Integrability of classical affine W-algebras

We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f its nilpotent element, are integrable, except possibly for one nilpotent conjugacy class in G_2, one in F_4 and five in E_8. By integrability we mean that the Lie algebra W(g,f)/dW(g,f) contains an infinite-dimensional abelian subalgebra.


Introduction
In order to define a Hamiltonian ODE one needs a Poisson algebra, that is a commutative associative algebra P, endowed with a Poisson bracket P ⊗ P → P, a ⊗ b → {a, b}, and an element h ∈ P, called a Hamiltonian function. Then the ODE du dt = {h, u} , u = u(t) ∈ P , is called Hamiltonian. Recall that, by definition of a Poisson algebra, the bracket {· , ·} should satisfy the Lie algebra axioms, and, for every f ∈ P, the map u → {f, u} should be a derivation of the associative product on P (Leibniz rule).
In a similar fashion, in order to define a Hamiltonian PDE one needs a Poisson vertex algebra (abbreviated PVA), which is a differential algebra, that is a commutative associative algebra V with a derivation ∂, endowed with a PVA λ-bracket V ⊗ V → V, a ⊗ b → {a λ b}, and an element h ∈ V/∂V, called a local Hamiltonian functional. (One denotes by the canonical map V → V/∂V since it is the universal map satisfying integration by parts: (∂f )g = − f ∂g.) Then the PDE is called Hamiltonian. In (1.1) u = u(x, t) should be viewed as a function in the space variable x and the time t, ∂ = ∂ ∂x is the partial derivative with respect to x, while d dt defines the time flow of the system. Recall that, by definition, the PVA λ-bracket should satisfy the Lie conformal algebra axioms, see axioms (i)-(iii) from Section 3.1, similar to the Lie algebra axioms, and the Leibniz rules (iv) and (iv') from Section 3.1. In particular, due to the first sesquilinearity axiom (i), the RHS of equation (1.1) is well-defined (i.e. it does not depend on the choice of the representative of the coset h).
A key property of a PVA V is that the formula { f, g} = {f λ g}| λ=0 (1.2) produces a well-defined Lie algebra bracket on V/∂V. The notion of a PVA appears naturally in the theory of vertex algebras as their quasiclassical limit, cf. [DSK06], in the same way as a Poisson algebra appears naturally as a quasiclassical limit of a family of associative algebras.
However, the theory of Hamiltonian PDEs was started 15 years before the advent of the vertex algebra theory, in the work of Faddeev and Zakharov [ZF71], who attribute the construction to C. S. Gardner, which, in algebraic terms, is as follows.
, . . . , ℓ}, n ∈ Z ≥0 ] be the algebra of differential polynomials in ℓ differential variables, with the derivation ∂ defined by ∂u {u j ∂ u i } → i,j∈I , where the arrow means that ∂ should be moved to the right, see [BDSK09]. Thus PVAs provide a coordinate free approach to the theory of Hamiltonian PDEs.
The simplest example of a Hamiltonian PDE is the celebrated KdV equation the only one studied in [ZF71]. It is Hamiltonian for the algebra of differential polynomials in one variable V = F[u, u ′ , u ′′ , . . . ], the local Hamiltonian functional h = 1 2 u 2 , and the λ-bracket, defined on the generator u by {u λ u} = (∂ + 2λ)u + cλ 3 , (1.6) and uniquely extended to V by the PVA axioms, which corresponds to the Poisson structure H(∂) = u ′ + 2u∂ + c∂ 3 . (1.7) In fact, equation (1.5) is Hamiltonian also for another choice of the local Hamiltonian functional and the Poisson structure: h 1 = 1 2 (u 3 + cuu ′′ ), H 1 (∂) = ∂, which makes the KdV equation a bi-Hamiltonian PDE.
For an arbitrary evolution PDE du dt = P , P ∈ V, an integral of motion is a local functional f ∈ V/∂V which is conserved by virtue of this equation. In particular, for the Hamiltonian equation (1.1) this is equivalent to the property that h and f are in involution (see (1.2)): The Hamiltonian PDE is called integrable if it has an infinite set of linearly independent integrals of motion in involution; equivalently, if h is contained in an infinite-dimensional abelian subalgebra of the Lie algebra V/∂V with Lie bracket (1.2). On the other hand, given such a subalgebra of V/∂V, choosing its basis { h j } j∈J , we obtain a hierarchy of compatible integrable Hamiltonian PDEs (1.8) It is easy to see from the axioms of a PVA that "these flows commute", meaning that d We call a PVA V integrable if the Lie algebra V/∂V with bracket (1.2) contains an infinite-dimensional abelian subalgebra.
It was the paper [GGKM67] of Gardner, Green, Kruskal and Miura, where they proved that the KdV equation (1.5) admits infinitely many linearly independent integrals of motion, which initiated the whole theory of integrable PDEs. It follows from [ZF71] that these integrals are in involution with respect to the bracket (1.4), where H(∂) is given by (1.7). This leads to the whole KdV hierarchy as in (1.8).
In the paper [GD76], Gelfand and Dickey, using the idea of Lax operator [Lax68], constructed, for each integer N ≥ 2, an integrable hierarchy of PDEs, called the N th KdV hierarchy, and in [GD78] they showed that these hierarchies are Hamiltonian. (The corresponding bracket was conjectured earlier by Adler [Adl79].) Their 2nd KdV hierarchy coincides with the classical KdV hierarchy.
In the paper [Mag78] Magri proposed a simple algorithm, called nowadays the Lenard-Magri scheme, which allows one to prove that integrals of motion of a Hamiltonian PDE (1.3) are in involution, provided that the same equation can be written using a different Poisson structure H 1 (∂) in place of H(∂), and a different local funtional h 1 in place of h. In such a case one obtains a bi-Hamiltonian hierarchy of PDEs. See [BDSK09] for details.
In their seminal paper [DS85] Drinfeld and Sokolov constructed the Lie algebra of local functionals obtained in [GD78] for the N th KdV hierarchy via classical Hamiltonian reduction for the affine Kac-Moody algebra sl N . This Drinfeld-Sokolov reduction was developed by them for an affine Kac-Moody algebra g, attached to an arbitrary simple Lie algebra g, which led to the construction of an integrable bi-Hamiltonian hierarchy of PDEs, called the Drinfeld-Sokolov hierarchy, attached to an arbitrary affine Kac-Moody algebra.
The Drinfeld-Sokolov reduction is based on a principal nilpotent element f of the simple Lie algebra g. It was extended in [FORTW92] to the case of an arbitrary nilpotent element f of a simple Lie algebra g. The construction of a Drinfeld-Sokolov hierarchy in this generality turned out much more difficult, however.
The construction of the original Drinfled-Sokolov hierarchy in [DS85] is based on Kostant's theorem that cyclic elements f + E ∈ g (see Definition 2.2(a)), attached to a principal nilpotent element f , are semisimple. In a series of papers in early 90's this construction was extended to other nilpotent elements admitting a semisimple cyclic element (see [DSKV13] for references).
The theory of integrable Hamiltonian hierarchies of PDEs has been naturally related to the theory of PVAs in [BDSK09]. An important class of PVAs, called the classical affine W -algebras and denoted by W(g, f ), where g is a simple Lie algebra and f is its nilpotent element, was considered in [DSK06] as the quasiclassical limit of quantum affine W -algebras. It was then shown in [DSKV13] that the PVA W(g, f ) can be obtained by a classical Hamiltonian reduction in the framework of PVA theory, analogous to the Drinfeld-Sokolov reduction. In the same paper it was proved that the Lie algebras of local functionals, constructed in [DS85] for principal f , and in [FORTW92] for general f , coincide with the Lie algebras W(g, f )/∂W(g, f ) with the bracket (1.2).
Furthermore, the Drinfeld-Sokolov hierarchies and their generalizations have been constructed in [DSKV13] for the W -algebras W(g, f ), using PVA techniques, provided that f ∈ g is a nilpotent element of semisimple type. The latter means that there exists a semisimple cyclic element f + E attached to f . This establishes integrability of the PVA W(g, f ) for f of semisimple type.
Unfortunately, the classification of cyclic elements in simple Lie algebras, obtained in [EKV13], shows that there are very few semisimple type nilpotent elements in classical Lie algebras, and only about half of the nilpotent elements in exceptional simple Lie algebras are of semisimple type.
The first basic idea of the present paper is that the Drinfeld-Sokolov method for constructing integrals of motion in [DS85], extended to the case of nilpotents f of semisimple type in [DSKV13], generalizes, after a simple modification, to all nilpotents f admitting a non-nilpotent cyclic element f + E (see Theorems 2.11 and 4.6).
The only nilpotent elements f which are left out from the above generalization are those of nilpotent type, i.e. such that all the cyclic elements f + E are nilpotent. According to [EKV13], there are altogether 15 conjugacy classes of nilpotent elements of nilpotent type in all exceptional Lie algebras, and, among classical Lie algebras, they exist only in so n with n ≥ 7, and correspond to partitions (p 1 > p 2 = p 1 − 1 ≥ . . . ), where p 1 is odd.
In order to treat the nilpotent elements f of nilpotent type, we use the idea of [DSKV13], another version of [FGMS95], that the Drinfeld-Sokolov method works also for those f which admit a semisimple quasi-cyclic element f +E (see Definition 2.2 (b)). In the present paper we show that this is also the case when f admits a non-nilpotent quasi-cyclic element (Theorems 2.14 and 4.6).
This establishes integrability of classical affine W -algebras W(g, f ) for all classical simple Lie algebras g and all their nilpotent elements f , and for all the exceptional simple Lie algebras g and their nilpotent elements, except, possibly, the following types (in the notation of [CMG93]): The contents of the paper are as follows. In Section 2 we define the notion of cyclic and quasi-cyclic elements (see Definitions 2.2 (a) and (b)) and discuss their properties. We show that non-nilpotent cyclic and quasi-cyclic elements give rise to integrable triples in g, see Definition 2.10. In Section 3 we recall the notions of a PVA and integrable Hamiltonian PDE, and discuss the Lenard-Magri scheme of integrability. In Section 4 we recall the construction of classical affine W -algebras and show that any integrable triple gives rise to an integrable generalized Drinfeld-Sokolov hierarchy (Theorem 4.6). This theorem implies integrability of all classical affine W -algebras, associated to classical Lie algebras, and all classical W -algebras W(g, f ), associated to exceptional Lie algebras g, except for the seven nilpotents f mentioned above.
Throughout the paper the base field F is an algebraically closed field of characteristic zero.
Acknowledgments. The first author was partially supported by the national PRIN grant "Moduli and Lie theory", and the University grant n.1470755. The second author was partially supported by the grant FR-18-10849 of Shota Rustaveli National Science Foundation of Georgia. The third author was partially supported by the Bert and Ann Kostant fund.
2. Cyclic and quasi-cyclic elements. Integrable triples 2.1. Setup. Let g be a reductive finite-dimensional Lie algebra and let f ∈ g be a non-zero nilpotent element. Recall that by the Jacobson-Morozov Theorem [CMG93, Thm.3.3.1], any non-zero nilpotent element f is part of an sl 2 -triple s = {e, h, f } in g, and by Kostant's Theorem [CMG93,Thm.3.4.10] all sl 2 -triples containing f are conjugate by the centralizer of f in G, the adjoint group for g. It follows that all the constructions of this paper depend only on the G-orbit of f , and not of the chosen sl 2 -triple.
We have the ad h 2 -eigenspace decomposition g = 2 Z, we shall use the notation g >j := k>j g k , and similarly for g ≥j , g <j and g ≤j .
Remark 2.1. The depth d of f is easy to compute by knowing the Dynkin characteristic of f , defined as follows. Choose a Cartan subalgebra h of g contained in g 0 and choose a set of simple roots α 1 , . . . , α r of g such that α i (h) ≥ 0, for all i = 1, . . . , r. If g is simple and θ is the corresponding highest root, In the general case of a reductive Lie algebra g the depth of f is equal to the maximum of the depths over all simple components of g. Recall from [CMG93, §3.5] that 1 2 α i (h) can have only the values 0, 1 2 and 1; the collection of these numbers is called the Dynkin characteristic of f . Traditionally, the Dynkin characteristic is the collection of integers α i (h), however, in the theory of W -algebras it is more natural to consider the halves of these integers.
Let (· | ·) be a non-degenerate symmetric invariant bilinear form on g. The subspace g 1 2 carries a skew-symmetric bilinear form ω, defined by It is non-degenerate since ad f : g 1 2 → g − 1 2 is an isomorphism (by sl 2 -representation theory).
is called a cyclic element attached to the nilpotent element f . (b) An element of g of the form is called a quasi-cyclic element attached to the nilpotent element f , if the centralizer of E in g 1 2 is coisotropic with respect to the skew-symmetric form ω (i.e., its orthocomplement in g 1 2 is isotropic).
The following lemma is obvious.
The subspace g d lies in the center of the subalgebra g >0 . (b) Let E ∈ g d− 1 2 and let l ⊥ be the centralizer of E in g 1 2 . Then E lies in the center of the subalgebra n := l ⊥ ⊕ g ≥1 .

2.2.
Classification of cyclic elements. The classification for reductive g easily reduces to the case when g is simple [EKV13], which we will assume in this subsection. We shall often use the well-known fact that an element of a reductive subalgebra in a reductive Lie algebra g is semisimple (resp. nilpotent) if and only if it is semisimple (resp. nilpotent) in g. Let z(s) (resp. Z(s)) be the centralizer of the sl 2 -triple s in g (resp. in the adjoint group G). More generally, for a subalgebra q of g we denote by z(q) its centralizer in g. Note that if a subalgebra q of g is normalized by s, then the ad x-grading of g induces that of q by letting q j = q ∩ g j . . Claim (c) follows from discussions in Section 4 of [EKV13] for classical Lie algebras, and Section 5 of [EKV13] for exceptional Lie algebras. Recall that an element of a reductive subalgebra of a reductive Lie algebra is semisimple in the subalgebra if and only if it is semisimple in the whole Lie algebra.
2.3. Quasi-cyclic elements. The classification of quasi-cyclic elements in reductive Lie algebras is discussed in [DSJKV20]. In this subsection we shall discuss only the quasi-cyclic elements in a simple Lie algebra g, associated to a nilpotent element f of nilpotent type.
Theorem 2.6 ([EKV13]). (a) There are no nilpotent elements of nilpotent type in sl n and sp n . (b) All nilpotent elements of nilpotent type in so n correspond to partitions for which the largest part p 1 is odd and has multiplicity 1, and the next part p 2 equals p 1 −1 and has even multiplicity. (c) All nilpotent elements f of nilpotent type in exceptional Lie algebras and their depths are listed in Table 1.
(a) For all nilpotent elements of nilpotent type in so n there exists a non-nilpotent quasi-cyclic element. (b) There are no non-nilpotent quasi-cyclic elements for the following nilpotent elements of nilpotent type (see Table 1): (c) For all other nilpotent elements of nilpotent type in exceptional simple Lie algebras there exists a non-nilpotent quasi-cyclic element (see Table 1).
Proof. (a) follows from Example 2.12 and Remark 2.15 below. (b) and (c) follow from the discussion preceding Table 1, Example 2.13 and Remark 2.15 below.
Example 2.8. Let g = G 2 and let f be the nilpotent element denoted byÃ 1 as in the last row in Table 1. This is a nilpotent element of nilpotent type, so all cyclic elements are nilpotent. We claim that there are no quasi-cyclic elements attached to f . Recall that the set of positive roots for g is R + = {α, β, α + β, α + 2β, α + 3β, 2α + 3β}, where α and β are simple roots and β is a short root. Choose root vectors e γ , γ ∈ R = R + ∪ (−R + ). Then, for the grading (2.1), we have: deg e α = 0, deg e β = 1 2 (see Table 1), so that, for this grading, Example 2.9. Let g be a simple Lie algebra, different from sp n , and let f = e −θ , the lowest root vector. In this case, the depth is d = 1, and there exists a unique E ∈ g 1 2 , up to a non-zero constant factor and action of Z(s), such that f + E is a semisimple quasi-cyclic element, see [DSKV14,Prop.8.4].
In Table 1 we list the Dynkin characteristics of all nilpotent elements f of nilpotent type in exceptional simple Lie algebras g (the notation is the same as in [CMG93]). In the third column we list the depth d of f . In the fourth column we list the image in End g d− 1 2 of the action of the Lie algebra z(s) on g d− 1 2 , and in the fifth the rank of this action, defined as dim g d− 1 2 / / Z(s) . Here for a linear reductive group G|V we use the standard notation V / /G = Spec F[V ] G . Finally, the last column says whether there exists a quasi-cyclic element attached to f which is semisimple or non-nilpotent. It follows from Table 1 that there are no quasi-cyclic elements attached to f if and only if dim g d− 1 2 = 1. By st a we denote the standard representation of the Lie algebra a (it is 26dimensional for a = F 4 and 7-dimensional for a = G 2 ); 1 stands for the trivial 1-dimensional representation; ⊕ stands for the direct sum of linear Lie algebras: The results of Table 1 are obtained as follows (see [DSJKV20] for details). Denote g d− 1 2 by m. First, we compute z(s)|m using the SLA package in the GAP computer algebra system. We note that, by Table 1, Z(s)|m is a polar representation with a Cartan subspace m 0 (see [DK85] for the definitions). In particular, any closed orbit of Z(s) in m intersects m 0 non-trivially [DK85]. It follows that if m 0 contains no elements E 0 , such that f + E 0 is quasi-cyclic, then m contains no elements E, such that f + E is non-nilpotent quasi-cyclic. Indeed, in the contrary case the orbit of minimal dimension in Z(s)E is closed and non-zero, hence there exists E 0 ∈ m 0 , which lies in this orbit, such that f + E 0 is quasi-cyclic. This allows us to restrict consideration of quasi-cyclicity of f + E to E ∈ m 0 . Then, one uses again SLA as well as Mathematica to study non-nilpotence or semisimplicity of quasi-cyclic elements f + E with E ∈ m 0 through minimal polynomials of their matrices in a faithful representation of a semisimple subalgebra containing them.
2.4. Integrable triples. Let g be a finite-dimensional Lie algebra, let (· | ·) be a non-degenerate symmetric invariant bilinear form on g, let f ∈ g be a non-zero nilpotent element contained in an sl 2 -triple s ⊂ g, and consider the corresponding 1 2 Z-grading (2.1). 1 never quasi-cyclic 1 never quasi-cyclic 10. An integrable triple associated to f is (f 1 , f 2 , E), where f 1 , f 2 ∈ g −1 and E ∈ g ≥ 1 2 is a non-zero homogeneous element, such that the following three properties hold: (i) f = f 1 + f 2 and [f 1 , f 2 ] = 0, (ii) [E, g ≥1 ] = 0 and the centralizer of E in g 1 2 is coisotropic with respect to the bilinear form (2.2).
(iii) f 1 + E is semisimple and [f 2 , E] = 0. In this case E is called an integrable element for f . Note that for an integrable triple (f 1 , f 2 , E) the decomposition f + E = (f 1 + E) + f 2 is a Jordan decomposition of f + E, and that E is a central element of the subalgebra n := l ⊥ ⊕ g ≥1 , Theorem 2.11. Let f be a nilpotent element of integer depth d of a reductive Lie algebra g. a) If f is of semisimple type, then there exists E ∈ g d , such that (f, 0, E) is an integrable triple in g. b) If f s , f n and E ∈ g d are the elements, constructed in Theorem 2.5(c), then (f s , f n , E) is an integrable triple in g.
Proof. It follows immediately from Theorem 2.5(c).
Let f 1 = p j=1 E (1,1,j+1),(1,1,j) ∈ g and f 2 = f − f 1 . It is immediate to check that [f 1 + E, f 2 ] = 0 and that f 2 lies in g −1 (hence, it is nilpotent). Moreover, the minimal polynomial of f 1 + E is p(λ) = λ(λ p − 2). Hence, f 1 + E is semisimple and (f 1 , f 2 , E) is an integrable triple associated to f . Example 2.13. In simple Lie algebras of exceptional types there are three cases of nilpotent orbits of nilpotent type admitting non-nilpotent quasi-cyclic elements but no semisimple quasi-cyclic elements. Namely, these are nilpotent elements with label 2A 2 + A 1 in E 6 , E 7 , and E 8 (see Table 1). For them we find integrable triples computationally, using the SLA package of the GAP system.
We choose the following representatives f : for E 6 , f 001 Thus, since f n ∈ g −1 , we have f s = f − f n ∈ g −1 . Hence, (f s , f n , E) is an integrable triple.
Additionally, let us remark that in all three of these cases f s has label 2A 2 and f n has label A 1 . Note also that the subalgebra generated by f and E is the direct sum of an sl 3 and a 1-dimensional center spanned by f n . Moreover f s is a principal nilpotent in this sl 3 .
Theorem 2.14. Let f be a nilpotent element of depth d of nilpotent type in a simple Lie algebra g, such that there exists E ∈ g d− 1 2 , for which f + E is a non-nilpotent quasi-cyclic element. Proof. Part (a) is clear. The cases, not covered by (a) are as follows. First, it is g = so N , which is covered by Example 2.12. Second, it is 2A 2 + A 1 in all algebras of type E, which is covered by Example 2.13.
Remark 2.16. It is clear from Example 2.8 that for g = G 2 and f a short root vector there are no integrable triples.

Poisson vertex algebras, Hamiltonian equations and integrability
3.1. PVA and Hamiltonian PDE. Recall (see e.g. [BDSK09]) that a Poisson vertex algebra (abbreviated PVA) is a commutative associative algebra V with a derivation ∂, endowed with a λ-bracket As a consequence of skew-symmetry and the left Leibniz rule, we also have the (iv') right Leibniz rule: {ab λ c} = {a λ+∂ c} → b + {b λ+∂ c} → a, where → means that ∂ is moved to the right.
Example 3.1. The most important example for this paper will be the affine PVA V(g, E), where g is a Lie algebra with a symmetric invariant bilinear form (· | ·) and E ∈ g. It is defined as the differential algebra V(g) = S(F[∂]g), the algebra of differential polynomials over the vector space g, with the PVA λ-bracket given by and extended to V by sesquilinearity axioms and the Leibniz rules.
Let V be a PVA. We denote by : V → V/∂V the canonical quotient map of vector spaces. Recall that (see [BDSK09]) V/∂V carries a well-defined Lie algebra structure given by { f, g} = {f λ g}| λ=0 , and we have a representation of the Lie algebra V/∂V on V given by A Hamiltonian equation on V associated to a Hamiltonian functional h ∈ V/∂V is the evolution equation The minimal requirement for integrability is to have an infinite collection of linearly independent integrals of motion in involution: In this case, we have the integrable hierarchy of Hamiltonian equations 3.2. Bi-Poisson vertex algebras and Lenard-Magri scheme of integrability. Let {· λ ·} 0 and {· λ ·} ∞ be two λ-brackets on the same differential algebra V.
We can consider the pencil of λ-brackets As above, we say that V is a bi-PVA if {· λ ·} z is a PVA λ-bracket on V for every z ∈ F. The affine PVA defined in Example 3.1 is in fact a bi-PVA. Let V be a bi-Poisson vertex algebra with λ-brackets {· λ ·} 0 and {· λ ·} ∞ . A bi-Hamiltonian equation is an evolution equation which can be written in Hamiltonian form with respect to both PVA λ-brackets and two Hamiltonian functionals h 0 , h 1 ∈ V/∂V: The most common way to prove integrability for a bi-Hamiltonian equation is to solve the so called Lenard-Magri recurrence relation (see [Mag78]): (3.5) The Lenard-Magri recurrence relation (3.5) produces local functionals in involution.
As a special case of Lemma 3.2, if V is a bi-PVA and h 0 , h 1 , · · · ∈ V/∂V satisfy the Lenard-Magri recurrence (3.5), then they are in involution: A := Span{ h n } ∞ n=0 ⊂ V/∂V is an abelian subalgebra with respect to both Lie algebra brackets {· , ·} 0 and {· , ·} ∞ . In this way, we get the corresponding hierarchy of bi-Hamiltonian equa- If moreover and g 0 , g 1 , · · · ∈ V/∂V is any other sequence satisfying (3.5), then A = Span{ h n , g n } ∞ n=0 ⊂ V/∂V is also an abelian subalgebra.

Classical affine W -algebras and generalized Drinfeld-Sokolov hierarchies
4.1. Definition of classical affine W -algebras. Let g be a reductive Lie algebra as in Section 2.1, (· | ·) a non-degenerate symmetric invariant bilinear form on g, f its non-zero nilpotent element, and consider the corresponding 1 2 Z-grading (2.1) and the skew-symmetric non-degenerate bilinear form ω on g 1 2 defined by (2.2). Fix an isotropic (with respect to ω) subspace l ⊂ g 1 2 and denote by l ⊥ = {a ∈ g 1 2 | ω(a, b) = 0 for all b ∈ l} ⊂ g 1 2 its orthogonal complement with respect to ω. Throughout the paper we consider the following nilpotent subalgebras of g: Fix an element E ∈ z(n) (the centralizer of n in g) and consider the affine bi-PVA V(g, E) from Example 3.1: it is the differential algebra V(g) = S(F[∂]g), with its two PVA λ-brackets {· λ ·} 0 and {· λ ·} ∞ (defined on a, b ∈ g by {a λ b} 0 = [a, b] + (a|b)λ and {a λ b} ∞ = (E|[a, b]), cf. equations (3.1) and (3.4)). Consider also the differential algebra ideal I, generated by the set Note that, since E ∈ z(n), we have {a λ w} ∞ = 0 for all a ∈ n and w ∈ V(g). Consider the space , for every a ∈ n ⊂ V(g, E) .
Lemma 4.1. W ⊂ V(g, E) is a bi-PVA subalgebra of V(g, E) and I ⊂ W is its bi-PVA ideal.
Proof. Straightforward, see e.g. [DSKV13]. Remark 4.3. Recall from [DSKV13] that the classical affine W -algebra depends only on the Lie algebra g and on the nilpotent orbit of f : for different choices of f in its nilpotent orbit, of the sl 2 -triple s containing f , and of the isotropic subspace l ⊂ g 1 2 , we get isomorphic W -algebras. Let p ⊂ g be a subspace complementary to m in g: g = m ⊕ p. We assume that p is compatible with the grading (2.1), so that g ≤0 ⊂ p ⊂ g ≤ 1 2 . Clearly, we can identify, as differential algebras V(g)/I ≃ V(p). Hence, the classical W -algebra can be viewed as a differential subalgebra of V(p), the algebra of differential polynomials over p: W(g, f, E) = W/I ⊂ V(g)/I ≃ V(p) .
(4.2) 4.2. Integrable triples and generalized Drinfeld-Sokolov hierarchies. In [DSKV13] a generalized Drinfeld-Sokolov hierarchy was constructed, using the Lenard-Magri recurrence relation (3.5), under the assumption that the element f + E ∈ g is semisimple. In this section we extend this result by constructing a generalized Drinfeld-Sokolov hierarchy for any integrable triple associated to f . Let (f 1 , f 2 , E) be an integrable triple associated to f (cf. Definition 2.10), and let l ⊥ be the centralizer of E in g 1 2 , with l isotropic with respect to (2.2). Setting as above n = l ⊥ ⊕ g ≥1 , we have by definition that [E, n] = 0.
Let K = F((z −1 )) be the field of formal Laurent series in z −1 over F. Consider the Lie algebra g((z −1 )) = g ⊗ F K. Since, by Definition 2.10, f 1 + E ∈ g is semisimple, f 1 + zE ∈ g((z −1 )) is also semisimple. Indeed, for a non-zero element t ∈ K we have the Lie algebra automorphism ϕ t of g((z −1 )) acting as t i on g i . Hence, if E ∈ g k , we have tϕ t (f 1 + E) = f 1 + t k+1 E. Then, working over the field extension of K, containing z 1 k+1 , we conclude that f 1 + zE is semisimple in g((z −1 )). We thus have the direct sum decomposition where h := Ker ad(f 1 + zE) and h ⊥ := Im ad(f 1 + zE) . (4.4) The notation h ⊥ relates to the fact that Im ad(f 1 + zE) is the orthogonal complement of Ker ad(f 1 + zE) with respect to the non-degenerate symmetric invariant bilinear form (· | ·) on g((z −1 )), extending the form (· | ·) on g by bilinearity. (But we will not use this fact.) Lemma 4.4. For every element A(z) ∈ g((z −1 )), there exist unique h(z) ∈ h and Proof. According to the direct sum decomposition (4.3), we can write, uniquely, If E ∈ g k , we extend the 1 2 Z-grading (2.1) to g((z −1 )) by letting z to have degree −k − 1, so that f + zE and f 1 + zE are homogeneous of degree −1. Then where g((z −1 )) i ⊂ g((z −1 )) is the space of homogeneous elements of degree i, and the direct sum is completed by allowing infinite series in positive degrees, cf.
Using the bilinear form (· | ·) on g we get the isomorphism p * ≃ m ⊥ . Let {q i } i∈P be a basis of p, and let {q i } i∈P be the dual (with respect to (· | ·)) basis of m ⊥ , namely, such that (q j |q i ) = δ ij . We denote (4.9) The next result is a generalization of [DSKV13,Prop.4.5] to the case of an integrable triple associated to the nilpotent element f .
The main result of this section is the following theorem, which allows us to construct an integrable hierarchy of bi-Hamiltonian equations for classical affine W -algebras.
Theorem 4.6. Let g be a reductive Lie algebra with a non-degenerate symmetric invariant bilinear form (· | ·) extended to g((z −1 )) by bilinearity. Let f be a nilpotent element of g, and let (f 1 , f 2 , E) be an integrable triple for f and consider the decomposition (4.3) of g((z −1 )). Let U (z) ∈ V(p) ⊗ g((z −1 )) >0 and h(z) ∈ V(p) ⊗ h >−1 be a solution of equation (4.10). Let c(h) be the center of h, and let a = a ∈ c(h) [a, f 2 ] = 0 ⊂ g((z −1 )) . (4.13) For a ∈ a, let g a (z) = (1 ⊗ a|h(z)) = n∈Z ≥0 g a,n z N −n , (4.14) where N is the largest power of z appearing in (1⊗a|h(z)) with non-zero coefficient, and (· | ·) is defined in (4.8). Let W = W(g, f, E) ⊂ V(p) be the classical affine W -algebra with its compatible PVA structures {· λ ·} 0 and {· λ ·} ∞ introduced in Definition 4.2. Then A = Span{ g a,n | a ∈ a, n ∈ Z ≥0 } is an infinite-dimensional abelian subalgebra of W/∂W ⊂ V(p)/∂V(p) (with respect to both 0 and ∞-Lie brackets), defining a hierarchy of bi-Hamiltonian equations dw dt a,n = {g a,n λ w} 0 λ=0 = {g a,n+1 λ w} ∞ λ=0 , w ∈ W , a ∈ a , n ∈ Z ≥0 . (4.15) is an abelian subalgebra, and, by the discussion in Section 3.2, we get the hierarchy of bi-Hamiltonian equations (4.15). Finally, we are left to show that if a is not in the center of g((z −1 )), then the local functionals g a,n , n ∈ Z ≥0 , span an infinite-dimensional subspace. This follows verbatim from the results in [DSKV13,Sec.4.7] noticing that ad(f + zE) and ad a commute. Since f 1 + zE ∈ a, we see that A is infinite-dimensional, as claimed.
Remark 4.7. Theorem 4.6 holds in the more general setting of a finite-dimensional Lie algebra g with a non-degenerate symmetric invariant bilinear form (· | ·), an sl 2 -triple {e, h, f } in g, and an integrable triple (f 1 , f 2 , E) for f .