Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras

We review main differential-algebraic structures lying in background of analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and Leibniz type algebraic structures are derived, a new nonassociative “Riemann” algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we revisited the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.


INTRODUCTION
We present a short review of main differential-algebraic structures lying in background of analytical constructing multicomponent Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative and noncommutative algebras. During the last decades there were discovered [38,22,20,70] many integrable Hamiltonian systems, whose internal symmetyry structure was analytical nature was understood owing to the Lie-algebraic properties of their internal hidden symmetry structures. A first account of the Hamiltonian operators and related differential-algebraic structures, lying in the background of integrable systems, was given by I. Gelfand and I. Dorfman [43,34] and later was extended by B. Dubrovin and S. Novikov [36,37], and also by A. Balinsky and S. Novikov [11,14,12,13]. Also some new special differential-algebraic techniques [80] were devised for studying the Lax integrability and the structure of related Hamiltonian operators for a wide class of the Riemann type hydrodynamic hierarchies. Just recently considerable work [8,10,9,73] has been done devoted to the finite dimensional representations of the Balinsky-Novikov algebra. Their importance Hamiltonian operators and nonassociative noncommmutative algebras) 3 for constructing integrable multi-component nonlinear Camassa-Holm type dynamical systems on functional manifolds was demonstrated by I. Strachan and B. Szablikowski in [91], which in part suggested the Lie-algebraic imbedding of the Balinsky-Novikov algebra into the general Lie-Poisson orbits scheme of classifying Lax integrable Hamiltonian systems. It is also worth of mentioning the related work [46] by Holm and Ivanov in which integrable multicomponent nonlinear Camassa-Holm type dynamical systems on functional manifolds were constructed.
In our work here we describe a differential-algebraic reformulation of the classical Lie algebraic scheme and develop an effective approach to classification of the underlying algebraic structures of integrable multicomponent Hamiltonian systems. In particular, we have devised a simple algorithm allowing to construct new algebraic structures within which the corresponding Hamiltonian operators exist and generate integrable multicomponent dynamical systems. We show, as examples, that the well-known Balinsky-Novikov algebraic structure, obtained in [43,11] as a condition for a matrix differential expression to be Hamiltonian and in [19,27,50,61] as a flat torsion free left-invariant affine connection on affine manifolds, affine structures and convex homogeneous cones, appears in our approach as a derivation on the Lie-algebra naturally associated with a suitably constructed differential loop algebra. As a direct generalization of this example we obtain two new differentiations, whose underlying algebraic structures coincide, respectively, with the well-known [3,40] right Leibniz algebra, introduced in [23,24,59], and with a new "Riemann" algebra, which naturally generate different Hamiltonian operators describing a wide class of multicomponent hierarchies [21,80] of integrable Riemann hydrodynamic systems. As the compatible Hamiltonian operators, important for studying integrable multicomponent Hamiltonian systems, are constructed by means of suitable central extentions of the adjacent weak Lie algebras, determined by the right Leibniz and Riemann type nonassociative and noncommutative algebras, their description requires a detailed investigation both of the structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations.
In a supplement the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras, is briefly revisited. In particular, its natural and simple generalization appeared to be useful [5,6,94,96,62,66,67] for describing a wide class of Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, initiated first in [25,35,76,78] in case of the noncommutative operator algebras and continued later in [62,51,52,53,62,66,67,71] in case of general associative noncommutative algebras.

THE HAMILTONIAN OPERATORS AND RELATED ALGEBRAIC STRUCTURES VIA THE DIFFERENTIAL-ALGEBRAIC APPROACH
Let (A;˝) be a finite-dimensional algebra of dimension N = dim A P Z + (in general noncommutive and nonassociative) over an algebraically closed field K. Using the algebra A one can construct the related loop algebra r A of smooth mappings u : S 1 Ñ A and endow it with a suitably generalized natural convolution x¨,¨y on r A˚ˆr A Ñ K, where r A˚is the corresponding adjoint to r A space. First, we shall consider a general scheme of constructing nontrivial ultralocal and local [38] Poisson structures on the adjoint space r A˚, compatible with the internal multiplication in the loop algebra r A. Consider a basis te s P A : s = 1, N u of the algebra A and its dual tu s P A˚: s = 1, N u with respect to x¨,¨y on A˚ˆA, that is @ u j , e i D := δ j i for all i, j = 1, N , and such that for any u(x) = ř and let ϑ˚: r A^r A Ñ r A be a skew-symmetric bilinear mapping. Then the expression tu i (x), u j (x)u := xu(x), ϑ˚(e i^ej )y (2.1) defines for any x, y P S 1 and all i, j = 1, N an ultra-local linear skewsymmetric pre-Poisson bracket on r A˚. Since the algebra r A possesses its internal multiplicative structure "˝", the important problem arises: Under what conditions is the pre-Poisson bracket (2.1) Poisson and compatible with this internal structure on r A? To proceed with elucidating this question, we define a co-multiplication ∆ : r A˚Ñ r A˚b r A˚on any element u P r A˚by means of the relationship x∆(u), (a b b)y := xu, a˝by (2.2) for arbitrary a, b P r A. Note that the co-multiplication ∆ : r A˚Ñ r A˚b r A˚, defined this way, is a homomorphism of the algebra r A˚with respect to the natural multiplication of functionals, and the linear pre-Poisson structure t¨,¨u (2.1) on r A˚is called compatible with the multiplication "˝" on the algebra r A, if the following symbolic invariance condition ∆tu i (x), u j (x)u = t∆(u i (x)), ∆(u j (x))u (2.3) holds for any x P S 1 and all i, j = 1, N . Taking into account that multiplication in the algebra A is given for any i, j = 1, N by the condition where the quantities σ s ij P K for all i, j and k = 1, N are constants, the related co-multiplication ∆ : r A˚Ñ r A˚b r A˚acts on the basic functionals u s P r A˚, s = 1, N , as Additionally, if the mapping ϑ˚: r A^r A Ñ r A is given, for instance, in the simple linear form where quantities c s ij P K are constant for all i, j and s = 1, N , then for the adjoint to (2.6) mapping ϑ : Symm( r A˚) Ñ r A˚^r A˚one obtains the expression For the pre-Poisson bracket (2.1) to be a Poisson bracket on r A˚, it should must also satisfy the Jacobi identity. To find the corresponding additional constraints on the internal multiplication "˝" on the algebra r A, define for any u(x) P r A˚the skew-symmetric linear mapping where the skew-symmetric mapping ϑ˚: r In particular, if we assume that the coefficients c ks ij = σ k ij α s for some constant numbers σ k ij and α s P K for all i, j and k, s = 1, N , where then the pre-Poissson bracket (2.20) yields a very compact form  [14], which we shall not consider here. Now, let r A(u) Ă r A denote the polynomial differential ideal generated by an element u P r A and its derivatives D n x u P r A, n P Z + . The corresponding space of polynomial functions r A(u) Ñ K, constructed by means of some 8 Orest Artemovych, Alexander Balinsky, Anatolij Prykarpatski scalar form on r A(u), will be denoted by F r A (u). Then the basic ultra-local and linear, with respect to an independent element u(x) P r A, x P S 1 , pre-Poisson bracket (2.1) is easily generalized to a local pre-Poisson bracket for arbitrary functions f, g P F r A (u) : tf, gu(u) = xu(x), ϑ˚(∇f (u(x))^∇g(u(x)y , (2.23) in which the mapping is invariantly reduced on the subspace r A(u)^r A(u) and depends nontrivially on the derivation D x : r A Ñ r A. In (2.23) we denoted the usual linear gradient mapping from F Keeping in mind the problem of finding constraints on the multiplicative structure of the algebra r A under which the pre-Poisson bracket (2.23) is Poisson, it is very interesting to construct nontrivial examples of linear local pre-Poisson brackets on F r A (u), compatible with the multiplication "˝" on A and nontrivially depending on the usual differential operator D x : r A Ñ r A for x P S 1 . In particular, for arbitrary functions f, g P F r A (u) one can consider the following nontrivial and simplest linear local pre-Poisson bracket tf, gu(u) := xu(x), ϑ˚(∇f (u(x))^∇g(u(x))y , (2.24) where ϑ˚: for any x P S 1 , and some arbitrarily chosen constant quantities c s jk P K for all j, k and s = 1, N . If one also assumes that these constant quantities satisfy the condition (2.16), that is c s ij = σ s ij for all i, j and s = 1, N, the mapping (2.25) can be recast as Hamiltonian operators and nonassociative noncommmutative algebras) 9 providing the pre-Poisson bracket (2.24) for arbitrary functions f, g P F r A (u) with the canonical Lie-Poisson form which was recently presented in [91]. Thus, if the Lie structure for any a(x), b(x) P r A, x P S 1 , generates the weak adjacent Lie algebra L r A , the pre-Poisson bracket (2.27) will automatically be Poisson on the space F r A (u). Moreover, the expression (2.27), rewritten in the tensor form naturally defines a related bilinear form (¨,¨) on the weak adjacent Lie algebra L r A , allowing to determine the corresponding Hamiltonian operator ϑ(u) : L r A Ñ L r A , whose matrix representation with respect to the basis te s P A | s = 1, N u is for any elements a(x), b(x) and c(x) P r A, x P S 1 , then the pre-Poisson bracket (2.27) equivalent to (2.29), being of the Lie-Poisson type, will be a priori Poisson. As the second criterion is easier to check, after some simple calculations one obtains the well-known [11,43]  The next example of the bilinear, nonlocal (pseudodifferential) and weakly skew-symmetric mapping where D x D´1 x := 1 : r A Ñ r A is the identity mapping, generates the weak adjacent Lie algebra L for any a(x), b(x) P r A. It is easy to check that the commutator structure (2.34) satisfies the weak Jacobi identity (2.31) iff the multiplicative structure of the algebra A satisfies the so called [59] right Leibniz algebra constraints: ) for arbitrary elements a, b P A. The corresponding matrix integro-differential Hamiltonian operator on the space F r A (u) with respect to the basis te s P A | s = 1, N u for this case equals for any u(x) P r A˚, x P S 1 . Consider now the bilinear, nonlocal and weakly skew-symmetric mapping (2.37) which naturally generates the adjacent Lie algebra L Then it is easy to check that the commutator structure (2.38) satisfies the weak Jacobi identity (2.31), iff the following so called Riemann algebra A multiplicative structure holds for arbitrary elements a, b P A. For the related Hamiltonian operator on the functional space F r A (u) with respect to the basis te s P A | s = 1, N u one easily obtains from (2.37) the integro-differential expression Hamiltonian operators and nonassociative noncommmutative algebras) 11 for any u(x) P r A˚, x P S 1 . The above results can be reformulated as the following theorem. Thus, all the operators (2.30), (2.36) and (2.40) are Hamiltonian and a priori satisfy the Schouten-Nijenhuis condition (2.10), as easily follows from Theorem 2.4. It is also clear that in contrast to the simple Hamiltonian operator criterion formulated in this theorem, direct and very cumbersome checking of the Schouten-Nijenhuis condition as in [43] for the cases considered above, would be required for the multiplicative structures on the algebra A coinciding with (2.32), (2.35) and (2.39).

HAMILTONIAN OPERATORS AND RELATED ALGEBRAIC STRUCTURES VIA THE LIE-ALGEBRAIC APPROACH
Assume now that the loop algebra ( r A;˝) allows a weak adjacent Lie algebra extension (L Owing to the construction [1,4,20,17,38], the Lie-Poisson bracket (3.4) satisfies a priori the classical Jacobi identity, and it can serve as a very powerful tool for constructing the related Hamiltonian operators on the functional space F r A (u). In particular, following [43,68], determined for any f, g P F r A (u), satisfies the Jacobi identity. Taking into account that the canonical Lie-Poisson bracket (3.4) depends essentially on the loop Lie algebra structure of L r A , we proceed further to extending the Lie algebra structure on L r A by means of the standard [38] central extension technique. Namely, letL for any a, b P L r A and α, β P K, where the 2-cocycle ϖ 2 : L r AˆL r A Ñ K is a skew-symmetric bilinear form satisfying the Jacobi identity: for any a, b and c P L r A . It is evident that the existence of nontrivial central extensions on the Lie algebra L r A strongly depends on the algebraic structure of the algebra A underlying the whole construction presented above. Yet there exist some general algebraic properties which allow to proceed further with success. For example, assume that a smooth mapping for any a, b and c P L  for any a, b P L r A and u P Lr A proof is given by means of direct checking the Jacobi identity (3.5) and is omitted. A in the following scalar form: where ϑ˚: r A^r A Ñ r A is some bilinear skew-symmetric mapping.
As follows from the results of Section (2) the right-hand side of expression (3.8) allows the equivalent form where the bracket [a, b]D defines for any a, b P r A a new adjacent Lie algebra structure on the loop algebra r A, a priori compatible with the basic Lie structure [¨,¨] D . In the case when these Lie structures coincide, that is for any f, g P F r A (u). Moreover, as follows from (3.9) and the ad-invariance property (3.1), the mapping D u (¨) =´[u,¨] D for any u P Lr A is automatically a derivation of the weak adjacent Lie algebra L r A . Example 3.4. As a natural example of the derivation D u : L r A Ñ L r A one can take the mapping denote the convolution operators of the co-multiplication ∆ : r A˚Ñ r A˚b r A˚with respect to its first and second tensor components, respectively. In particular, we have Orest Artemovych, Alexander Balinsky, Anatolij Prykarpatski as we have assumed by definition, that (u i , e j ) := δ i j , i, j = 1, N . If now the weak Lie algebra L r A is generated by the commutator Lie structure (3.4 for any a, b P L r A , it easy to check that the mapping (3.11) is a skewsymmetric with respect to the bilinear form (¨,¨) on r A weak derivation of the Lie algebra L r A . Moreover, the related weak Lie algebraic structure There are also other strictly algebraic tools for constructing Poisson brackets on the functional space F r A (u). For instance, as a simple consequence of Proposition 3.2 the following result [34,69,89] holds.

Proposition 3.5. Suppose that a nondegenerate linear skew-symmetric
for any a, b and c P L For any function f P F r A (u) consider its differential δf P Λ 1 ( r A(u)) and define for a chosen element h P r A(u)˚the vector field ξ h P Γ( r A(u)) via the equality δf (ξ h ) = (∇f (u), h).
As symplectic forms on the phase space r A(u) are dual objects to the Poisson brackets on the functional space F r A (u), one easily obtains the following [69,85,89] proposition.
for any a, b and c P L r A . Then differential 2-forms where is also a Hamiltonian operator.
for any a, b and c P L r A . A proof of the second part of the proposition consists in direct checking the closedness of the 2-forms ω (2) 2 P Λ 2 ( r A(u)), which is equivalent to the Yang-Baxter condition (3.15).
Concerning their compatibility, we observe that the Hamiltonian operator ϑ 2 : L r A Ñ L r A , corresponding to the expression (3.17), is representable as the composition ϑ 2 = ϑ 1 (ϑ´1 0 ϑ 1 ), where the Hamiltonian ope- . From this representation one easily derives [22,20,69,43,89] the compatibility of the Hamiltonian operators ϑ 2 and ϑ 1 on r A(u), following from the evident compatibility of operators ϑ 0 and ϑ 1 on r A(u), owing to 2-cocycle property of the bilinear form (3.14).
for any x P S 1 , which satisfies the weak Yang-Baxter commutator condition (3.13). Then it is easy to check that the inverse mapping R´1 = d/dx, x P S 1 , is the natural skew-symmetric derivation of the weak Lie algebra L r A , generating a Poisson structure compatible with that of (3.10).
In a manner similar to the above [69,57,85,89] one verifies the existence of the following so called "quadratic" Poisson brackets. Namely, the next proposition holds.
defined for any f, g P F r A (u), are Poisson and compatible on r A(u).
As an interesting and also useful consequence of the R-matrix construction, one has the fact that the following subspaces are Lie subalgebras of L r A , which is equivalent to the condition that the mappings are homomorphisms [89] of the Lie algebras L for all u P r A˚and arbitrary functions f, g P F r A (u). Moreover, from the analysis provided above we know that if the Hamiltonian operator (2.29) corresponds to some 2-cocycle on the Lie algebra L r A , then it will be a priori Hamiltonian. Moreover, owing to and to the related multiplicative Balinsky-Novikov algebra structure (2.32) on A for any a, b and c P A. Similarly, the skew-symmetric structure (2.34) and to the related multiplicative right Leibniz algebra structure (2.32) on A Moreover, the skew-symmetric structure (2.34) for any a, b P L r A on the weak adjacent Lie algebra L r A gives rise to the Hamiltonian operator (2.38) on r A(u) and to the related multiplicative Riemann algebra structure (2.39) on A for all a, b, c P A.

Remark 4.2.
Just as in Section 2, one can construct for all a, b, c P A dual Balinsky-Novikov for any a, b P L r A . As mentioned above, simultaneously we have shown that the expressions (4.1), (4.3) and (4.4) are true Hamiltonian operators on the functional space F r A (u) satisfying the Schouten-Nijenhuis condition (2.10). Using the algebraic scheme in [91] and the right Leibniz algebra (4.3) and the new Riemann algebra (4.4), one can describe a wide class of multicomponent completely integrable dynamical systems containing, as follows from the recent results in [80], infinite hierarchies of the multicomponent hydrodynamical Riemann type systems.
As the expressions (2.30), (2.36) and (2.40) are true Hamiltonian operators on the functional space F r A (u) satisfying the Schouten-Nijenhuis condition (2.10), following the algebraic scheme of [91] mentioned above and using the results of [80] and the right Leibniz algebra (2.35) and the new Riemann algebra (2.39), one can describe a wide class of multicomponent completely integrable dynamical systems containing the infinite hierarchies of multicomponent Riemann hydrodynamical flows. For instance, consider the generalized completely integrable Riemann type dynamical system on the functional space r A(u) for some nonassociative and noncommutative finite-dimensional algebra A, where x P S 1 , N P Z + , which was recently studied in detail in [77,80]. , , on the space F r A (u) for the Riemann type dynamical system (4.6), whose Hamiltonian representation holds for the Hamiltonian function H 2 P F r A (u), equal to Proceeding similarly for the case N = 3, one easily obtains from (2.40) the skew-symmetric three-dimensional matrix Hamiltonian operator coinciding, modulo the trivial constant 2-cocycle determined for a suitable symmetric bilinear form f 3 : r Aˆr A Ñ K and all a, b P L r A with the Hamiltonian operator for the Riemann dynamical system (4.6), whose Hamiltonian representation holds for a suitably constructed Hamiltonian function H 3 P F r A (u). There is also an interesting observation concerning an infinite hierarchy [80] of the generalized Riemann hydrodynamic systems on the functional space F r A (u), where s, N P Z + , with the algebra A generated by the constraints (2.39). For the case s = 2 and N = 3 the above skew-symmetric three-dimensional matrix Hamiltonian operator on the functional space F r A (u) for the Riemann type dynamical system (4.7), whose Hamiltonian representation  (a, b), determined for any a, b P L r A by means of two suitably symmetric f bilinear forms, which naturally generates the (4.8)-compatible Hamiltonian operatorθ holds for the Hamiltonian function H It is worth noting here, as was already remarked in [81], that the generalized Riemann hydrodynamic system (4.7) for s = 3, N = 3 reduces to the well-known integrable Degasperis-Processi dynamical system [32,31] for the function u := u 1 : Also, for s = 2 and N = 3, the system (4.7) for the function u := u 1 reduces to the well-known [29] integrable Camassa-Holm dynamical system u t´uxxt + 3uu x´2 u x u xx + uu xxx = 0, whose multicomponent extensions were recently extensively studied in [39,46,30,91]. Now, returning to the case N = 2 of the system (4.6), it reduces under the substitutions u 1 := u, u 2 := D´1 x u 2 x /2 to the well-known [47,44,74, 79] Hunter-Saxton nonlinear dynamical system Orest Artemovych, Alexander Balinsky, Anatolij Prykarpatski on the functional manifold r A(u), u P r A˚, describing propagation of shortwaves in a relaxing medium with spatial memory effects. As shown in [74,79,75], the dynamical system (4.9) is a completely integrable bi-Hamiltonian flow on the functional manifold r A(u), u P r A˚, with respect to the compatible pair of scalar Hamiltonian operators As we are interested in the corresponding multicomponent generalization of the dynamical system (4.9), we need to consider the functional space r A(u), u P r A˚, generated by a finite-dimensional noncommutive and nonassociative algebra A, and construct the Poisson operators on F r A (u) in the form (2.36), related to the right Leibniz algebra structure (2.35) and reducing at N = 1 to the scalar Hamiltonian operator ϑ 2 (u) : T˚( r A(u)) Ñ T ( r A(u)) from the pair (4.10).
Moreover, as the compatible Hamiltonian operators are generated by means of suitable central extentions of the adjacent weak Lie algebra, the problem of description them, as was noted above, requires a detailed investigation of the structural properties and finite-dimensional representations of the right Leibniz algebras defined by the constraints (2.35). In what will follow, we stop mostly on the structural properties of the right Leibniz algebras, defined by the constraints (2.35), in particular, we characterize in detail the related derivation algebras.

PRELIMINARY ALGEBRAIC SETTING
An algebra (L, +,¨) over a field K is called a (right) Leibniz algebra if it satisfies the identity x(yz) = (xy)z´(xz)y for any x, y, z P L. Any Lie algebra is clearly a Leibniz algebra.
Leibniz algebras were introduced by A.M. Bloh [23,24] and rediscovered by J.-L. Loday [59]. Recall that a subalgebra H Ď L is said to be an ideal InnL := tr a | a P Lu of all inner derivations of L is an ideal of the Lie ring DerL (see e.g. [41]. A general theory for inner derivations in nonassociative algebras is given in [88]. If L is a Leibniz algebra, then Leib(L) := spatx 2 | x P Lu is the smallest ideal of L such that the quotient algebra L/Leib(L) is a Lie algebra (see e.g. [15,33] A linear mapping F : L Ñ L is called a generalized derivation of a Leibniz algebra L associated with a derivation δ P DerL (in the sense of Brežar [26] if F (xy) = F (x)y + xδ(y) for all x, y P L. Denote by GDerL the set of all generalized derivations of L. We will write (F, δ) P GDerL if and only if F is a generalized derivation of L associated with δ P DerL. Since (δ, δ) P GDerL for any δ P DerL, one concludes that InnL Ď DerL Ď GDerL.
A generalized derivation F of L that is associated with an inner derivation r a P InnL is called a generalized inner derivation of L. In what follows, let D = DerL, G = GDerL, ∆ be a nonempty subset of D (respectively G). If I is an ideal of L and δ(I) Ď I for all δ P ∆, then I is called a ∆-ideal of L. Inasmuch (x + d(x))(x + d(x)) P Leib(L) for any x P L and d P D, we deduce that d(x 2 ) P Leib(L) and so Leib(L) is a D-ideal of L. For a Leibniz algebra (L, +,¨), define the derived sequence as follows: A Leibniz algebra L is called nilpotent if there exists a positive integer s such that L (s) = 0 (see e.g. [2,33,42,16]). For a (Lie or Leibniz) algebra and it only has the following X-ideals: 0, Y and A (here 0 and Y are not necessarily different), ‚ X-primary if, for any X-ideals T, Q of A, the condition T Q Ď Y implies that T Ď Y or Q m Ď Y for some positive integer m.
In particular, a X-semisimple (respectively X-prime, X-simple or Xprimary) Lie (or Leibniz) algebra L is called semisimple (respectively prime, simple or primary) if X = 0 and a 0-ideal is an ideal of L. A Leibniz algebra L is semisimple (respectively prime, simple or primary) if and only if the Lie algebra L/Leib(L) is the ones. If L is a simple Leibniz algebra, then L/Leib(L) is a simple Lie algebra, but the opposite is not true. It is not difficult to check that if a Leibniz algebra L is prime (respectively semisimple or simple), then rannL = Leib(L). Every semisimple (respectively prime, simple or primary) Leibniz algebra is D-semisimple (respectively D-prime, D-simple or D-primary).
The Leibniz algebras are very popular in physics. Many authors have investigated derivations of Leibniz algebras in the context of geometric study of algebras (see e.g. [54,72,83,82,84]) and representations of Leibniz algebras (see e.g. [42,41,60,63] [54] have proved that a finite-dimensional Leibniz algebra L with a nonsingular derivation is nilpotent, I. S. Rakhimov, K. K. Masutova and B. A. Omiro [83] have proved, in particular, that any derivation of a simple finite-dimensional Leibniz algebra over a field of zero characteristic can be represented as sum of three derivations of special form. In this paper we study connections between Leibniz algebras L, their derivation algebras DerL and generalized derivation algebras GDerL. Our first result subject to these topics is the following proposition which will be proved in Section 6: We also prove an analogue of the result of S. Tôgô [93] that is a finitedimensional Leibniz algebra L such that L ‰ L 2 and Z(L) ‰ 0 has an outer derivation (see Proposition 6.5 below).
Obviously, a finite-dimensional Leibniz algebra L is semisimple if its maximal solvable ideal is equal to Leib(L). Semisimple Leibniz algebras have studied in [2,33,45,83] and others. In this way we prove in Section 8 the next result. Moreover, ML is an ideal of the Lie ring GDerL. In Section 8 we will prove the following Theorem 5.3. Let L be a Leibniz algebra. Then the following hold: (1) if DerL/ADerL is a semisimple (respectively prime, simple or primary) Lie algebra, then L/rannL is a G-semiprime (respectively G-prime, Gsimple or G-primary) Lie algebra, (2) if InnL/AInnL, where AInnL = ADerL X InnL, is a semisimple (respectively prime, simple or primary) Lie algebra, then L/rannL is a semisimple (respectively prime, simple or primary) Lie algebra.

PROPERTIES OF DERIVATION ALGEBRAS
At first we will present some information about derivation algebras. If A Ď L, then Inn A L := tr a | a P Au and, in particular, InnL = Inn L L.
Lemma 6.1. Let L be a Leibniz algebra, A its ideal. Then the following hold: Proof. By routine calculations. □ Lemma 6.2. Let A be a Leibniz algebra and Φ an ideal of D. Then we have: Proof. (i) Immediately.

Corollary 6.3. Let L be a Leibniz algebra. Then InnL is a simple (respectively prime, semisimple or primary) Lie algebra if and only if L is a simple (respectively prime, semisimple or primary) Leibniz algebra.
Proof of Proposition 5.1. (1)ñ(2). Let D be a simple Lie algebra. If InnL = 0, then L 2 = 0 and any endomorphism of the additive group L + is a derivation of L. If p is a prime, then (2)ñ(1). By Corollary 6.3, InnL is a simple Lie algebra. Consider three cases. (iv) IGDerL = ML + InnL, where ML is an ideal of IGDerL, and Proof. Assume that (F, δ), (K, d) P GDerL, T P ML and x, y P L. (δ´F )(xy) = δ(x)y + xδ(y)´F (x)y´xδ(y) = (δ´F )(x)y it follows that δ´F P ML. If g P DerL X ML, then g(x)y = g(xy) = g(x)y + xg(y).
(iv) By the same argument as in the part (iii).
(ii) Since F (xy) = F (x)y, we deduce that F P ML.
(iii) We have DerL Ď GDerL and therefore the assertion holds. Proof. It is easy to check by using the same argument as in the proof of Lemma 7.2. □ Let Φ Ď GDerL, Γ Ď DerL,  we conclude that a´b, δ(a), ta, at P Σ Φ and therefore Σ Φ is an ideal of L.
(2)-(3) By the same argument as in the part (1). □ Lemma 7.6. Let L be a Leibniz algebra and A its ideal. Then the following conditions are equivalent: Since ∆ Λ and ∆ Φ are D-ideals by Lemma 6.1(ix), we deduce that by the D-primeness of L. This gives that Λ = 0 or Φ = 0 by Lemma 6.1(iii). As a consequence of Lemma 6.2(iii), the quotient Lie algebra D/ADerL is prime.
(b) If L is a D-semisimple Leibniz algebra, then we can obtain that D/ADerL is semisimple analogously as in (a).
(c) Assume that L is a D-simple Leibniz algebra and Ψ is an ideal of D. Then rannL = Leib(L). If Φ := Ψ X InnL, then ∆ Φ is a D-ideal of L and so ∆ Φ Ď rannL. By Lemma 6.1(iii), Φ = Inn ∆ Φ L = 0 and Ψ Ď ADerL by Lemma 6.1(iii). Thus D/ADerL is a simple Lie algebra. (1) if GDerL/ML is a semisimple (respectively prime, simple or primary) Lie algebra, then L/rannL is a G-semiprime (respectively G-prime, G-simple or G-primary) Lie algebra; (2) if IGDerL/ML is a semisimple (respectively prime, simple or primary) Lie algebra, then L/rannL is a semiprime (respectively prime, simple or primary) Lie algebra.
Proof. (1a) Assume that GDerL/ML is a prime Lie algebra and A, B are G-ideals of L such that AB Ď rannL. Then Then the derivation condition (9.4) can be equivalently rewritten [1,68,20,70,78] as the strong Lie derivative along the vector field K(u) = ϑ(u)φ(u) P T (M ) at any u P M for all "selfadjoint" elements φ P T˚(M ). Equivalently, a given linear skew-symmetric operator ϑ(u) : T˚(M ) Ñ T (M ), u P M , is Hamiltonian iff the Lie derivative (9.5) vanishes for all "self-adjoint" elements φ P T˚(M ). Moreover, as was observed in [64], it suffices to check the condition (9.5) only on the subspace of elements φ P T˚(M ) satisfying the condition φ 1 (u) = 0 for any u P M .
As an example, one can check that a skew-symmetric matrix-differential operator on M of the form Similarly, one can verify that the skew-symmetric inverse-differential operator σ s ij e s holds for any i, j = 1, n. The skew-symmetric inverse-differential operator (9.7) can be naturally generalized to the expression which can be rewritten as The condition (9.5) for the operator (9.8) to be Hamiltonian reduces to the constraints on the related nonassociative algebra A : =span K te j : j = 1, nu exactly coinciding with that of (4.4), and analyzed in some detail in Section 3.
As it was already mentioned, based on the matrix representations of the right Leibniz algebra (4.3) and the new nonassociative Riemann algebra (4.5), one can construct many nontrivial Hamiltonian operators on the associated weak Lie algebra L r A , related with diverse types of nonassociative algebras A. These Hamiltonian operators prove to be very useful [21,80,81] for describing a wide class of multicomponent hierarchies of integrable Riemann type hydrodynamic systems and their various physically reasonable reductions. 9.2. Poisson structures on manifolds generated by associative noncommutative algebras. Proceed now to a slightly generalized construction of Hamiltonian operators on a phase space, generated by associative noncommutative algebra A-valued matrices, which was first studied in [25,35,76,78] in case of the noncommutative operator algebras and continued later in [62,51,52,53,62,66,67,71] in case of general associative noncommutative algebras. This natural and simple generalization appeared to be very useful [5,6,94,96,62,66,67] for describing a wide class of new Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, interesting for diverse applications in modern quantum physics.
We start here with a free associative noncommutative algebra A = Kxu 1 , u 2 , . . . , u m y, generated by a finite set of elements tu j P A : j = 1, mu, and define its "abelianization" A 6 := A/[A, A] and the projection π : A Ñ A 6 , where [A, A] := tuv´vu P A : u, v P Au. Consider now a naturally related with A n-dimensional matrix Lie algebra G := gl(n; A) over the field K with entries in A subject to the usual matrix commutator [a | b] := ab´ba for all a, b P G. Being first interested in the Lie-algebraic studying [22,20,38,85] of co-adjont orbits on the adjoint space G˚, let us construct a bi-linear form x¨|¨y : GˆG Ñ A 6 on the Lie algebra G by means of the trace-type expression for any a, b P G. The following important lemma holds.
Lemma 9.3. The bilinear form (9.9) on G is symmetric, nondegenerate and ad-invariant.
Proof. Symmetricity. We have: for any a, b P G. Nondegeneracy. Assume that xa | by = 0 6 P A 6 for a fixed a P G and all b P G. To state that a = 0, let us put then b = a and obtain xa | ay = ÿ i,j=1,n π(a ij a ij ) = 0 6 .
Taking into account that the associative algebra is generated by the finite set of elements tu j P A | j = 1, mu, it is easy to deduce from n 2 expansions of elements from A that the sum ÿ k=1,n π(c k c k ) = 0 6 (9.11) iff c k = 0 for all k = 1, n 2 . Indeed, the sum of (9.11) under the π-mapping can be now rewritten, respectively, as  with some D-coefficients from K for all σ j P S n , depending quadratically on coefficients of expansions, staying at uniform and symmetric basis elements of the algebra A. As the π-mapping sends all of them, by definition, to zero, the resulting system (9.11)  Proof. Really, from the symmetry property (9.10) one easily obtains that (9.12) modulo π-mapping for any elements a, b and c P G. As the bilinear form (9.9) is non-degenerate, one has G˚» G, that jointly with the ad-invariance property (9.12) means that the Lie algebra G is metrized. □ Being interested in constructing integrable noncommutative dynamical systems on the algebra A, we need to introduce into our analysis a "spectral" parameter λ P C, responsible for the existence of infinite hierarchies of the corresponding dynamical systems invariants, guaranteeing their integrability. This wil be done in next Section, devoted to the Lie-algebraic analysis on loop-Lie-algebras, related with the Lie algebra G, introduced above.
Consider now the Lie algebra tG, [¨,¨]u, constructed above, and the related loop Lie algebra of the corresponding G-valued Laurent series with respect to the parameter λ P C,G and define on it the corresponding to (9.9) modulo π-mapping bilinear form (¨|¨) :GˆG Ñ A: (ã |b) := res λ xma |by (9.13) for any elementsã,b PG. It is easy to observe that the bilinear form (9.13) is also symmetric and non-degenerate. Thus, the following proposition holds.
As the loop Lie algebraG allows natural direct sum splittingG =G + 'Gí nto two Lie subalgebrasG + andG´, wherẽ their adjoint spaces with respect to the bilinear form (9.13)) split the adjoint loop spaceG˚=G+ 'G˚and satisfy the equivalences Let now a linear endomorphism R :G ÑG equals R = (P +´P´) /2, where, by definitions, P˘:G ÑG˘ĂG are the projections on the corresponding subspacesG˘ĂG. It is a well known property [22,20,38,85] that for anỹ a,b PG the deformed Lie product satisfies the Jacobi condition and generates on the loop Lie algebraG a new Lie algebra structure. Within the classical Adler-Kostant-Symes Lie-algebraic approach, or its R-matrix structure generalization [22,20,38,85], the adjoint loop spacẽ G˚is then endowed with the modified Lie-Poisson structure tl(ã),l(b)u 6 for any basic functionalsl(ã),l(b) P D(G˚) subject to which the whole set ,ã]) = 0 6 ,ã PG˚( of smooth Casimir functionals onG˚is commutative with respect to the deformed Lie-Poisson structure (9.14) onG˚, that is tγ, µu 6 = 0 6 P A 6 for all γ, µ P I(G˚) and, by definition, The latter makes it possible to construct integrable Hamiltonian flows on the associative algebra A as Poissonian flows on the co-adjoint orbits on the adjoint spaceG˚, generated by a suitable loop Lie algebraG of Casimir gradient elements. Namely, if an elementl PG˚is fixed, the corresponding Hamiltonian flow onG˚subject to the deformed Poisson bracket (9.14) and a Casimir functional γ P I(G˚) possesses the well known Lax type [55,65,85] representation dl/dt = [P + gradγ(l),l], (9.15) where t P K is a related evolution parameter. The example of this construction and its Lie algebraic properties are discussed in the next Subsection.
9.6. Kontsevich type integrable systems on unital finitely generated free associative noncommutative algebras. Let a free unital finitely generated associative non-commutative algebra A := Kxu˘, v˘y be the corresponding group algebra of a group Gtu, vu, generated by two elements u, v P G. The algebra A is infinite dimensional with the countable basis following the result obtained in [96]. As a first important task, we will calculate the corresponding Poisson structure on the related A-algebra valued phase space M (0 | 2) A (l), generated by coefficients, presented in the expression (9.17). To do this, we need to take into account that the phase space M (0 | 2) A (l), being endowed with the R-modified Poisson structure (9.14), is strongly reduced via the Dirac scheme [38,78] subject to the set one can obtain a complete set of π-commuting to each other conservation laws of the Kontsevich dynamical system (9.18), thus proving its generalized integrability. Moreover, choosing both another group algebra and orbit elementsl PG˚, one can construct the same way many other integrable Hamiltonian systems on the associative noncommutative phase space A, that is planned to be a topic of a next investigation.

CONCLUSION
In this work we succeeded in formal tensor and differential-algebraic reformulating the criteria [43,90,64] for a given differential expression to be Hamiltonian and developed an effective approach to classification of the algebraic Poisson structures lying in the background of the integrable multicomponent Hamiltonian systems. We have devised a simple algorithm allowing to construct new algebraic structures within which the corresponding Hamiltonian operators exist and generate integrable multicomponent dynamical systems. We also showed, as examples, that the well known Balinsky-Novikov algebraic structure, obtained before in [43,11] as a condition for a matrix differential expression to be Hamiltonian, appears within the devised approach as a derivation on the adjacent Lie algebra, naturally associated with a suitably constructed differential loop algebra. By means of a direct generalization of this example it is obtained new Lie algebraic relationships, whose background algebraic structures coincide, respectively, with the right Leibniz algebra, introduced in [23,24,59] and with a new Riemann type nonassociative algebra. The constructed Hamiltonian operators describe a wide class of multi-component hierarchies [21,80] of integrable multicomponent hydrodynamic Riemann type systems. Their reductions appeared to be closely related both to the integrable Camassa-Holm and with the Degasperis-Processi dynamical systems, and are of special interest from the equivalence transformation point of view, devised recently in [95].
Taking into account that the compatible Hamiltonian operators, important for studying integrable multicomponent Hamiltonian systems on functional manifolds, are constructed by means of suitable central extentions of the adjacent weak Lie algebras, determined by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. Subject to these important aspects we stopped in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We added also a Supplement in which we revisited the classical Poisson manifolds approach to Hamiltonian operators on functional noncommutative manifolds, as well as presented it simple and natural realization, generated by associative noncommutative group algebra. The latter appeared to be very useful for describing a wide class of new Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, interesting for diverse applications in modern quantum physics.