Non-associative structures of commutative algebras related with quadratic Poisson brackets

There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson brackets of Lie–Poisson type, examples of Balinsky–Novikov and Leibniz algebras are discussed. The non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie, and Zinbiel algebras are studied in detail.

structed Casimir invariants on the adjoint space to the corresponding symmetry Lie algebra. A first formal account of the related Hamiltonian operators and differentialalgebraic structures, lying in the background of such integrable systems, was given by Gelfand and Dorfman [19,26] and later was extended by Dubrovin and Novikov [20,21], and also by Balinsky and Novikov [10][11][12][13]. There were also devised some new special differential-algebraic techniques [41] for studying the Lax type integrability and the structure of related Hamiltonian operators for a wide class of Riemann type hydrodynamic hierarchies. Just recently a lot of works appeared [7][8][9]40] being devoted to the finite-dimensional representations of the Novikov algebra. Their importance for constructing integrable multi-component nonlinear Camassa-Holm type dynamical systems on functional manifolds was demonstrated by Strachan and Szablikowski in [47], where there was suggested in part the Lie-algebraic imbedding of the Novikov algebra into the general Lie-Poisson orbits scheme of classification Lax type integrable Hamiltonian systems.
In the work we succeeded in formal differential-algebraic reformulating the classical Lie algebraic scheme and developed an effective approach to classification of algebraic structures lying in the background of integrable multicomponent Hamiltonian systems. We have devised a simple Lie-algebraic algorithm, allowing to construct new algebraic structures within which the corresponding linear and quadratic Hamiltonian operators, generated by the corresponding Lie-Poisson structure on the co-adjoint orbits, exist and describe the related integrable multicomponent dynamical systems. In these cases an interesting problem of describing the Balinsky-Novikov and Leibniz type algebras, whose multiplicative structures satisfy some additional tensor r -structure type relationships naturally arises. We studied also the non-associative structures of commutative algebras related with Balinsky-Novikov, Leibniz, Lie and Zinbiel algebras, having diverse important applications both in theory of integrable dynamical systems and to modern problems of communication technology.

Quadratic Poisson brackets: compatibility and related algebraic structures
Let (A, +, • ) be a finite-dimensional non-associative and non-commutative algebra of dimension N = dim A ∈ Z + over an algebraically closed field K. To the algebra A one can naturally relate the loop algebra A of smooth mappings u : S 1 → A and endow it with a suitably generalized natural convolution · , · on A * × A → K, where A * is the corresponding adjoint to the space A. First, we shall consider a general scheme of constructing non-trivial ultra-local and local [24] quadratic Poisson structures [11,12,14,45] on the loop space A, compatible with the internal multiplication in the algebra A. Namely, let {e s ∈ A : s = 1, N } be a basis of the algebra A and its dual {u s ∈ A * : s = 1, N } with respect to · , · on A * × A, that is u j , e i . . = δ j i for all i, j = 1, N , and such that for any u(x) = N s=1 u s (x; u)e s ∈ A, x ∈ S 1 , the quantities u s (x; u) . . = u s (x), u ∈ K for all s = 1, N , x ∈ S 1 . Denote by A * ∧ A * . . = Skew( A * ⊗ A * ) and let ϑ * : A * ∧ A * → Symm( A * ⊗ A * ) be a skewsymmetric bilinear mapping. Then for linear on A functions a(u) . . = a, u and b(u) . . = b, u for any a, b ∈ A * the expression defines an ultra-local quadratic skew-symmetric pre-Poisson bracket on A * . Since the algebra A possesses its internal multiplicative structure " • , the important problem [11,12] arises: Under what conditions is the pre-Poisson bracket (1) Poisson and compatible with this internal structure on A? To proceed with elucidating this question, we define a co-multiplication : A * → A * ⊗ A * on an arbitrary element c ∈ A * by means of the relationship for arbitrary w, v ∈ A. Note that the co-multiplication : A * → A * ⊗ A * , defined this way, is a homomorphism of the tensor algebra T 1 ( A * ) into T 2 ( A * ) and the linear pre-Poisson structure { ·, ·} on A * [see (1)] is called compatible with the multiplication " • " on the algebra A, if the following invariance condition: holds for all a, b ∈ A * and arbitrary u ∈ A. Now, taking into account that multiplication in the algebra A can be represented for any i, j = 1, N by means of the relationship where the quantities σ s i j ∈ K for all i, j and k = 1, N are constants, the comultiplication : A * → A * ⊗ A * acts on the basic functionals u s ∈ A * , s = 1, N , as Additionally, if the mapping ϑ * : A * ∧ A * → Symm( A * ⊗ A * ) is given, for instance, in the simple linear form the quantities c i j sk ∈ K are constant for all i, j and s, k = 1, N and chosen to be symmetric in their lower indices, then for the adjoint to (5) mapping ϑ : Recall that a linear mapping ϑ : A → B from an algebra A to the A-bimodule B is called a derivation if for any λ, μ ∈ A there holds the Leibniz property The following theorem [11] gives an effective compatibility criterion for the multiplication in the algebra A.
Proof The idea of the proof consists in checking the relationships on the corresponding coefficients following both from (2) and (6) for basis elements λ, μ ∈ Symm( A ⊗ A).
Observe now that the pre-Poisson bracket (1) can be equivalently rewritten as giving rise, owing to the arbitrariness of elements a, b ∈ A * , to the following tensor equality: with the derivation ϑ. As was remarked in [11,12], the following natural commutator expression: for any λ ∈ Symm( A ⊗ A) and a fixed skew-symmetric constant tensor r ∈ A ⊗ A is an inner derivation of the algebra Symm( A ⊗ A). Thus, one can consider a class of pre-Poisson brackets (7) in the following commutator tensor form: and pose a problem of finding conditions on the tensor r ∈ A ⊗ A under which the pre-Poisson bracket (8) becomes a Poisson one.
If the algebra A is non-commutative and associative, the adjacent Lie algebra LÃ A makes it possible to construct the related formal Lie group GÃ . . = 1 + A, whose tangent space at the unity can be identified with the Lie algebra LÃ of the right-invariant vector fields on GÃ. For a fixed element u ∈ GÃ one can denote by ρ u , ξ u : LÃ → T u (GÃ) the differentials of the right and left shifts on GÃ, respectively. Let ρ * u , ξ * u : T * u (GÃ) → L * Ã be dual mappings, respectively. Then the following theorem, stated in [45], holds.

Theorem 2.2
The following bracket: for any a, b ∈ T * u (GÃ) is Poisson if the homomorphism R : A → A, naturally related with the tensor r ∈ A ⊗ A, is skew-symmetric and satisfies the modified Yang-Baxter relationship for all α, β ∈ LÃ subject to the Lie commutator structure in LÃ.
If to take into account that in this case there hold the expressions for any c ∈ A * , where the mappings 1 and 2 stand for convolutions of the co-multiplication : A → A ⊗ A with the first and second tensor components, respectively, that is for any α ∈ A, the bracket (9) will become for any a, b ∈ T * u (GÃ), which can be easily enough computed, if to take into account the relationship (4).
The following result [14,45] is a simple consequence of Theorem 2.2 in the case of the associative matrix algebra A and is almost classical.

Theorem 2.3
Let the algebra A be matrix associative with respect to the standard multiplication, and endowed both with the natural commutator Lie structure [ ·, ·] and with the trace-type symmetric scalar product · , · . . = Tr( · ·). Define also for the tensor r i j e i e j , α (12) for any α ∈ A. Then the pre-Poisson bracket (11) is Poisson if the R-homomorphism (12) is skew-symmetric and satisfies the modified Yang-Baxter relationship (10). Moreover, the Poisson bracket (11) can be equivalently rewritten in the following simplified form: for any a, b ∈ A * .

Remark 2.4
The Yang-Baxter relationship (10) is basic for finding the corresponding internal multiplication structure of the algebra A, allowing the quadratic Poisson bracket (11). If, for example, to assume that the adjacent loop Lie algebra LÃ allows splitting into two subalgebras, , then the homomorphism R = P + −P − solves the relationship (10), where, by definition, the mappings P ± : LÃ → L ± A ⊂ LÃ are the suitable projections. If to assume, that the adjacent loop Lie algebra LÃ is generated by the associative multiplication " * " of the Balinsky-Novikov loop algebra A, then the related Lie structure is given by the commutator for the derivation D x = d/dx and any α, β ∈ A, giving rise to the ultra-local quadratic Poisson bracket (11). To our regret, we do not know whether the Lie structure for any α, β ∈ A and all x ∈ S 1 , suitably determining the adjacent loop Lie algebra LÃ, can be generated by some associative multiplication on the loop Balinsky-Novikov algebra, with respect to which the Lie structure (14) could entail the local quadratic Poisson bracket (11).

Problem 2.5
Concerning the algebraic structures discussed above the interesting problem arises: Classify associative Balinsky-Novikov loop algebras A, whose adjacent Lie algebras LÃ allow splitting into two non-trivial subalgebras subject to the Lie structure (13).

Remark 2.6
In the case of basic Leibniz loop algebra A, it is well known that the usual commutator structure (13) does not generate the adjacent loop Lie algebra LÃ, yet the following inverse-derivative Lie structure: suitably determined for any α, β ∈ A and all x ∈ S 1 , already does. Yet, we do not know whether the Lie structure (15) can be generated by some associative multiplication " * " on the loop Leibniz algebra A.

Quadratic Poisson structures: the Lie-Poisson type generalization
Assume as above that (A, +, • ) is a finite-dimensional algebra of the dimension N = dim A∈Z + (in general non-associative and non-commutative) over an algebraically closed field K. Based on the algebra A one can construct the related loop algebra A of smooth mappings u : S 1 → A and endow it with the suitably generalized natural convolution · , · on A * × A → K, where A * is the corresponding adjoint space to A. First, we will consider a general scheme of constructing non-trivial ultra-local and local Poisson structures on the adjoint space A * [24] compatible with the internal multiplication in the loop algebra A. Consider a basis {e s ∈ A : s = 1, N } of the algebra A and its dual {e s ∈ A * : s = 1, N } with respect to the natural convolution · , · on A * × A, that is e j , e i . . = δ j i , i, j = 1, N , and such that for any be a skew-symmetric bilinear mapping. Then the expression defines for any a, b ∈ A an ultra-local linear skew-symmetric pre-Poisson bracket on A * . If the mapping ϑ * : A ∧ A → Symm( A) is given, for instance, in the simple linear form where quantities c s i j ∈ K are constant for all i, j and s = 1, N , then for the adjoint to (17) mapping ϑ : A * → A * ∧ A * one obtains the expression For the pre-Poisson bracket to be a Poisson bracket on A * , it should satisfy additionally the Jacobi identity. To find the corresponding additional constraints on the internal multiplication " • " on the algebra A, define for any u(x) ∈ A * the skew-symmetric linear mapping called by the Hamiltonian operator [26], via the identity holds for any basis element e i ∈ A, i = 1, N , the resulting pre-Poisson bracket (16) is equal to for any u(x) ∈ A * and all a, b ∈ A. If now to define on the algebra A the natural adjacent Lie algebra structure to the algebra A for any basis elements e i , e j ∈ A, i, j = 1, N , the expression (22) yields for all a, b ∈ A the well-known [1,4] classical Lie-Poisson bracket Concerning the adjacent Lie algebra structure condition (23), it can be easily rewritten as the set of relationships whose obvious solution is for any i, j, s = 1, N . As the bracket (24) is of the classical Lie-Poisson type, for the Hamiltonian operator (21) to satisfy the Schouten-Nijenhuis condition (20) it is enough to check only the weak Jacobi identity for the loop Lie algebra LÃ, adjacent to the algebra A via imposing the Lie structure (23), taking into account the relationships (25). For instance, if the commutator of the adjacent loop Lie algebra LÃ is given by the expression the corresponding algebra A coincides with the well-known Balinsky-Novikov algebra, determined by means of the following relationships: where, by definition, R a b = b • a = L b a for any a, b ∈ A. If, for instance, the commutator of the adjacent loop Lie algebra LÃ is given by the expression for a suitably determined inverse-derivation mapping D −1 x : A → A, the corresponding algebra A coincides with the well-known right Leibniz algebra, described by the relationships for any a, b ∈ A. As a consequence of reasonings above one can formulate the following general theorem. Similarly, one can consider a simple ultra-local quadratic pre-Poisson bracket on A * in the form for any a, b ∈ A, where the skew-symmetric mapping ϑ * : is given for any i, j = 1, N in the quadratic form In particular, if to assume that the coefficients c ks i j = σ k i j α s for some constant numbers σ k i j and α s ∈ K for all i, j and k, s = 1, N , where, by definition, the multiplications coincides with that of the algebra A, then the pre-Poissson bracket (30) yields for any a, b ∈ A a very compact form generalizing the classical Lie-Poisson expression (24) and parametrically depending on the constant vector Thus, for the pre-Lie-Poisson bracket (31) one can formulate suitable constraints on the algebraic structure of A. For instance, if the weak algebraic structure on the adjacent Lie algebra LÃ is given, respectively, either by the Lie commutator (26) or by (28), then the corresponding multiplicative structures of the algebra A are generated, respectively, by the Balinsky-Novikov (27) and Leibniz (29) algebras relationships, augmented with the following common tensor multiplicative constraint: which holds for any a ∈ A and arbitrary but fixed element α ∈ A. So, one can formulate the following theorem.

Theorem 3.2 The quadratic pre-Lie-Poisson bracket (31) on A * is Poisson if and only
if the internal multiplicative structure of the algebra A is compatible both with the weak Lie algebra structure on the adjacent loop Lie algebra LÃ and with the tensor multiplicative relationships (32).
In these cases there arises an interesting problem of describing the Balinsky-Novikov and Leibniz algebras, whose multiplicative structures additionally satisfy tensor relationships (32). Such and related algebraic structure problems are planned to be studied in detail elsewhere. In the next section we proceed to study general algebraic structures related both with generalized Balinsky-Novikov and Leibniz algebras, and so-called Zinbiel algebras, having diverse important applications in communications technology.

Algebraic structures preliminaries
Let A be an associative commutative algebra over a field K of any finite or infinite dimension (with the addition "+" and multiplication " · ") not necessary with identity and δ its derivation, i.e., δ : A → A is a K-linear map satisfying the Leibniz rule. Then is a Balinsky-Novikov algebra (the so-called δ-adjancent or δ-associated Balinsky-Novikov algebra of A ) with respect to " * " defined by the rule (where ξ is a fixed element of A) and so for all a, b, c ∈ A. In particular, A δ,0 . . = A δ,ξ with ξ = 0. Balinsky-Novikov algebras were introduced in connection with the so-called Hamiltonian operators [26] and Poisson brackets of hydrodynamic type [13]. Note here, that the term "Novikov algebra" was suggested by Osborn in [40]. Moreover, is a Lie algebra (the so-called δ-adjancent or δ-associated Lie algebra of A) with respect to the Lie bracket "[−, −]" defined by the rule Every -prime algebra is -semiprime and every -simple algebra is -prime. If = {θ } and D is a -simple (respectively, -prime or -semiprime), then we say that D is θ -simple (respectively, θ -prime or θ -semiprime). Moreover, if = {0}, then a -simple (respectively, -prime or -semiprime) algebra is simple (respectively, prime or semiprime). Any unexplained terminology is standard as in [27,28,31,34].
The purpose of this paper is also to study relationships between associative commutative algebras A, their δ-associated Balinsky-Novikov algebras A δ,ξ and δ-associated Lie algebras A δ,L . Connections between properties of an associative commutative algebra A and its δ-associated algebra A δ,L have been investigated by Ribenboim [43], Jordan, Jordan [29,30], and Nowicki [37]. Xu [49] found some classes of infinite-dimensional simple Balinsky-Novikov algebras of type A δ,ξ . Bai and Meng [6] proved that, if A is a finite-dimensional associative commutative algebra and 0 = δ ∈ Der A, then A δ,0 is transitive (i.e., r a : A x → x * a = x · δ(a) ∈ A is a nilpotent right transformation operator of A δ,0 for any a ∈ A) and A δ,L is a solvable Lie algebra [32]. In [48,Proposition 2.8] it is proved that the Balinsky-Novikov algebra A δ,ξ is simple if and only if an associative commutative ring A is δ-simple. As noted in [5], there is a conjecture: the Balinsky-Novikov algebras N can be realized as the algebras A δ,0 , where A is a suitable associative commutative algebras, and their (compatible) linear deformation. Recall that a binary operation G 1 : N × N → N of a Balinsky-Novikov algebra (N , +, * ) is called its linear deformation if algebras (N , +, g t ), where g t (a, b) = a * b + tG 1 (a, b), are Balinsky-Novikov algebras for every t. If G 1 is commutative, then it is called compatible.
As noted in [5], a "good" structure theory for algebraic systems means an existence of a well-defined radical and the quotient by the radical is semisimple. Our result in this direction is the following. Theorem 4.1 Let A be an associative commutative algebra with 1, char K = 2, 0 = δ ∈ Der A and ξ ∈ A. Then the following conditions are equivalent: (i) A is a δ-semiprime (respectively, δ-prime or δ-simple) algebra, (ii) A δ,ξ is a semiprime (respectively, prime or simple) Balinsky-Novikov algebra, (iii) A δ,L is a semiprime (respectively, prime or simple) Lie algebra.
A triple (Z , +, • ) is called a Zinbiel algebra (or a dual Leibniz algebra) if for all x, y, z ∈ Z . As a consequence, (x • y) • z = (x • z) • y. If x y = x • y + y • x for all x, y ∈ Z (see Lemma 5.1), then (Z , +, ) is an associative commutative algebra (the so-called adjacent or associated associative algebra Z A of a Zinbiel algebra Z ). Zinbiel algebras were introduced by Loday in [33,34] and are very popular in the control theory (in context of "chronological" algebras, see e.g. [2,3,31,44]) and in the theory of Leibniz cohomology [35]. Zinbiel rings can be defined by analogy.
Some interesting properties of Zinbiel algebras were obtained by Dzhumadil'daev, Tulenbaev [22,23], and Omirov [39]. In particular, Dzhumadil'daev [22] proved that any finite-dimensional Zinbiel algebra over the complex numbers field is nilpotent. We prove the next result. Proposition 4.2 Let Z be a Zinbiel algebra over a field K. If the characteristic char K = p > 0 is prime, then the associated associative algebra Z A is nil of bounded degree p.

An associative commutative structure of a Zinbiel algebra
Recall that a (Zinbiel or associative) algebra (A, +, ) is called reduced if the implication a a = 0 ⇒ a = 0 is true for any a ∈ A. An additive subgroup I of a Zinbiel ring (respectively, algebra) Z is said to be an associative ideal of Z if I Z ⊆ I . It is easy to see that I is an associative ideal of Z if and only if it is an ideal of Z A . A Zinbiel ring Z is called 2-torsion-free if, for any x ∈ Z , 2x = 0 implies that x = 0.

Lemma 5.2
Let Z be a Zinbiel ring (respectively, algebra over a field K), ∅ = ⊆ Der Z and a ∈ Z . Then the following properties hold: if X is a non-empty subset of Z , then the right annihilator rann X . . = {t ∈ Z : X • t = 0} of X is an associative ideal of Z and rann X ⊆ rann(X • Z ), (iii) if I is a right -ideal of Z , then Z • I and I + (Z • I ) are -ideals of Z , (iv) if X is an associative -ideal of Z , then the left annihilator lann X . . = {t ∈ Z :

then Z is reduced if and only
if Z A is reduced, (vi) if Z is 2-torsion-free (respectively, char K = 2) and I , J are commutating ideals of Z (i.e., i • j = j • i for i ∈ I and j ∈ J ), then I • J ⊆ rann Z, Proof Let z, t ∈ Z .
(i) Clearly a • Z is a subgroup of the additive group (Z , We also see that (v) It follows from z • z = 0 ⇔ z z = 0.
(vi) Assume that i ∈ I and j ∈ J . Then and from this On the other hand, Then (33) and (34)

Corollary 5.4 Let Z be a Zinbiel ring (respectively, algebra) and ∅ = ⊆ Der Z. If Z A is -simple (respectively, -prime or -semiprime), then Z is the same.
Proof If ⊆ Der Z , then ⊆ Der Z A .
Simplicity. Since every -ideal of Z is a -ideal of Z A , the simplicity of Z A implies that Z is simple. Semiprimeness. By analogy as in the prime case.

Proof of Proposition 4.2
Let a ∈ Z . By a n we denote the n-th power of a in the algebra Z A (n is a positive integer). We have • (a • a)).

Lemma 6.3 Let δ be a surjective derivation of an associative commutative algebra A with 1. If I is a right ideal of a Balinsky-Novikov algebra A δ,ξ , then I is an ideal of A.
Proof Indeed, if i ∈ I and a ∈ A, then I i * a = i · δ(a) + ξ · i · a and therefore ξ · i = i * 1 ∈ I . Since δ is surjective, we have i · δ(a) = i * a − ξ · i · a ∈ I and so i · A ⊆ I .
It is easy to see that e * e = 0 for any idempotent e 2 = e ∈ A. Lemma 6.4 Let A be an associative commutative algebra, δ ∈ Der A, and ξ ∈ A. Then the following properties hold:  1 ∈ A and B is an ideal of A δ,ξ , then ξ · B, δ(B)

contains a non-zero δ-ideal of A, (ii) if I is a δ-ideal of A, then I is an ideal of the Balinsky-Novikov algebra A δ,ξ , (iii) if K is an additive subgroup of a Balinsky-Novikov algebra A δ,ξ and δ(K ) ⊆ K , then I A (K ) . . = {k ∈
Proof (i) For the proof see [37].
(ii) Indeed, i * a = i · δ(a) + ξ · i · a ∈ I and a * i = a · δ(i) + ξ · i · a ∈ I for any i ∈ I and a ∈ A.
(iii) Assume that k ∈ I A (K ) and x ∈ A.
(iv) It is easy to see that (v) For any a ∈ A and c ∈ C we see that C a * c = a · δ(c), whence δ(C) ⊆ I A (C).
(vi) If i ∈ I and a ∈ A, then [i, a] = i · δ(a) − a · δ(i) ∈ I , the claim follows.
(xi) Let a, x ∈ A and s ∈ T A (S). Then (xii) For any i ∈ I , (xiii) By (x), W is an ideal of the Lie algebra A δ,L and so W w * a = w · δ(a) + ξ · w · a and W [w, a] = w * a − a * w = w · δ(a) − a · δ(w) for any w ∈ W and a ∈ A. Then W w * a − [w, a] = (ξ · w + δ(w)) · a. Hence δ(w) + ξ · w ∈ I A (W ).
Lemma 6.5 Let A be an associative commutative algebra, δ ∈ Der A, and x, ξ ∈ A. Then the following properties hold: [a,b] for any a, b ∈ A δ,ξ , (vi) L(A δ,ξ ) . . = {l a : a ∈ A} is a Lie algebra.
(iii) By the same argument as in (ii).
(iv)-(vi) Obvious. Further we shall need the following result.
Proof By the same argument as in the proof of Lemma 6.9. Lemma 6.11 [17]  Lemma 6.12 Let A be an associative commutative algebra with 1, char K = 2, 0 = δ ∈ Der A, and ξ ∈ A. If A is a δ-prime algebra, then Z (A δ,ξ ) = 0.

Lie properties
Lemma 7.1 Let A be an associative commutative algebra with 1, ξ ∈ A, and δ ∈ Der A. If δ(A) P for any minimal δ-prime ideal P of A, then: (i) every abelian ideal I of the Lie algebra A δ,L is contained in the δ-prime radical P δ (A) . . = { P : P is a δ-prime ideal of A}, (ii) the Lie algebra A δ,L is not solvable.
Proof (i) Let I be a non-zero abelian ideal of the Lie algebra A δ,L . If I P δ (A), then there exists a minimal δ-prime ideal of P of A such that I P. Obviously, : A/P a + P → δ(a) + P ∈ A/P is a non-zero derivation of the quotient algebra A/P. Since A/P is a -prime algebra, (A/P) ,η is a prime Lie algebra, where η = ξ + P. Hence (I + P)/P is zero, a contradiction.
(ii) It follows from (i).

Lemma 7.2 [37, Theorem 3.3] Let A be an associative commutative algebra with 1 and 0 = δ ∈ Der A. Then A is a δ-simple algebra if and only if A δ,L is a simple Lie algebra.
Proof By the same argument as in the proof of Lemma 6.9.
Proof By the same argument as in the proof of Lemma 7.4.
We specified some of interesting properties of an associative commutative algebra A and its δ-associated algebra A δ,L , which earlier were investigated by Ribenboim [43], C.R. Jordan, D.A. Jordan [29,30], and Nowicki [37]. Moreover, as follows from the results stated above, there exist deep and very interesting relationships between associative commutative algebras A, their δ-associated Balinsky-Novikov algebras A δ,ξ and δ-associated Lie algebras A δ,L .