Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

Let L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.


Introduction
The study of polynomial identities of non-associative algebras plays a crucial role in the investigation of genetic populations. In 1939 Etherington (see [25]) introduced the so called genetic algebras that are algebras arising from the multiplication table given by the composition of the gametic or zygotic (and many other) types of some population. Genetic algebras are, in general, non-associative algebras and, depending on some special identities they satisfy, they give rise to peculiar "genetic facts". For instance, see the Bernstein algebras that are baric algebras satisfying the identity (x 2 ) 2 = ω(x)(x 2 )x 2 . The Bernstein algebras describe the equilibrium state of a certain population in future generations.
A complete understanding of the equilibrium states of any population is strictly related to the study of the varieties generated by Bernstein algebras. In this paper, following this line of research, we would like to study the varieties generated by some special Leibniz algebras that are non-associative algebras carrying on the defining identities of Lie algebras but the anticommutativity. We recall the important role played by Lie algebras in genetics. The genetic code is usually represented mathematically as a Lie algebra and new such structures are studied by several groups of biomathematicians (see for instance [47]). In particular, here we study varieties of null-filiform Leibniz algebras under the action of a big class of Hopf algebras. Although we still have not a literature containing the application in genetics of the action of any Hopf algebra, we strongly believe this should be a necessary step in founding a bridge between mathematics and genetics. Nevertheless we know that abelian gradings on Lie algebras (that is a very particular example of an action of a Hopf algebra on a Lie algebra) and the knowledge of their representations, identities, etc. can be useful in the description of degeneracies of a genetic code (see [26] and [32]).
It is worth saying Leibniz algebras were introduced by Bloh in [10] and Loday in [40], and they have many applications either in pure and applied mathematics or in physics. Because of this, many known results of the theory of Lie algebras as well as combinatorial group theory have been spread to Leibniz algebras during the last two decades, (see, for instance, [8,13,16,21,38] and [50]). It is well known that a Leibniz algebra decomposes into a semidirect sum of the solvable radical and a semisimple Lie algebra [8]. Therefore, restricting the study of solvable Leibniz algebras to nilpotent ones, it is sufficient studying nilpotent Leibniz algebras with certain conditions such as the conditions on the index of nilpotency, various types of gradings and characteristic sequence (see, for example, [2,15] and [37]). Regarding the index of nilpotency, we recall that in the theory of Lie algebras a finite dimensional filiform Lie algebra over field K is a nilpotent Lie algebra L whose nil index is maximal and equal to dim(L) − 1. The notion of p-filiform Lie algebras makes sense for p ≥ 1 (see [11]), while for p = 0 it is not possible to define that, since a Lie algebra has at least two generators. In the case of Leibniz algebras the class of null-filiform Leibniz algebras and their properties was originally introduced in [5]. The maximal index of nilpotency of n-dimensional Leibniz algebra is equal to n + 1, whereas, the maximal index of nilpotency of an n-dimensional Lie algebra is n.
In the classical and purely mathematical literature towards varieties of algebras (as well as polynomial identities of algebras), one of the main problems is the so-called Specht problem which concerns the existence of a finite basis of identities for any subvariety of a given variety. It is known that every variety of associative algebras over a field of characteristic 0 satisfies the Specht property [36]. In the case of Lie algebras this problem is still open in general even in the case the ground field is of characteristic 0. Partial cases were studied and solved by Iltyakov in [33] including the case of finite dimensional Lie algebras. The study of polynomial identities for Leibniz algebras was started by Drensky and Piacentini Cattaneo in [24]. Drensky et al. described the free metabelian Leibniz algebras and provided a complete list of left-nilpotent of class 2 varieties of Leibniz algebras. Abanina and Mishchenko in [1] studied the variety of Leibniz algebras determined by the identity x(y(zt)) ≡ 0 and proved many properties similar to the variety of abelian-by-nilpotent Lie algebras of class 2. Several years later Mishchenko and Valenti in [43] studied the variety of Leibniz algebras defined by identity y 1 (y 2 y 3 )(y 4 y 5 ) ≡ 0 and provided a description of multilinear identities in the language of Young diagrams through representation theory of symmetric groups. In [49] Vais and Zelmanov proved that any finitely generated Jordan algebra in characteristic 0 satisfies the Specht property by showing that it has the same identities of a finite dimensional generalized Jordan pair. Unfortunately, we do not know yet whether the answer is positive nor negative in case of infinitely generated Jordan algebras. Some special cases of the Specht property for graded Jordan algebras can be found in [19].
The following pages will contain a full description of the variety of a finite dimensional null-filiform Leibniz algebras under the action of a finite dimensional pointed cocommutative Hopf algebra first and under the action of a Taft algebra too. This means we will furnish information such as the description of the action, the polynomial identities, the S nstructure of the relatively free algebras (cocharacters, codimensions, Hilbert series). As a consequence we get an H -variety generated by a null-filiform Leibniz algebra always satisfies the Specht property if H is a Taft's algebra whereas it satisfies the Specht property only if the underlying group of H is finite in the case H is pointed cocommutative.

Preliminaries
Although almost all of the arguments below can be performed for any algebra (associative, Lie, Jordan, etc.) every algebra in the sequel will be a Leibniz algebra. We recall a Leibniz algebra is a vector space L over a field K with bilinear product (−, −) which satisfies the Leibniz identity Indeed we have the notion of Leibniz subalgebras, ideals, homomorphisms, gradings. On this purpose, we would like to construct a free set in the class of graded Leibniz algebras. Hence let G be a group and let {X g | g ∈ G} be a family of disjoint countable sets. Set X = g∈G X g and denote by L X|G the free Leibniz algebra freely generated by the set X. An indeterminate (or variable) x ∈ X is said to be of homogeneous G-degree g, written deg(x) = g, if x ∈ X g . We always write x g if x ∈ X g . The homogeneous G-degree of a monomial m = x i 1 x i 2 · · · x i k is defined to be deg(m) = deg(x i 1 ) · deg(x i 2 ) · · · deg(x i k ). For every g ∈ G we denote by L X|G g the subspace of L X|G spanned by all monomials having homogeneous G-degree g. Notice that L X|G g L X|G g ⊆ L X|G gg for all g, g ∈ G. Thus L X|G = g∈G L X|G g and L X|G is a G-graded algebra. We refer to the elements of L X|G as G-graded polynomials or just graded polynomials. a 2 , . . . , a n ) = 0 for all a 1 , a 2 , . . . , a n ∈ g∈G A g such that a k ∈ A deg(x k ) , k = 1, . . . , n. We denote by I d G (A) the ideal of all graded polynomial identities of A. It is a T G -ideal of L X|G in the sense that it is invariant under all graded homomorphism of L X|G . We shall call substitution with elements of A any graded homomorphism L X|G → A and we sometimes use the notation x = a ∈ A in order to denote explicitly such evaluation of the variable x.
Given a subset S ⊆ L X|G one can talk about the least T G -ideal of L X|G containing the set S. Such T G -ideal will be denoted by S T G and will be called the T G -ideal generated by S. We say that elements of S T G are consequences of elements of S, or simply that they follow from S. If I d G (A) = S T G , we say that S is a basis for the graded polynomial identities of A.
Let K be a field of characteristic 0, and let H be a Hopf algebra over K with comultiplication : H → H ⊗ H . Here we use Sweedler's notation (h) = h (1) ⊗ h (2) .

Definition 2.1 A K-algebra A is called an H -module algebra or an algebra with an Haction, if A is a left H -module with action
We shall construct a free object inside the class of H -module algebras too. Let H be a Hopf algebra with unit 1 and let us consider a countable set of indeterminates X := {x 1 , x 2 , . . .}; we set x j := x 1 j . We choose a linear basis (γ β ) β∈ in H and we denote by L X|H the free Leibniz algebra over K generated by the free formal generators x where only a finite number of α β 's is non-zero. We refer to the elements of L X|H as H -polynomials. Note that here we do not consider any H -action on L X|H .
Let A be a Leibniz algebra with an H -module algebra structure. Any map ψ : X → A has a unique homomorphic extension ψ : It is worth noticing that if we consider the trivial Hopf algebra H = F , then we are simply studying ordinary polynomial identities and we shall omit any index or super-index to refer to its H -identities or related stuffs whereas, if H = KG and G is abelian, we are studying G-graded polynomial identities. For further lectures about polynomial identities of associative algebras we strongly recommend the books [23] by Drensky and [29] by Giambruno and Zaicev. We also address the reader to the book [6] by Bahturin that is more focused on identities of Lie algebras.
Denote by P H n the space of all multilinear H -polynomials in x 1 , ..., x n , n ∈ N, i.e., The symmetric group S n acts on the left on the space P H n by σ ( is a left S n -module. This leads us to consider the S n -character of P H n (A), namely χ H n (A), which is called n-th cocharacter of polynomial H -identities or the n-th H -cocharacter of A. By the classical theory of representations of the symmetric group (see for instance the book by Sagan [46]), the irreducible S n -characters are in oneto-one correspondence with the partitions of the non-negative integer n (which carries a Young Tableau) because the ground field is of characteristic 0. In particular, if χ λ denotes the irreducible S n -character corresponding to the partition λ, then we are allowed to write where m H λ ≥ 0 is the multiplicity of the irreducible character χ λ in the decomposition of χ H n (A). Moreover the non-negative integer is As we said above, if we specialize H with the dual algebra of the group algebra F G, where G is a finite abelian group, we get the notion of G-graded identities, codimension, exponent, etc. The existence of the exponent in the graded case when A is supposed to be associative and over a field of characteristic 0, has been studied by several authors as Giambruno and Zaicev in [28] when G is the trivial group, Benanti, Giambruno and Pipitone in [9] when G = Z 2 , by Aljadeff, Giambruno and La Mattina in [4] in the case A is affine and G is abelian, by Giambruno and La Mattina (see [27]) if A is any G-graded algebra and G is abelian and in general by Aljadeff and Giambruno in [3]. In the general case of an H -algebra only partial results are known about the existence of such exponent. If H is finite dimensional and semisimple acting on an associative algebra over a field of characteristic 0, then Karasik proved in [35] the H -exponent exists and is an integer. It is easy to see Taft's algebras are not semisimple algebras. Also the result by Gordienko [30] is another good step in this direction. In fact, he proved the existence of the exponent for finite dimensional algebras over an algebraically closed field of characteristic 0 that are simple under the action of a Taft algebra. We recall Taft's algebras are non-commutative, non-cocommutative and not semisimmple Hopf algebras. In [45] the authors constructed the first example of a non-associative unital algebras whose PI-exponent does not exist whereas in [31] the author proved the existence of the exponent for finite dimensional Lie and associative algebras graded by any group. The first example of an infinite dimensional Lie algebra with a non-integer ordinary PI-exponent was constructed by Mishchenko and Zaicev in [44].
Finally, given an H -module algebra A, we would also give a new definition of H -Hilbert series of the relatively free algebra of A. Let us set the framework. We take H being finite dimensional and let us choose a linear basis B = {h 1 , . . . , h d } of H . We choose a natural number k ≥ 1 and the set of H -variables by H ilb(B, t) the Hilbert series of an algebra B in the variable t, we have the next definition which should be considered as a generalization to the Hopf algebra case of the graded Hilbert series of an algebra as appeared for the first time in [18]. See also the paper [17] for an interesting relation with the graded exponent.

Definition 2.2 Let
A be an H -module algebra and k ≥ 1 a natural number. We define the H -Hilbert series of A in k-variables as Let us show briefly a bridge between H -identities and graded identities that would be helpful for our purposes in this paper. The proof of Proposition 3.3.6 of [29] gives us a well known and nice duality between G-gradings and G-actions of finite dimensional algebras provided that G is a finite abelian group. However one can define G-polynomials as KG-polynomials, where the group algebra F G is endowed with its canonical Hopf algebra structure. Notice also in the proof we mentioned above the authors take an opportune basis of KG as a vector space (corresponding to "projections") so that the KG-polynomials correspond to G-graded polynomials, where the G-grading is constructed adequately. In few words, given a finite abelian group G and a finite-dimensional algebra A with a G-action we obtain a G-grading on A and viceversa; furthermore, the G-polynomial identities and the G-graded identities coincide, that is

I d KG (A) = I d gr (A).
Due to the results in [7] we can associate to every group grading a certain signature. We recall the definition of a signature. We say that a vector space A is an -algebra and is a signature of A, where = n≥0 n , if each ω n ∈ n defines an n-linear map ω n : A × · · · × A → A. For instance, our definition of algebra is a -algebra, where | 2 | = 1, and n = ∅, for n = 2. We can construct the free -algebra, so we can talk about -polynomials identities (see, for instance, [34,Chapter 2]). Let A be a G-graded algebra, where G is finite and define π g : A → A as the projection with respect to the decomposition A = g∈G A g . Hence we can consider the signature G = 1 ∪ 2 , where | 2 | = 1, and 1 = {π g | g ∈ G}. In [7], the authors prove that the elements π g (x) in the relatively free G -algebra correspond to graded variables of degree g. Thus

General Overview on Free Leibniz Algebras and Null-Fifliform Leibniz Algebras
We shall denote by L X the free Leibniz algebra freely generated by X over K. Then X is called set of free generators of L X . The next proposition shows that the Leibniz identity allows us to reduce every polynomial to a linear combination of left-normed monomials.
Proposition 3.1 [39,41,42] Every multilinear polynomial in free Leibniz algebra L X can be written as a linear combination of left-normed monomials.
Proof Let us denote by deg(m) the length (usual degree) of a monomial m. The proof will be performed by induction on the degree. Let x 1 , x 2 , x 3 ∈ X, then the Leibniz identity gives We have four types of monomials of length 4 as listed below: The monomials (a) and (b) are already in the required form. We note that in the second case (x i 1 x i 2 ) is considered as a single variable. Moreover, cases (c) and (d) are immediate consequence of the cases (a) and (c), respectively. Suppose the assertion true for monomials of length less than n − 1 ≥ 4 and let us prove it for monomials of degree n. Let w be a monomial so that deg(w) = n. Then w has one of the following forms: where deg(u) = n − 1 and deg(u) + deg(v) = n. If w = (u)x i n we are done because w is left-normed. We consider now the case w = x i 0 (u). We write and we denote y := (· · · ((x i 1 x i 2 )x i 3 ) . . . )x i n−2 , then we get by the Leibniz identity. We consider now the summand (x i 0 y)x i n−1 and we notice that it can be written as a linear combination of left-normed monomials because of induction hypothesis because deg(y) = n − 2. Now we consider (x i 0 x i n−1 )y and, as above, we set z := (· · · ((x i 1 x i 2 )x i 3 ) · · · )x i n−2 and we get, always by the Leibniz identity, Note that deg(z) = deg(y) − 1 = deg(u) − 2, then, as above, the first summand is a linear combination of left-normed monomials by induction hypothesis whereas the second one can be further decomposed as above. For a given Leibniz algebra L, the sequence of two-sided ideals defined as is said to be the lower central series of L.

Definition 3.2 A Leibniz algebra
L is said to be nilpotent, if there exists n ∈ N such that L n = 0. The minimal number n such that L n = 0 is said to be the nilpotency index of L.

Definition 3.3 An n-dimensional Leibniz algebra
Obviously, nilpotent null-filiform Leibniz algebras have maximal nilpotency index. In [5] the authors achieved the following characterization of null-filiform Leibniz algebras. The support of a G-grading is the set Supp(G) = {g ∈ G | A g = 0}. It has been proved in [12] that any G-grading on a null-filiform Leibniz algebra, where G is an abelian group, is cyclic. In particular, the next result holds.

Actions of Pointed Cocommutative Hopf Algebras on Null-Filiform Leibniz Algebras
In this section we shall describe the actions of finite dimensional pointed cocommutative Hopf algebras on null-filiform Leibniz algebras NF n . We shall also give a complete description of identities and the multilinear and homogeneous structure of the relatively free algebra of NF n . We consider the following structure generated by the action of a Hopf algebra H on any algebra (associative, Lie, Leibniz, etc.).

Definition 4.1 Let
A be an H -module algebra over a field K. Then the smash product algebra H #A is defined as follows: as a vector space H #A = H ⊗ A and we write h#a instead of h ⊗ a while the multiplication is given by It is easy to see A ∼ = 1#A and H ∼ = H #1. We also recall for a given Hopf algebra (H, , ) we define the set of group-like elements as while we define the set of primitive elements as Notice that if g is a Lie algebra and U(g) is its universal enveloping algebra, then P (U(g)) = g.
Moreover a Hopf algebra is said to be pointed if every simple subcoalgebra has dimension 1 whereas it is said to be connected if the sum of its simple subcoalgebras has dimension 1. We have a nice description of pointed cocommutative Hopf algebras attributed to Cartier and Gabriel in [22] and to Konstant in [48].

Theorem 4.2 Let H be a Hopf algebra with G = G(H ), then if H is pointed cocommutative we get F G#H
Hopf algebra of H containing the unit element.
In the connected case we have the next result due to the independent works by Cartier [14] and Kostant which remained unpublished.

Theorem 4.3 Let H be a cocommutative connected Hopf algebra over a field K of characteristic 0. Then H ∼ = U(g) for g = P (H ).
Keeping in mind these last two classical results, we assume K is an algebraically closed field of characteristic zero and H = KG#H 1 , where G = G(H ) is a finite abelian group, H 1 is a KG-module algebra via g · h = ghg −1 for g ∈ G, h ∈ H 1 , and H 1 = U(g), where g = P (H 1 ) (the set of primitive elements).
We first note that G · g ⊆ g. Thus g is a G-graded algebra and, since G is abelian, this G-grading on g induces a G-grading on U(g). Now let A be a finite dimensional H -module algebra. Then A is a G-graded algebra because, as remarked before, KG can be identified as a subalgebra of H . Moreover, the Ggrading on A induces naturally a G-grading on End K (A) and then a G-grading on Der(A), the set of all derivations of A. Also, we get g acts as a set of derivations on A. This means we have a Lie homomorphism which is a graded homomorphism too. Hence, we have a graded homomorphism U(g) → End K (A).
Conversely, a G-grading on A and on g and a graded Lie homomorphism g → Der(A) defines a structure of H -module algebra on A. This is the content of the next result. Looking back at the previous result, we can also consider graded differential polynomials, that is, graded polynomials under the action of the graded Lie algebra g. In this case, using Theorem 4.4, the H -identities coincide with the differential graded polynomial identities. More precisely, we get the next result. Indeed a graded Lie algebra g acts on A as a graded derivation, i.e., if d ∈ g g is homogeneous in the grading and a ∈ A h , then d(a) ∈ A g·h .
In the sequel, we describe the graded derivations of NF n . Let B = {e 1 , . . . , e n } be a basis of NF n , then we have which means d is completely determined by d(e 1 ). If d(e 1 ) = n i=1 α i e i , then we can write d(e 1 ) as (α 1 id NF n + α 2 R e 1 + α 3 R 2 e 1 + · · · + α n R n−1 e 1 )(e 1 ), where R e 1 is the right multiplication by e 1 . Of course, for every i = 1, . . . , n − 1, we have R i e 1 = R e i . This means a derivation d is such that d = α 1 id NF n + α 2 R e 1 + α 3 R 2 e 1 + · · · + α n R n−1 e 1 . Because sum of derivations is again a derivation, we get any derivation d has the following presentation: d = α 2 R e 1 + α 3 R 2 e 1 + · · · + α n R n−1 e 1 as linear combination of derivations. Of course the R i e 1 's are linearly independent. This means the Lie algebra Der(NF n ) of derivations of NF n is one generated, abelian and is a vector space of dimension n − 1 and, of course, any Lie subalgebra of Der(NF n ) is a finite dimensional vector space of dimension less than or equal to n − 1.
The next result describes explicitly the action of a derivation on the basis elements of NF n taking into account the remarks above. and we are done.

H-Identities of Null Filiform Leibniz Algebras Under the Action of a Pointed Cocommutative Hopf Algebra
Given any Lie algebra g of dimension l and a group G = {g 1 , g 2 , . . .} with neutral element 1 G , by Proposition 4.5, in order to study the H = U(g)#G-graded polynomial identities of NF n , we have to study the differential graded polynomial identities of NF n . On this purpose, let B = {δ 1 , . . . , δ l } be a G-homogeneous basis of g. The elements of B act on NF n as a set of G-graded derivations. x g , g / ∈ Supp(NF n ), x g,δ g , (g, δ g ) / ∈ Supp Der (N F n ), Proof We will argue only for the identities of type (x δ k ,g 1 1 x h 2 because the remaining cases can be handled similarly. Observe the only non-trivial substitution of both (x δ k ,g 1 1 x δ i ,g 2 2 )x δ j ,g 3 3 and (x δ k ,g 1 1 x δ j ,g 3 2 )x δ i ,g 2 3 is (ee 1 )e 1 , where e is a non-zero element of δ k (N F g 1 n ) and we are done. both corresponding (up to specialize g with h) the set of H -polynomials: 2. (· · · ((x g x h )x h ) · · · )x h . The families of polynomials 1. and 2. are linearly independent. We need to prove the polynomials of the family 1. are linearly independent because the other case can be treated similarly. If they are not linearly independent there exist α δ ∈ K not all 0 such that Let e be a non-zero homogeneous element of NF g n and consider the substitution φ sending x g → e and x h → e 1 . Then the previous relation says (· · · ((e δ∈B α δ δ e 1 )e 1 ) · · · )e 1 = 0 that is δ∈B α δ δ(e) = 0. Without loss of generality, we may suppose all of the δ's of a certain fixed degree. Moreover, notice that (δ, g) ∈ Supp Der (N F n  As a consequence of Theorem 5.3 and Theorem 5.4, we have the following description of the G-graded identities and cocharacters of NF n , where G is any group.
Theorem 5.5 Let G be a finite group and let NF n be a G-graded n-dimensional nullfiliform Leibniz algebra. Then Moreover, we have c G m (N F n ) = |Supp(NF n )|m if m ≤ n and 0 otherwise. Then, consequently, exp G (N F n ) = 0. We also have Theorem 5.6 Let G be a finite group and m ∈ N, then

The Case of Taft's Hopf Algebras
In this section we shell exploit the actions of a Taft Hopf algebra on the null-filiform Leibniz algebra NF n and we shall show up the set of its identities and the description of its relatively free algebra.
First we shall give a small account on Taft's algebras (see also [20]). Let K be a field containing a primitive p-th root of the unit γ for some positive integer p. Let (H p , , , S) be the Hopf algebra so that as an algebra with comultiplication such that Moreover the antipode S is such that Thus, H p is an p 2 -dimensional algebra which is neither commutative nor cocommutative. This algebra is known as the Taft's Hopf algebra of order p. A particular case of a Taft's algebra occurs when p = 2 and the latter algebra is known as the Sweedler's Hopf algebra. From now on let A be a finite dimensional (associative, Lie, Jordan, etc.) algebra over a field K. Notice that the element c acts as a homomorphism of algebras on A. Moreover, since c p = 1, we obtain that c acts as an automorphism of A of order p. Using the same idea, x acts as a c-derivation (also known as a skew-derivation), that is, it satisfies Moreover, the actions of x and c are related by the following relation xc = γ cx. On the other hand, the choice of an automorphism α of A of order p and an α-derivation d satisfying d p = 0, and dα = γ αd, defines an H p -action on A. In fact it is sufficient to consider the F -algebra F α, d which turns out to be a Hopf algebra isomorphic to H p . Hence we get the next result. Proposition 6.1 Let A be a finite dimensional algebra over a field K containing a p-th primitive root of unit γ , assume that its characteristic does not divide p, then an action of H p on A is completely determined by a choice of: (1) an automorphism α of A of order p, (2) an α-derivation d of A such that d p = 0, and αd = γ dα.
Equivalently, the structure of H p -module algebra on A is uniquely determined by a choice of: (1) a Z p -grading A = i∈Z p A i , (2) an α-derivation d (where α defines the Z p -grading above) such that d(A i ) ⊆ A i−1 , and d p = 0.
Note that if we consider a linear basis B 1 of the subalgebra c of H p generated by c, then {x i β | β ∈ B 1 , i = 0, 1, . . . , p − 1} is a basis of H p . Let A be an H p -module algebra. The proof of [29,Proposition 3.3.6] gives us a basis {χ 1 , . . . , χ p } of c such that each χ i corresponds to a projection of a certain Z p -grading on A. So B = {x j χ i } is a basis of H p and = 1 ∪ 2 is a signature, where 1 = B, and | 2 | = 1. Let D p = F x = span F {1, x, x 2 , . . . , x p−1 } . By [7] again we have the variables x i (χ j (x)) correspond to graded variables under the action of x j . In few words the H p -polynomials correspond to Z p -graded polynomials with the action of D p and the polynomial identities coincide. Hence we can establish the following. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.