Reduced Pre-Lie Algebraic Structures , the Weak and Weakly Deformed Balinsky – Novikov Type Symmetry Algebras and Related Hamiltonian Operators †

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.


Introduction
A left pre-Lie algebra (A, +, •) is a vector space A over an algebraically closed field F with a bilinear map • : A ⊗ A → A, satisfying the relation for any a, b, c ∈ A. This is just the invariance of the associator (a, b, c) = (a • c) • b − a • (c • b) under the interchange of b, c ∈ A. Hence, every associative algebra is also a pre-Lie algebra, as the associator vanishes identically.It follows from (1) that usual anti-symmetrization yields a Lie bracket on A for arbitrary a, b ∈ A. However, not every Lie algebra arises from a pre-Lie algebra.These algebras have been used, under various names, for a long time.As is known [29,83], they were called left-symmetric algebras in the work of Vinberg [86] on convex homogeneous cones, and so were dubbed as Vinberg algebras in some papers.They also appear in the study of affine manifolds, named as right-symmetric algebras [59].It was proposed in [29] to adopt the name pre-Lie algebras, which had been used by Gerstenhaber [44] as the Lie bracket on the Hochschild cohomology, which arises as a pre-Lie algebra structure on cochains.These pre-Lie algebras have applications in many fields, including perturbative quantum field theory [55,56], where insertion of Feynman graphs into each other equips them with a pre-Lie structure which controls the combinatorics of renormalization.
J. Moser pointed out the importance of connections between Lie algebraic structures and Hamiltonian dynamics, especially with regard to questions of integrability, in numerous contributions including [60][61][62].The fact that many of the integrable Hamiltonian systems discovered during the last several decades have been shown to depend intimately on the Lie-algebraic properties of their internal hidden symmetry structures [15][16][17]38,65], has more than served to confirm Moser's observations.A first account of the Hamiltonian operators and related differential-algebraic relationships, lying in the background of integrable systems and coinciding with reduced pre-algebraic structures, was given by Dorfman [32] and Gel'fand and Dorfman [43] and later extended by Dubrovin and Novikov [35,36], and also by Balinsky and Novikov [11].In addition, new special differential-algebraic techniques were devised [3,71,72] for studying the Lax integrability and the structure of related Hamiltonian operators for a wide class of the Riemann type hydrodynamic hierarchies.Recently, much work [3,[6][7][8][9][10]66] has been devoted to the finite-dimensional representations of the reduced pre-Lie algebraic structures now called the Balinsky-Novikov algebras.Their importance for constructing integrable multi-component nonlinear Camassa-Holm type dynamical systems on functional manifolds was demonstrated by Strachan and Szablikowski [81].Moreover, they suggested in part the Lie-algebraic imbedding of the Balinsky-Novikov algebra in the general Lie-Poisson orbits scheme of classifying Lax integrable Hamiltonian systems.It is worth mentioning the related work [47] by Holm and Ivanov, where integrable multi-component nonlinear Camassa-Holm type dynamical systems were also constructed.
We have devised a formal differential-algebraic recasting of the classical Lie algebraic scheme and developed an effective approach to classification of the underlying algebraic structures of integrable multi-component and multi-dimensional Hamiltonian systems.In particular, we have devised simple algorithm, based on the Lie-Poisson structure analysis on the adjoint space to toroidal Lie algebras, rigged with non-associated and noncommutative algebras, which enables singling out new algebraic pre-Lie algebraic structures, containing the corresponding Hamiltonian operators, which generate integrable multi-component and multidimensional dynamical systems.The theory of these systems was recently started in [21][22][23]33,37,53,54,78,84,85,88] and developed in [46,79].In particular, we studied nonassociative and noncommutive algebras over C and the related Lie-algebraic symmetry structures on the torus T n for n ∈ N, generating via the Adler-Kostant-Symes scheme multi-component and multi-dimensional Hamiltonian operators.The latter serve for describing integrable heavenly type equations, whose theory has been just recently started in [21,75,76,78,[80][81][82]84,85] and advanced in [3,14,15,46,73].In the case of multidimensional torus T n for n ∈ N\{1} we have constructed a new weak Balinsky-Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations.
We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky-Novikov type algebra A h on the real axis R, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds.Namely, we have stated that the current algebra A h := {a(x We also show that the well-known Balinsky-Novikov algebraic pre-Lie algebraic structures, obtained in [11,43] as a condition for a matrix differential system to be Hamiltonian and in [13,24,52,69] as that on a flat torsion free left-invariant affine connection on affine manifolds, affine structures and convex homogeneous cones, arise as a derivation on the Lie-algebra associated with a differential loop algebra.
Using the theory of Zelmanov [89], in particular, the theory of nonassociative and associative left-symmetric algebras, we described algebraic properties of new Balinsky-Novikov type algebras, including their fermionic and related multiplicative and commutative Lie versions.

Pre-Lie algebraic structures and related Hamiltonian operators
Let (A, +, •) be a finite-dimensional algebra (in general noncommutative and associative) over an algebraically closed field F endowed with a nondegenerate symmetric trace-like [77,81] bilinear form •, • : A ⊗ A→ F. We shall require that A allows a natural Lie algebra extension for any a, b, c ∈ L A .Using A, one can construct the related toroidal algebra A of smooth mappings T n → A of the n-dimensional torus T n , n ∈ Z + , and endow it with the suitably generalized commutator operation [•, •] : A ⊗ A → A subject to the natural pointwise multiplication operation • : A ⊗ A → A. The corresponding loop Lie algebra L A will be naturally rigged with a generalized symmetric nondegenerate bilinear ad-invariant form (•, and for any a, b, c ∈ L A .The form (4) makes possible the natural identification A * A; in particular, for a linear functional u * ∈ A we also define its adjoint action on A as for a fixed u ∈ A and any a ∈ A. Now, one can naturally identify the space L * A , adjoint with respect to the form (4) to L A , with itself and consider further the space D(L *

A
) of smooth scalar functions on L * A together with its related Lie-Poisson bracket: for any f , g ∈ D(L *

A
), where the weak gradient map Owing to its definition [1,2,16,38], the bracket (7) satisfies the classical Jacobi condition, so it is a powerful tool for constructing the related Hamiltonian operators on D(L *

A
) and satisfies the Jacobi identity.
As the canonical Lie-Poisson bracket (7) does not involve essentially the loop Lie algebra structure of L A , we proceed further to a new Lie algebra structure on L A via its central extension.Namely, let L A := L A ⊕ F denote the centrally extended Lie algebra L A endowed with the extended Lie bracket for any for any a, b ∈ L A and α, β ∈ F, where the 2-cocycle ω 2 : L A × L A → F is a skew-symmetric bilinear form and satisfies the Jacobi identity: for any a, b, c ∈ L A .It is evident that the existence of nontrivial central extensions on L A strongly depends on the underlying structure of the algebra A as presented above.Yet there are some algebraic properties that allow us to proceed.Namely, assume that a smooth map D u : L A → L A defines for a fixed u ∈ L A a weak derivation of L A , that is for any a, b, c ∈ L A .Then the following important result holds [64,77].A proof simply requires verifying (11) and is omitted.
There are many ways to construct a priori nontrivial derivations on L A such as the following simple consequence of Proposition 1.1 [32,64,77]: Let a nondegenerate skew-symmetric endomorphism R : L A → L A satisfy the well known Yang-Baxter commutator condition: for any a, b ∈ L A .Then the inverse map R −1 : L A → L A is a skew-symmetric derivation of the Lie algebra L A and defines a 2-cocycle on L A for any a, b ∈ L A .
Remark 1.3.An interesting consequence of Theorem 1.2 is that the subspaces are Lie subalgebras of L A , splitting it into the direct sum In particular, the R-structures on L A can be used for constructing additional Hamiltonian operators on L * A .More precisely, we endow, following [43], L * A with the natural differential algebraic structure assuming it to be a polynomial differential algebra A(u), generated by an element u ∈ A and its derivatives u (j) ∈ A (j ∈ Z + ) with respect to the standard derivation D x := ∂/∂x, x ∈ S 1 , on A. On A(u) one can naturally define the space of linear uniform gradient-wise derivations Γ A (u) as where [∇ h , D x ] = 0 for any h ∈ A(u) and the expression acts on any f ∈ A(u) as Taking into account the action of the derivations − A (u) on the differential algebra A(u), one can equip it with a natural Lie algebra structure where the element {h, g} is written by means of the standard Fréchet derivative on A(u) : for any h, f ∈ A(u).Following [43], on A(u), supplemented with a unit element, one can determine a space of functionals F A (u) as the set of equivalent elements f ∼ h ∈ A(u) for which f − h ∼ D x g for some element g ∈ A(u).Such functionals can be denoted as the integrals On F A (u) there exists a natural differential δ : where the conjugation mapping " * " is taken with respect to the bilinear form (4) on A introduced above.Owing to the relationship f , * (u)(1) for all u ∈ A, (21) can be rewritten as Using (22), one can successively determine the whole Grassmann algebra Λ( A(u)) of differential forms on A(u), generated u ∈ A. In particular,suppose a closed nondegenerate differential 2-form ω (2) ∈ Λ 2 ( A(u)), δω (2) = 0, is given on A(u).Then from [1,2,16] { f , g} ω (2) := −ω (2) (∇ f , ∇ g), (23) where for any f , g ∈ F A (u) the maps on the algebra A(u) determine for any u ∈ A * A the corresponding Hamiltonian operator ϑ(u) : Whence, we are led [64,74,77] to the following result.
Proposition 1.4.Suppose that L A allows a skew-symmetric nondegenerate R-structure homomorphism R : L A → L A , satisfying the generalized Yang-Baxter condition for any a, b ∈ L A and α ∈ F. Then differential 2-forms ω (2) j ∈ Λ 2 ( A(u)) (j = 1, 2) on the algebra A(u) defined as and for any f , g ∈ F A (u) are closed.Moreover, the corresponding Hamiltonian operators, determined from ( 27) and (28) via the identifications are compatible; that is, the sum A is also a Hamiltonian operator for arbitrary λ, µ ∈ F.

85
Sketch of a proof.As (27) is closed a priori, a proof of the proposition consists in checking the closedness of the 2-forms ω (2) , which is equivalent to (26).Taking into account (27) and (28) and the representation of ϑ 2 : L A → L * A as the composition ϑ 2 = ϑ −1 1 ϑ 0 , where the Hamiltonian operator ϑ 0 : L A → L * A is naturally determined from (7) as for any f , g ∈ F A (u).This is equivalent to the compatibility of the Hamiltonian operators ϑ 1 and ϑ 2 on A(u) [16,17,43,64,77].In case when the parameter α = 0, the expression (27) determines a 2-cocycle on L A owing to the fact that the inverse map R −1 : L for any a, b ∈ L A , presenting a 2-cocycle on L A .Consequently, λϑ 0 + µϑ 1 : L A → L * A is Hamiltonian for arbitrary λ, µ ∈ F, which also yields the compatibility of ϑ 1 and ϑ 2 on A(u).
Proposition 1.5.Let a skew-symmetric R-structure R : L A → L A on L A satisfy (26).Then and defined for any f , g ∈ F A (u), are Poisson and compatible on A(u).

Lie-Poisson brackets, skew-symmetric derivations and Balinsky-Novikov type algebraic structures
Here we consider for any u for any a, b ∈ L A , parameterized by an arbitrary yet fixed u ∈ A * and modeling the Hamiltonian operator, analyzed in [11,43] and used in [81].To verify that ( 34) is a weak derivation of L A , it suffices to check that the tri-linear Leibniz type relationship holds for any a, b and c ∈ L A .Following simple calculations, taking into account that u ∈ L A and ∂ ∂x u ∈ L A are functionally independent, one finds that ( 34) is a skew-symmetric weak derivation of L A iff the following algebraic constraints are imposed on A: and which hold for any a, b, c ∈ A, and were derived in a similar context by Gel'fand and Dorfman [43] and Balinsky and Novikov in [11].As already mentioned, the algebra, defined by ( 36) and (38), is a reduced pre-Lie algebra A, which was first introduced in [44,86].In [66] this algebra was also called a Novikov algebra.In particular, commutative BNAs are associative.
It is worth observing that the linearity of ( 35) with respect to u ∈ A A * allows the canonical Lie-Poisson [1,2,74] representation: for any a, b ∈ A, where is a new skew-symmetric commutator structure imposed on A. Moreover, since the bracket (39) needs here no symmetry and invariance properties ( 4) and ( 5), we simply state that it is Poisson iff the commutator (40) generates a weak Lie algebra structure on A, that is for any fixed u ∈ A * and arbitrary elements a, b, c ∈ A. It follows from ( 41) that A coincides with the BNA algebra (38).
Example 1.6.Having defined the Lie bracket for any a, b ∈ A, one easily deduces from the weak Jacobi condition (41) the reduced pre-Lie algebra structure: Example 1.7.In the case of the one-dimensional loop algebra A, a commutator Lie structure, defined for any elements a, b ∈ A as where the inverse acting as Example 1.8.It was recently shown in [3] that the spatially one-dimensional skew-symmetric bilinear map imposed on A for any a, b ∈ A, generates an adjacent Lie algebra L A iff the following Riemann type reduced pre-Lie algebra structure holds for all a, b ∈ A.
[r a , generates an adjacent Lie algebra L Ã iff the following degenerate constraint l a•b = 0 holds for all a, b ∈ A. Similarly, the Lie-commutator structure generates no pre-Lie algebraic structure on the algebra A related to a Hamiltonian operator on A * .

Weak and weakly deformed Balinsky-Novikov type algebras
In this Section we study nonassociative and noncommutive algebras over C and the related Lie-algebraic symmetry structures on the torus T n for n ∈ N, generating via the Adler-Kostant-Symes scheme multi-component and multidimensional Hamiltonian operators.The latter serve for describing integrable heavenly type equations, whose theory has been just recently started in [21,75,76,78,80-82, 84,85] and advanced in [3,14,15,46,73].In the case of multidimensional torus T n for n ∈ N\{1} we have constructed a new weak Balinsky-Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations.
We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky-Novikov type algebra on the real axis R, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds.

A weak Balinsky-Novikov type symmetry algebra
Let (A; +, •) be a finite dimensional nonassociative and, in general, noncommutative algebra over the field C, endowed with a nondegenerate symmetric and invariant bilinear form < •, • >, that is < a, b >=< b, a > and < a, b • c >=< a • b, c > for any a, b and c ∈ A. Let also G± := di f f ± (T n ) ⊗ A, n ∈ N, be A-valued loop subalgebras of the algebra G : = G+ ⊕ G− on the torus T n , holomorphic, respectively, inside D 1 + and outside D 1 − of the unit disk D 1 ⊂ C 1 , such that for any ã(λ) ∈ G− the value ã(∞) = 0.The loop algebra G can be naturally identified with a dense subspace of the dual space G * through the pairing Here we put, by definition [1,45], a A-valued loop vector field ã and introduced, for brevity, the gradient operator ∂/∂x := (∂/∂x 1 , ∂/∂x 2 , ..., ∂/∂x n ) T and the standard bilinear form < •, • > E n in the Euclidean space E n .The algebra G can be further equipped with the Lie bracket Let us assume now that this way obtained A-valued vector field algebra G is a weak Lie algebra L G , that is As a result of easy enough calculations for the case of the A-valued loop vector field Lie algebra L G on the torus T n of dimension n ∈ N\{1} one finds that the algebra A should satisfy the following algebraic constraints: for any ã, b ∈ L G and arbitrary element l ∈ L * G satisfies the Jacobi identity (55).r Remark 2.1.In case of the A-valued loop vector field Lie algebra L G Γ(S 1 ) ⊗ A on the one-dimensional circle S 1 the corresponding algebraic constraints reduce to the following less strong dual Balinsky-Novikov algebra A expressions: for any a, b ∈ A. Thereby, the obtained algebra A will be naturally called a weak Balinsky-Novikov type algebra.
Summarizing the reasonings above, we can now formulate the obtained above result as the following theorem.
Theorem 2.2.The canonical Lie-Poisson bracket (57) on the co-adjoint space G * Λ1 (T n ) ⊗ A in the case n ∈ N\{1} is compatible with the internal algebraic structure of the algebra A iff it satisfies the weak Balinsky-Novikov algebraic constraints (56).
Observe now that owing to pairing (53), the corresponding dual spaces G * + and G * − satisfy the relationships G * where for any l(λ) ∈ G * − one can impose the dual constraint l(0) = 0. Having defined the projections one can construct a classical R-structure [38,74,77] on the Lie algebra G as the endomorphism R : G → G, where which allows to determine on the vector space G the new Lie algebra structure for any ã, b ∈ G, satisfying the standard Jacobi identity.
Let D( G * ) denote the space of smooth functions on G * .Then for any f , g ∈ D( G * ) one can write the general canonical [16,38,70,74] Lie-Poisson bracket where l ∈ G * is a seed element and ∇ f , ∇g ∈ G are the standard functional gradients at l ∈ G * with respect to the metric (53).The related to (62) space of Casimir invariants is defined as the set where for any A-valued seed element the gradients ∇h j ( l) := ∇h j (l), ∂/∂x and the coadjoint action ( 63) can be equivalently rewritten, for instance, as for any j = 1, n.If to take two smooth functions h (y) , on the space G * vanishes, that is at any seed element l ∈ G * .Since the functions h (y) , h (t) ∈ I( G * ), the following coadjoint action relationships hold: which can be equivalently rewritten as ∂/∂x, (l∇h (y) (l)) and similarly ∂/∂x, (l∇h (t) (l)) Consider now the following Hamiltonian flows on the space G * : where h (y) , h (t) ∈ I( G * ) and y, t ∈ R are the corresponding evolution parameters.Since h (y) , h (t) ∈ I( G * ) are Casimir elements, the flows (72) commute.Thus, taking into account the representations (70), one can recast the flows (72) as Lemma 2.3.The compatibility of commuting flows ( 73) is equivalent to the Lax type vector fields relationship which holds for all y, t ∈ R and arbitrary λ ∈ C.
Proof.The compatibility of commuting flows (73) implies that ∂ 2 l/∂t∂y − ∂ 2 l/∂y∂t = 0 for all y, t ∈ R and arbitrary λ ∈ C. Taking into account the expressions (72), one has for any A-valued vector field ), where we have denoted From ( 75) we obtain that ad * ϕ( l) l = 0 for all y, t ∈ R, ϕ( l) ∈ G and arbitrary λ ∈ C. Now based on the arbitrariness of Z ∈ G and analyticity of the A-valued vector field expression (76), one easily shows [22] that ϕ( l) = 0, thus finishing the proof.
For finding the exact representatives of the functions h (y) , h (t) ∈ I( G * ) it is necessary to solve the determining equation ( 66), taking into account that if the chosen element l ∈ G * is singular as |λ| → ∞, the related expansion for the gradients holds, where the degree p ∈ Z + can be taken as arbitrary.Upon substituting (77) into (66) one can find recurrently all the coefficients ∇h(l) j , j ∈ Z + , and then construct gradients of the Casimir functions h (y) , h (t) ∈ I( G * ) reduced on G+ as ∇h (t) (l) + = (λ p t ∇h(l)) + , ∇h (y) (l) + = (λ p y ∇h(l)) + (78) for some positive integers p y , p t ∈ Z + .
Remark 2.4.As mentioned above, the expansion ( 77) is effective if a chosen seed element l ∈ G * is singular as |λ| → ∞.In the case when it is singular as |λ| → 0, the expression (77) should be replaced by the expansion for an arbitrary p ∈ Z + , and the reduced Casimir function gradients then are given by the expressions for some positive integers p y , p t ∈ Z + .Then the corresponding flows are, respectively, written as The above results, owing to 2.3, can be formulated as the following main proposition.
Take an A-valued loop vector field l ∈ G * and let h (y) , h (t) ∈ I( G * ) be Casimir functions subject to the metric (•, •) on the A-valued loop Lie algebra G and the natural coadjoint action on the A-valued loop co-algebra G * .Then the following dynamical systems are commuting to each other Hamiltonian flows for all y, t ∈ R.Moreover, if H is a faithful representation vector space for the weak Balinsky-Novikov algebra A, the compatibility condition of these flows is equivalent to the vector fields representation where ψ ∈ C 2 (R 2 × T n ; H) and the A-valued loop vector fields ∇h (t) ( l) + , ∇h (y) ( l) + ∈ G+ , given by the expressions ( 73) and ( 78), satisfy the so called Lax-Sato compatible relationship (74) for any λ ∈ C.

A weakly deformed Balinsky-Novikov type symmetry algebra
Consider a finite-dimensional noncommutative and non-associative algebra (A h ; +, •) over C, endowed additionally with a commutative family of automorphisms {∆ x : A h → A h : x ∈ R}, depending smoothly on a real parameter x and satisfying the following weak continuity condition: for any x, which makes it possible to construct a "current algebra" with the naturally compatible pointwise multiplication: for any a(x), b(x) ∈ A h , x ∈ R. The algebra A h can be additionally rigged with the Lie structures: is an endomorphism A h , specified by the parameter h ∈ R\{0}.
Remark 2.5.It is easy to see that the corresponding operator limit, lim h→0 Da(x) = da(x)/dx for any a(x) ∈ A h and x.
Now we pose a problem: what conditions should be imposed on A h for A h to become a Lie algebra?For an answer it, is enough to check the Jacobi identity for any a, b and c ∈ A h .
Observe that mapping (89) satisfies for any ã, b ∈ A h the property where a (h) := ∆ h a ∈ A h .Having defined the usual right R a and left L a shifts on the algebra A h as R a b := b•a, L a b := a•b for arbitrary a, b ∈ A, respectively, one easily proves the following result.
Proposition 2.6.The current algebra A h is a Lie algebra L A h iff the following conditions hold for all a, b ∈ A h : for the Lie bracket (87), and for the Lie bracket (88).
As a dBNA A h is assumed to be finite-dimensional, one can naturally determine [16][17][18] the adjoint space L * , where gradγ(u) ∈ L A h is the standard gradient of this functional.Moreover, if L A h allows the central extension LA h := (L A h ; C) by means of a Maurer-Cartan bilinear form where (•, •) s is a symmetric bilinear form on L A h and for any (a; α), (b; β) ∈ LA h , and α(∆ h ) : A h → A h is some skew-symmetric constant map, then the Hamiltonian operator where (a, α(∆ h )b) s := (α(∆ h ) * a, b) s for any a, b ∈ L A h , is compatible [38] for any λ ∈ C.This makes it possible to generate [16,17,38] an infinite hierarchy of mutually commuting smooth independent functionals γ j ∈ D(L * A h ), j ∈ Z + , with respect to both ϑ(u) and α(∆ h ) * : and giving rise to an infinite system of mutually commuting completely integrable Hamiltonian flows du/dt j = −ϑ(u) gradγ j (u) on L * A h with respect to independent evolution parameters t j ∈ R, j ∈ Z + .

The Riemann type reduced pre-Lie algebra isomorphism and related algebraic properties
Observe that the algebra A with relationships (50), generated by the two-dimensional toroidal pre-Lie algebra structure (48), is close to the Riemann type pre-Lie algebra structure ( 47), yet generated by the spatially one-dimensional skew-symmetric structure (46).Moreover, the following result holds.
Theorem 3.1.The algebra A, generated by the relationships (50), is isomorphic to the reduced Riemann type pre-Lie algebra (47).
Proof.Assume that a, b, x ∈ A. For (i), we simply note that To verify (ii), we compute that

An additive mapping δ :
for all a, b ∈ A. Let DerA be the set of all derivations of A. Moreover,  for any a, b ∈ A, then the following statements hold: Proof.Assume that a, b, x ∈ A, then (i) follows from Property (ii) follows from (111) with b = a, so x(aa) = (xa)a = (aa)x and therefore a 2 ∈ Z(A).
To prove (iii) we use Next, the property (iv), as a consequence of the property (i), gives rise to the relationship thus ensuing the property (v) directly from (i).
Proposition 3.5.Let A be a non-commutative reduced Riemann type algebra over a field F satisfying (110) and F = 2. Then the following hold: Proof.If (A, A) = 0, then 2a 2 = (a, a) = 0 for any a ∈ A, which proves (i).
for any x, y, z ∈ A. Hence, for any a, b ∈ A on the n-dimensional toroidal algebra A, (41), includes some new BNA-type pre-Lie algebra structures on the basic nonassociative algebra A, which can be useful for applications in the multi-dimensional integrability theory started in [21][22][23]33,37,53,54,78,84,85,88] and developed in [46,79].They are strongly based on differential-algebraic and related analytical techniques and make it possible to construct new algebraic structures on the corresponding nonassociative algebras, within which the corresponding Hamiltonian operators generate integrable multi-component and multidimensional dynamical systems.In what follows, we investigate the underlying algebraic structures of non-associative BNA-type pre-Lie algebras by focusing on the basic Balinsky-Novikov algebra and its fermionic modification.

The Balinsky-Novikov algebra and its fermionic modification
Recall that (N, +, •) is a left-symmetric algebra (LSA), i.e. (N, +) is an F-linear space with a bilinear product (x, y) → xy := x • y satisfying (38) for all a, b, c ∈ N. Every BNA is an LSA.Moreover, is a Lie algebra, where [x, y] = xy − yx for any x, y ∈ N ( the associated Lie algebra of an LSA N).LSAs play a fundamental role in theory of affine manifolds (cf.[4]).Obviously, N L is abelian if and only if N is abelian.An algebra N satisfying (38) and modifying (37) for all a, b, c ∈ N, is called a fermionic BNA.A (nonassociative or associative) algebra A is called: semiprime if, for any ideal T of A, the condition T 2 = 0 implies that T = 0; prime if, for any ideals T, Q of A, the condition TQ = 0 implies that T = 0 or Q = 0; and simple if A 2 = 0 and its only ideals are 0 and A.
It is easy to see that every simple BNA is prime and every prime BNA is semiprime.The theory of BNAs was started by Zelmanov [89].He proved that a finite-dimensional simple BNA over an algebraically closed field of characteristic 0 is one-dimensional.Osborn [67] proved that for any finite-dimensional simple BNA N over a perfect field of characteristic p > 2, the associated Lie algebra N L is isomorphic to the rank-one Witt algebra.Simple BNAs have also been investigated by Osborn [67], Osborn and Zelmanov [68]and Xu [87].Many authors have investigated the Lie structure of BNAs.BNAs N with abelian (respectively nilpotent, solvable) associated Lie algebras N L have been studied by Burde and de Graaf [28], Burde, Dekimpe and Vercammen [27] and Burde and Dekimpe [26].The class of commutative associative algebras (CAA) equals the class of BNAs with abelian associated Lie algebras.CAAs (real and complex) of dimension 3 were characterized in [28] (see e.g. Baehr, Dimakis and Müller-Hoisson [5]) and Balinsky-Novikov C-algebras N of dimension 4 with the nilpotent associated Lie algebras N L were characterized in [7,8].In [4] it is proved that a complete LSA is always solvable.Recall that a BNA N is complete (or transitive) if the right multiplication operator r a is a nilpotent linear map for any a ∈ N. Burde [25] investigated a Lie algebra N L of a simple LSA N.
We investigate properties of semiprime BNAs (see Proposition 6.5) leading to a proof of the following result.
Theorem 4.1.Let N be a BNA.Then the following statements hold: (1) Suppose N is non-commutative and char F = 2.If N is simple (respectively prime), then it contains a commutative Lie ideal A that contains every commutative Lie ideal of N and N L /A is a simple (respectively prime) Lie algebra.
(2) If N L is a simple (respectively prime or semiprime) Lie algebra, then N is a simple (respectively prime or semiprime) BNA.
Let R(N) := {r a | a ∈ N} and L(N) := {l a | a ∈ N} and R(N) be the Lie algebra generated by R(N).If N is a fermionic BNA and (see [10,Claim1]).Let L(N) be the Lie algebra generated by all r x and l y , where x, y ∈ N. By [10, Claim 2]: Let AR(N) denote an associative algebra generated by R(N) (with respect to two operations: addition and composition of operators).Moreover, left multiplication operators of a BNA N forms a Lie algebra L(N) with respect to the pointwise addition "+"and the pointwise Lie multiplication "[−, −]"given by the rules [N, N] ⊆ lann N (and so N is at most 2-step nilpotent).We have the following result.Jacobson [48] initiated an investigation of (associative) multiplicative algebras of nonassociative finite-dimensional algebras A (see e.g.[39,42] and others).Bai and Meng [7] classified complete Balinsky-Novikov C-algebras with nilpotent associated algebras.Recall that an LSA A is right-nilpotent of length ≤ n, where n ≥ 1 is a fixed integer, if r a 1 r a 2 . . .r a n−1 (a n ) = 0 for all a 1 , a 2 , . . ., a n−1 , a n ∈ A. By [41,Theorem 2], a BNA N of bounded index over a field of characteristic 0 is nilpotent.It is also known [10, Corollary 1] that every fermionic BNA N is right-nilpotent because r 2 x = 0 and r x r y = −r y r x (114) for any x, y ∈ N. We shall prove that in any finite-dimensional fermionic BNA N over F of characteristic = 2, AR(N) is a finite-dimensional nilpotent associative algebra of NI ≤ 1 + dim F N (see Proposition

Elementary properties of fermionic BNAs
Here Proof.Assume that x, y ∈ N, z ∈ Z(N), i, t ∈ I and j ∈ J.
The proof of (iii) follows from x(ij) = (xi)j + i(xj) − (ix)j ∈ I J and (ij)x = −(ix)j ∈ I J, and (iv) Every ideal of a BNA N is a Lie ideal of N.An ideal B of N is called noncentral if BZ(N).To prove (ii), we note that if b ∈ I N (U) and N is a fermionic BNA, then bn, nb ∈ U and   For (iii), from (ab + ba)Z(N) = 0 and (ab − ba)Z(N) = 0, we obtain (2ab)Z(N) = 0 and the result follows.
Proof.Verification of (i), (iii), (v) and (vi) is straightforward.Proof.Let N be a fermionic BNA and a, x, y ∈ N. Then (see [10]) [r a , r y ] = 2r a r y , [r a , r x r y ] = 0, ( Proof.Let (e 1 , . . ., e m ) be a basis of the F-linear space N and r i := r e i (i = 1, . . ., m).
for some β ij ∈ F, where α l ∈ F and a j ∈ N (j = 1, . . ., t).If s is a positive integer, If s > m, there exists an integer p (1 ≤ p ≤ q) such that s p > 1, so there is an integer h Thus b s = 0.By [57,Theorem], AR(N) is a nilpotent algebra.Obviously AR(N) is finite-dimensional.
To prove (i), one can observe that from (115)-( 118) it follows that [r a r x , r y r z ](t) = r a r x r y r z (t) − r y r z r a r x (t) = 0.
and so implying [[X N , X N ], X N ] = 0.The rest holds in view of (113) and Lemmas 5.5, 5.6.

An additive subgroup
Lemma 6.2.Let N be a BNA, A its ideal and U its Lie ideal.Then: Hence bn ∈ I N (U), which proves (ii).

(i) the left annihilator lann
To prove (iii), assume that [u, v] = 0 for some u, v ∈ U. Then  (c) If A is a nonzero ideal of N and N L is semiprime, then A 2 = 0 because [A, A] ⊆ A 2 .Thus, the proof is complete.

Conclusion
We proved that an algorithm based on Lie-Poisson structure analysis on the adjoint space to toroidal Lie algebras allows us to construct new algebraic structures within which the corresponding Hamiltonian operators exist and generate integrable multicomponent and multidimensional dynamical systems.We also showed that the well-known Balinsky-Novikov algebraic structure, obtained as a condition for a matrix differential expression to be Hamiltonian, arises in our approach as a derivation on the Lie algebra, naturally associated with a differential loop algebra.We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky-Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds.Using the theory of nonassociative and associative left-symmetric algebras, we described algebraic properties of new Balinsky-Novikov type algebras, including their fermionic and important related multiplicative and commutative Lie versions.
for the Lie bracket [a, b] D := a • Db − b•Da for any a, b ∈ A h , where the map D := (∆ h − 1)/h for any h ∈ R\{0}.

Proposition 1 . 1 .
Let a smooth map D u : L A → L A be a skew-symmetric weak derivation of L A , where u ∈ L A L * A .Then ω 2 (a, b) := (a, D u b) (13) for any a, b ∈ L A and u ∈ L * A L A defines a nontrivial 2-cocycle on L A .
where for any a, b ∈ A l a (b) := a • b and r a (b) := b • a denote, respectively, the left and right shifts on the A. The commutator expressions (36) imposed on A coincide with those that determine the well-known Balinsky-Novikov algebra (BNA) by means of ..)dy , generates a weak Lie algebra structure iff the following hold[3] for arbitrary a, b ∈ A :

Example 1 . 9 .Remark 1 . 10 .
For the two-dimensional toroidal algebra A one can define for any a, b ∈ A the following new commutator structure [a, b] D := ∂a ∂x which generates a weak Lie algebra L A iff [r a , r b ] = 0 = [l a , l b ] (49) and [r a , l b ] = 0 (50) hold for any a, b ∈ A. Note that the similar to (51) Lie commutator structure [a, b] D ) for any a, b ∈ A, where, by definition, R a b := b • a is the right shift and L a b := a • b is the left shift on the algebra A. This, in particular, means that the canonical [1,16,17,38,74] Lie-Poisson bracket {( l, ã), ( l, b)} := ( l, [ ã, b] D )

[
a, b] D := (Da) • b − (Db)•a (87) and [a, b] D := a • Db − b•Da (88) for arbitrary a := a(x), b := b(x) ∈ A for any x ∈ R, where the map A h to the adjacent current Lie algebra L A h as a set of linear continuous functionals u : L A h → R on L A h via the expression u(a) := (u, a) s for some symmetric bilinear form (•, •) s on L A h , and to construct on it the canonical Lie-Poisson structure {u(a), u(b)} := u([a, b] D ) (96) for any linear functions u(a), u(b) ∈ D(L * A h ), a, b ∈ L A h , with arbitrary u ∈ L * A h , satisfying the Jacobi identity owing to (92) and (93).As the expression (96) can be rewritten as {u(a), u(b)} := (ϑ(u)a)(b) (97) for any u ∈ L * A h and a, b ∈ L A h , where the linear map ϑ(u) : L A h → L * A his called[16,17,38,43] a Hamiltonian operator.This operator makes it possible to construct, for any smooth functional γ ∈ D(A * h ), the Hamiltonian system du/dt = −ϑ(u)gradγ(u)

lLemma 3 . 2 .
ab = l a l b (104) and [l a , r b ] = 0. Recall that an algebra A is called a Riemann algebra if (xb)a = (xa)b (105) and x(ab) = (ab)x = (ba)x, (106) and a Balinsky-Novikov algebra if (ab)c = (ac)b and (ab)c − a(bc) = (ba)c − b(ac) for any a, b, x ∈ A. In the sequel, we shall denote the center of A by Z(A), the commutator by [a, b] = ab − ba and the commutator subgroup by [A, A].If A is an algebra, then the following statements hold: (i) if A satisfies (104), it is associative, (ii) if A satisfies (103) and (104), then [A, A] • A = 0 and A 2 ⊆ Z(A) (and so A is at most 2-step Lie nilpotent),

3. 1 .Lemma 3 . 4 .
A general Riemann type pre-Lie algebra structure Let (A, A) be the additive subgroup of A generated by Jordan commutators (a, b) := ab + ba, where a, b ∈ A. An associative algebra D is nilpotent if D n = 0 for some positive integer n; the least such n is called the nilpotency index (NI) of D. If A is a general Riemann type pre-Lie algebra over a field F satisfying r a r b = l ab = l a l b (110) 111) and (xb)a = r a (r b (x)) = l a (r b (x)) = a(bx), which imply that (ab)x = a(bx), so A is associative.Moreover, x(ba) = (xb)a = (ab)x and x(ab) = (xa)b = (ba)x which implies that (x, [a, b]) = 0.
(l a + l b )(x) = l a (x) + l b (x) and [l a , l b ](x) = l a (l b (x)) − l b (l a (x)) for all a, b, x ∈ N. A map δ : N L → N L is called a derivation of N L if δ(a + b) = δ(a) + δ(b) and δ([a, b]) = [δ(a), b] + [a, δ(b)]for all a, b ∈ N L .The set Der(N L ) of all derivations of N L is a Lie algebra over F.An LSA N is a derivation algebra if its left multiplications l x or its right multiplications r x are derivations[9] of the associated Lie algebra N L .By [9, Corollary 2.1] a BNA N is a derivation algebra if and only if

Proposition 4 . 2 .
Let N be a fermionic BNA.Then the following statements hold:(i) the Lie algebra L(N) = L(N) + X N is a sum of a subalgebra L(N)and an ideal X N := R(N) + R(N)R(N), where L(N) and N L / lann N are isomorphic, where lann N := {x ∈ N | xN = 0} and X N is at most 2-step Lie nilpotent, (ii) if L(N) ⊆ Der(N L ), then [N, N] ⊆ lann N and L(N) = L(N) + R(N) ⊆ Der(N L ).

Lemma 5 . 1 .
(a, b) := ab + ba for any a, b ∈ N and (V, V) is an additive subgroup of N generated by the set {(u, v) | u, v ∈ V}, where V ⊆ N. As usual, an additive subgroup A of an algebra N is called anideal if AN, N A ⊆ N.An additive subgroup U of a BNA N is a Lie ideal of N if [U, N] ⊆ U. Clearly,U is a Lie ideal of N if and only if U is an ideal of the associated Lie algebra N L .We shall need the following analogs of [27, Lemmas 2.1,2.2,2.7].Let N be a fermionic BNA, with ideals I and J. Then: (i) Z(N) is an ideal of N; (ii) if U is a Lie ideal of N, then Z(U) is the units; (iii) I J is an ideal of N; and (iv) if I is commutative, then I 2 ⊆ Z(N).

Lemma 5 . 2 .
If N is a fermionic BNA, with ideal A and Lie ideal U, then: (i) the left annihilator lannA := {n ∈ N | nA = 0} of A in N is an ideal of N; (ii) if char F = 2, then I N (U) := {u ∈ U | uN + Nu ⊆ U}is an ideal of N and I N (U) ⊆ U; and (iii) the centralizer C N (U) := {z ∈ N | zu = uz for any u ∈ U} of U in N is a Lie ideal of N. Proof.Assume that n, t, x ∈ N. If a ∈ A and b ∈ lann A, then (bn)a = −(ba)n = 0 and (nb)a = n(ba) − b(na) + (bn)a = (bn)a = −(ba)n = 0, which proves (i).

Lemma 5 . 6 .
If a, b ∈ B and r ∈ N, then a − b, [a, r] ∈ B and so l a − l b = l a−b , [l a , l r ] = l [a,r] ∈ L B (N), which proves (ii) Property (iv) follows from the fact that l a , l b ∈ Φ and l r ∈ L(N), then l a−b = l a − l b , l [a,r] = [l a , l r ] ∈ Φ and therefore a − b, [a, r]∈ ∆ Φ .If B is a left ideal of a fermionic BNA N, then R B (N) + R B (N)R(N) := {r b + ∑ x,t r t r x is a finite sum | b, t ∈ B, x ∈ N}is an ideal of the Lie algebra L(N) at most 2-step Lie nilpotent.

[Proposition 5 . 7 .
l x , r a r y ] = r a r xy + r xa r y .(118) If a ∈ B, then xa ∈ B and the assertion holds.If N is a finite-dimensional fermionic BNA over F, AR(N) is a finite-dimensional nilpotent algebra of NI ≤ 1 + dim F N.

( b )
Let N be a prime BNA and A, B be nonzero Lie ideals of N such that [A, B] = 0.If [A, A] = 0 and [B, B] = 0, then, by Lemma 6.2(iii), there exist non-central ideals A 0 and B 0 such that A 0 ⊆ A and B 0 ⊆ B and [A 0 , B 0 ] = 0, contradicting Lemma 6.3(i).Therefore, we assume that A is commutative.Let C := C N (A).If T(C) = N, then [N, N] ⊆ C,which leads to a contradiction in view of Lemma 6.3.Hence, T(C) is proper in N. If [T(C), T(C)] = 0, T(C) contains a non-central ideal I 0 of N by Lemma 6.3(iii) and so A ⊆ C N ([I 0 , N]).Since C N ([I 0 , N]) is an ideal of N by Lemma 6.1 and Lemma 6.2(viii), we obtain a contradiction in view of Lemma 6.3.Thus T(C) is commutative and C = T(C).Let K be an arbitrary Lie ideal ofN such that [K, K] = 0.If K ∩ A = 0, then K ⊆ C. Assume that K ∩ A is nonzero and C 1 := C N (K ∩ A).As above, C 1 = T(C 1 ) is commutative and therefore C 1 = C.Hence, N L /C is a prime Lie algebra.We prove(2) as follows: (a) Assume that N L is a simple Lie algebra.Then(2) follows because every ideal of N is its Lie ideal.(b) Let N L be a prime Lie algebra and A, B be ideals of N such that AB = 0. Since [A, B] ⊆ A ∩ B and [A ∩ B, A ∩ B] ⊆ AB, [A, B] = 0, A = 0 or B = 0.

c
2018 by the authors.Submitted to Journal Not Specified for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).