y Hol (p, τ)q , and by skewsymmetry in the conformal algebra C and the holomorphic vertex algebra K: and by multiplicativity of exponentials, Section 16,: proving skew-symmetry for Y Sing LC . To check commutator for LC we first note that it follows from Lemma 20.1 that Here p, q ∈ K are rational functions of τ, say. Then R K (τ 1 )p is rational in τ and holomorphic in τ 1 , and R S K τ 1 is the exponential operator acting on rational functions of τ 1 . We will use similar notation below without further elaboration.
Then we calculate The Lie bracket of L C is induced by the 0 bracket on LC. This is the coefficient of ∑ n∈Z T n 1 τ in (6.2), hence The Lie bracket on L C is then given by Define the generating series of elements of L C ("currents") for f ∈ C by (6.5) The current commutator is given by (6.6) EXAMPLE 6.2. Let Cg be the H T -conformal algebra of the Lie algebra g, see Example 2.5. Then let L g = L Cg be the corresponding Lie algebra. If {X α } is a basis for g, then L g is spanned by elements The corresponding currents have commutator Of course this looks like (2.4) and in the next example we explain the connection.
EXAMPLE 6.4. Let V be an H T -vertex Poisson algebra. This is in particular an H T -conformal algebra, and so we obtain a Lie algebra L V . Elements of L V are of the form f {p} , for f ∈ V and p ∈ K. The commutator of two such elements is given by The corresponding current is given by (6.6).
Recall form Section 5 that the coefficients of the classical field C( f ), for f ∈ V , can be identified with elements of L V : In fact we can let F ∈K, see (24.2), for instance are fields on V , they are in particular End(V )-valued distributions, and we can calculate their commutator distribution: this is the distribution on K ⊗ K with value at F ⊗ G If we have an expansion (infinite, in general) as in (28.3), then the coefficients c ℓ (τ) are also fields, by (28.4).
We say that fields are mutually rational if the distribution [ f (τ 1 ), g(τ 2 )] has rational singularities, i.e., is killed by multiplication by some m * 1⊗S (F), for F ∈ C pol Z . By Lemma 28.2 this implies that we have a finite sum whereD( f ⊗ g) is a W -valued differential difference operator (i.e., an element of WĤ T ) where W = End(V )[[τ ±1 ]] and the right hand side sum is finite with the c n,k (τ) End(V )-valued distributions (in fact fields) on K.

Normal Ordered Products and Dong's Lemma
In this section we collect some properties of fields on a vector space V that we will later need to investigate H T -vertex algebras (see Definition 5.1 below) and construct examples.
LEMMA 3.1. Let f (τ), g(τ) be fields on V . Then also are fields on V .
PROOF. By the product formula (4.8) we see that f Hol (τ)g Hol (τ) is a well defined element of End(V )[[τ ±1 ]]. The singular part of this distribution is zero so this product is a field for trivial reasons. Now for any v ∈ V the V -valued distribution g Sing (τ)v is rational, and we can multiply it with any End(V )-valued distribution, see section 24. So both f Hol (τ)g Sing (τ)v and f Sing (τ)g Sing (τ)v are well defined V -valued distributions on K. Their singular part is clearly rational. REMARK 3.2. In general the product f Sing g Hol is not well defined.
We use these results on products of holomorphic and singular parts to define normal ordered products. DEFINITION 3.3. Let f (τ) and g(τ) be fields on V, and F ∈ K. The F-th normal ordered product of f (τ) and g(τ) is We define in particular the normal ordered product, using Lemma 25.1 and the definitions of ρ( 1 τ ) and λ ( 1 τ ), (22.2), as We also refer to f (τ 2 ) [h] g(τ 2 ) as the hth normal ordered product of f and g.
In case F ∈ C pol Z the F-th normal ordered product is a component of the commutator. Indeed for F ∈ C pol Z we have R S (τ 1 )(F(τ 2 )) = L S (τ 2 )(F(τ 1 )) = m * 1⊗S (F), so that in this case We have in particular, see Lemma 28.1, To get the components of the finite expansion (2.7) in case f (τ 1 ) and g(τ 2 ) are mutually rational we need to take F inĈ pol Z , see (24.2): we have (3.7) c n,k (τ) = f (τ) {e * n,k } g(τ). By Lemma 3.1 the hth normal ordered product of fields is again a field, for all h ∈Ĥ T . For F ∈Ĉ pol Z the F-th normal ordered products are components of commutators of fields so also fields. LEMMA 3.4. Let V be anĤ T -module and f (τ), g(τ) fields on V , F ∈ K and h ∈Ĥ T with coproduct ∑ h ′ ⊗ h ′′ . Then LEMMA 3.5. (Dong's Lemma) Let a(τ), b(τ), c(τ) be fields on V that are mutually rational. Then also the Fth normal ordered product a(τ) {F} b(τ) and c(τ) are mutually rational, for all F ∈K.
We first need a simple Lemma on polynomials.
i.e., such that K is independent of y.
We will assume that also q 0 , r 0 belong to I, but for the rest they are arbitrary. Then assume we have found q j (x), r j (x) ∈ I such that the coefficient of v j in (3.8) vanishes, for j = 1, 2, . . ., s − 1. Then the v s term reads By assumption the double sum belongs to I, so that we can find q s , r s making (3.9) vanish. By multiplying the q j , r j , j < s, by an element of I, if necessary, we can assume q s , r s ∈ I, too, and the lemma follows by induction.
By definition of rationality there are f , g, h ∈ C pol Z such that (3.10) We need to find k ∈ C pol Z such that (3.11) It suffices to show that Indeed, taking Tr τ 1 of (3.12) gives (3.11). Now note that there is a G ∈ C pol Z such that FG ∈Ĉ pol Z and in that case we have R S (τ 2 )(F(τ 1 )G(τ 1 )) = L S (τ 1 )(F(τ 2 )G(τ 2 )) = m * 1⊗S (FG) = (FG) 12 . This implies that Also note that (3.14) h 13 g 23 A = h 13 g 23 B.

State-Field Correspondence and Vacuum Axioms
Let V be a vector space. A State-Field correspondence for V is a map that associates to all f ∈ V a field Y ( f , τ) on V , called the vertex operator of f . We also write f (τ) for Y ( f , τ). Now let V contain a distinguished vector 1 V , called the vacuum of V . We say that a state-field correspondence satisfies the Vacuum Axioms (with respect to 1 V ) in case with the constant term (see (2.5)) We fix from now on a state-field correspondence satisfying the vacuum axioms for some 1 V ∈ V . Let f ∈ V and define for all h ∈Ĥ T an operator At this point we don't know that, given a state-field correspondence, h → h V gives anĤ T -module structure to V . This is an extra assumption about the state-field correspondence.

Definition of H T -Vertex Algebras
We will need the notion of an antipodal vertex operator Y S ( f , τ) associated to f ∈ V . More generally if D is a distribution on K we put (5.1) D S (F) = D(S(F)), using the antipodal homomorphism on K given by S(τ) = −τ.
DEFINITION 5.1. An H T -vertex algebra is a vector space V with a vacuum vector 1 V ∈ V and a state-field correspondence f → f (τ) = Y ( f , τ), satisfying the vacuum axioms (4.1), (4.2) and such that furthermore • (Skew-Symmetry) • (Mutual Rationality) The vertex operators Y ( f , τ 1 ) and Y (g, τ 2 ) are for all f , g ∈ V mutually rational.
Note that from (5.2) it follows that the map h ∈ H T → h V ∈ End(V ) gives V the structure of H T -module.
The simplest examples of H T -vertex algebras are of course the holomorphic vertex algebras introduced in section 5, corresponding to commutative H T -Leibniz algebras, with as vertex operator Y ( f , τ) = Y Hol ( f , τ) = R V (τ 1 ) f . REMARK 5.2. We define similarly the notion ofĤ T -vertex algebra, by imposing instead of (5.2) aĤ T -covariance axiom. It seems that the non holomorphic H T -vertex algebras are all in factĤ T -vertex algebras.

First Properties of H T -vertex Algebras
From now on V is an H T -vertex algebra, with vacuum 1 V . LEMMA 6.1. (

PROOF. From the vacuum axioms and the definition (2.4) it follows that
Hence by definition (4.3) we find First some general remarks. The H T -covariance of the antipodal vertex operators is For the exponential operator R V (τ) (acting on a space of V -valued distributions, say), we have the following identities: by elementary identities in the Hopf algebra H T and (6.3).
As Y ( f , τ) is an End(V )-valued distribution on K, we get for all F ∈ K (in fact ∈K) a linear map We call f {F} g the F-product of f , g. Then Lemma 6.2 implies the Leibniz rule for F-products: Another reformulation of the Leibniz rule is the following ad-covariance of vertex operators. , τ)). PROOF. For any g ∈ V we have by H T -covariance and elementary identities PROOF. By taking τ = 0 in the holomorphic part of Lemma 6.3 we find

Uniqueness and Normal Ordered Products
THEOREM 7.1. (Goddard's Uniqueness) Let G(τ) be a field on an H Tvertex algebra that is mutually rational to all vertex operators Y ( f , τ) of V and such that G(τ)1 V = R V (τ)g, for some g ∈ V . Then G(τ) = Y (g, τ).
The proof is the same as in the usual case, using the vacuum axioms, skew symmetry and rationality, cf., [Kac98], Thm. 4.4. Now we will express the vertex operators of an H T -vertex algebra in terms of normal ordered products, see Definition 3.3. LEMMA 7.2. For f , g ∈ V and F ∈K we have More generally, if f 1 , f 2 . . . , f k ∈ V and F 1 , F 2 , . . ., F k ∈K, then We define the normal ordered product of more factors as usually from right to left: g(τ)) . . .) PROOF. We have, by the vacuum axioms, Lemma 6.1 and Lemma 6.4 so that the first part of the Lemma follows from Goddard's uniqueness. The second part follows by induction.
Note that the first sum on the RHS in the Lemma will be finite, the second infinite, in general.
PROOF. The first equality follows from the mutual rationality of vertex operators by combining (2.7), (3.7) and the previous Lemma 7.2. The second equality follows from (3.6) combined with Lemma 7.2.

Alternative Axiomatization
We took as one of the axioms of an H T -vertex algebra the H T -covariance axiom Y (h V f , τ) = h K .Y ( f , τ), and derived in Lemma 6.3 the ad-covariance τ). This gives a set of axioms similar to those for an H T -vertex Poisson algebra, see Definition 2.6, and of an H T -conformal algebra, see Definition 2.2. However, these axioms are not very efficient, in particular we assumed the skew-symmetry (5.3) property. In this section we give a different set of axioms, including the ad-covariance (8.1), which are easier to check and are analogous to those in [Kac98]. In particular we don't assume skewsymmetry as an axiom, but derive it.
The fields Y ( f , τ 1 ) and Y (g, τ 2 ) are for all f , g ∈ V mutually rational. Then V is an H T -vertex algebra as defined in Definition 5.1.
PROOF. We need to prove skew-symmetry (5.3) and H T -covariance (5.2). First we note that because the proof of Lemma 6.1 still works. We also have by ad h -covariance (8.1) and Lemma 19.1 (8.3) R S K (τ 1 )Y (g, τ 2 ) = R S V (τ 1 )Y (g, τ 2 )R V (τ 1 ). Finally, for any holomorphic distribution D Hol we have R S (τ 1 )D Hol (τ 2 )| τ 1 =0 = D S Hol (τ 1 ), where D S is the antipodal distribution defined by (5.1). Now by mutual rationality we have, for some Now the left hand side is manifestly holomorphic in τ 2 , whereas in the right hand side F 12 R S K (τ 1 )Y (g, τ 2 ) f can be assumed to holomorphic by suitable choice of F. So we can put τ 2 = 0 and cancel F 12 | τ 2 =0 so get skewsymmetry (5.3).
Next note that we have as the proof of 6.7 just uses ad h -covariance of fields. Now we use Goddard's uniqueness in our situation: if V has a state-field correspondence Y as in the statement of the theorem and we have a field G(τ) mutually rational to all Y ( f , τ) such that for some g ∈ V Y (g, τ)1 V = G(τ)1 V , then in fact G(τ) = Y (g, τ). We use this to conclude that for all f , g ∈ V In particular, for F = S(h) 1 τ , h ∈ H T and g = 1 V we find on the one hand, by compatibility of state-field correspondence with the H T -action , and on the other hand, by (8.4) τ) :, so that H T -covariance (5.2) follows from the vacuum axioms. (c) f α (τ)1 V = f α Hol (τ)1 V , and, defining f α ∈ V by f α (τ)1 V | τ=0 = f α , the map f α (τ) → f α is injective. (d) All fields f α (τ) are mutually rational. (e) V is spanned by f α 1

Affine H T -vertex algebras
Let g be a Lie algebra. Recall from Example 6.2 the notion of the conformal affinization L g associated to the conformal algebra Cg, see Example 2.5. Define a decomposition L g = L g cr ⊕ L g ann in creation and annihilation Lie subalgebras (andĤ T -submodules), where L g cr = Span(X {p} ; X ∈ g, p ∈ K Sing ), L g ann = Span(X {p} ; X ∈ g, p ∈ C pol Z ). Define a 1-dimensional trivial L g ann module C 0 = Cm 0 and let V (g) be the induced L g-module: Introduce an action of H T on V (g) by putting hv 0 = ε(h)v 0 and extending the H T action (5.3) on L g to U (L g) by the Leibniz rule. Define for {X α } a basis for g currents X α (τ) as in (6.5). These are fields on V (g) that are mutually rational, and the action of the coefficients of these fields on the vacuum v 0 obviously produces a spanning set for V (g). Also note that, if X α (τ)v 0 | τ=0 = X α 0 , the the map X α (τ) → X α 0 is injective. Furthermore, we have ad h -covariance of these fields.
LEMMA 10.1. For all h ∈ H T and all X α we have ad

PROOF. For all
Then is follows from Proposition 9.1 that V (g) is an H T -vertex algebra, called the affine H T -vertex algebra of g.

Toda Vertex Algebra
Recall from Example 6.3 the Lie algebra L Toda associated to the Toda conformal algebra of Example 2.4. Define as before a decomposition L Toda = Define a 1-dimensional trivial L Toda ann module C 0 = Cm 0 and let VToda be the induced L Toda-module: In exactly the same as for the affine vertex algebra we see that VToda is an H T -vertex algebra, called the Toda vertex algebra. It is a quantization of the Toda vertex Poisson algebra discussed in Chapter 4.
Note that VToda is non commutative, for the multiplication f ⊗ g → f {1} g, because the subalgebra L Toda cr is non-Abelian, see (6.7).