On the free metabelian Novikov and metabelian Lie-admissible algebras

In recent years, algebras with metabelian identity have become popular objects in ring theory. Metabelian identity also can be stated as the solvability of index $2$. Various types of classical algebras with metabelian identity, such as Lie, Leibniz, Malcev, Jordan, etc. are considered. The variety of metabelian Lie algebras has attracted significant attention, see [18, 21]. A basis of the free metabelian Lie algebra was constructed in [1]. In an analogical way, a basis of the free metabelian Leibniz algebra was constructed in [7]. Symmetric polynomials in the free metabelian Lie algebras were considered in [8]. The generators of symmetric polynomials in free metabelian Leibniz algebras were found in [24]. Other examples of Lie-admissible algebras are assosymmetric algebras [11]. A basis of the free metabelian Malcev algebra is constructed in [20]. For this reason, we add metabelian identity to a well-known class of algebras which are Novikov and Lie-admissible.

Novikov algebras were introduced in a study of Hamiltonian operators concerning the integrability of certain partial differential equations [12]. Later, they played a significant role in a study of Poisson brackets of hydrodynamic type [2].

It is well-known that, given a commutative algebra $A$ with a derivation $D$, the space $A$ under the product $x_{1}\circ x_{2}=D(x_{1})x_{2}$ forms a (right) Novikov algebra. Moreover, every Novikov algebra can be embedded into an appropriate commutative algebra with derivation $D$ [4]. Using rooted trees, the monomial basis of the free Novikov algebra in terms of $\circ$ was constructed in [10]. In terms of Young diagrams, the basis was developed in [9]. By utilizing commutative algebra with derivation $D$ and a well-defined order, an alternative monomial basis of the free Novikov algebra is presented in [15]. The issues of solvability and nilpotency of Novikov algebras were addressed in [23]. In section $2$, we construct a basis of the free solvable Novikov algebra with an index of $2$. In section $3$, we explicitly describe the symmetric polynomials for the multilinear part of the free metabelian Novikov algebra.

Let’s shift our focus to metabelian Lie-admissible algebras. In the realm of Lie-admissible algebra theory, additional conditions like flexibility or power-associativity are crucial, leading to numerous noteworthy outcomes in this context, see [3, 19]. Another important direction of the research on Lie-admissible algebras concerns the property that their associator satisfies relations defined by a natural action of the symmetric group of degree 3 [13, 14]. A basis of the free Lie-admissible algebra and the Gröbner-Shirshov base theory for Lie-admissible algebras is given in [5]. In addition, there is given an analogue of PBW-theorem for the pair of Lie and Lie-admissible algebras. A basis of the free Lie-admissible algebra in terms of commutator and anti-commutator is given in [17]. In Section $4$, we construct the basis of a free metabelian Lie-admissible algebra in terms of commutator and anti-commutator. In Section $5$, we explicitly describe the symmetric polynomials for the multilinear part of the free metabelian Lie-admissible algebra.

We consider all algebras over a field $\mathbb{K}$ of characteristic 0.

An algebra is called metabelian (right) Novikov if it satisfies the following identities:

Let $X=\{x_{1},x_{2}\ldots\}$ be a countable set of generators. We denote by $\operatorname{{Nov}}\langle X\rangle$ and $\operatorname{{M}}\operatorname{{Nov}}\langle X\rangle$ the free Novikov and free metabelian Novikov algebra, respectively.

Let us denote by $\mathcal{N}_{n}$ the set of monomials of degree $n$ of the following form:

where $n\geq 5$. We set

For degree $3$, $\mathcal{N}_{3}$ is defined as follows:

For degree $4$, we define the set $\mathcal{N}_{4}$ as follows:

Let $p(x_{1},x_{2},\ldots,x_{n})$ be a polynomial of the free metabelian Novikov algebra generated by a finite set $X=\{x_{1},x_{2},\ldots,x_{n}\}$. The polynomial $p(x_{1},x_{2},\ldots,x_{n})$ is called symmetric if it satisfies the following condition:

where $\sigma\in S_{n}$. Let us define a set of polynomials $\mathcal{P}$ in $\operatorname{{M}}\operatorname{{Nov}}\langle X\rangle$ as follows:

and

where $n\geq 5$. The multilinear part of the free metabelian Novikov algebra is a space consisting of all elements containing each $x_{i}$ exactly once.

An algebra is called metabelian Lie-admissible if it satisfies the following identities:

Let us consider the polarization of metabelian Lie-admissible algebra, i.e., we consider an algebra with two operations which is defined on metabelian Lie-admissible algebra as follows:

In this case, the defining identities of the variety of metabelian Lie-admissible algebras become to

As a consequence, we obtain

and

which hold in free metabelian Lie-admissible algebra.

Let us construct a basis of the free metabelian Lie-admissible algebra in terms of binary trees with two types of vertices $\bullet$ and $\circ$. We consider only trees of the following form:

We place on vertices of the tree $\bullet$ and $\circ$ in all possible ways. On the leaves of the tree, we place generators from the countable set $X$. This tree in a unique way corresponds to the sequence $(*_{x_{i_{1}}},*_{x_{i_{2}}},\ldots,*_{x_{i_{n-1}}},x_{i_{n}}).$

1) If there are several consecutive black dots in a sequence, then all corresponding generators of these vertices are ordered, i.e., for $(\ldots,\circ_{i_{k-1}},\bullet_{i_{k}},\bullet_{i_{k+1}},\ldots,\bullet_{i_{l-1}},\circ_{i_{l}},\ldots)$, we have $i_{k}\geq i_{k+1}\geq\ldots\geq i_{l-1}$;

2) If the rightmost vertex is white then the rightmost generator is less than the previous one, i.e., for $(\ldots,\circ_{i_{n-1}},x_{i_{n}})$, we have $i_{n-1}\leq i_{n}$;

3) If a given consecutive sequence of black dots continues to the rightmost vertex and the number of black dots is bigger than $2$, then all the generators of these vertices are ordered and the rightmost generator is bigger than the previous one, i.e., for $(\ldots,\bullet_{i_{k}},\bullet_{i_{k+1}},\ldots,\bullet_{i_{n-1}},x_{i_{n}})$, we have $i_{k}\geq i_{k+1}\geq\ldots\geq i_{n-1}<i_{n}$;

4) In condition $3$ if the number of black dots is not bigger than $2$, then the generators are ordered as in Lyndon-Shirshov words, i.e., the basis monomials of the free Lie algebra of degrees 2 and 3;

For every such tree, we define a monomial from free metabelian Lie-admissible algebra as follows: the tree with $n$ leaves is a right-normed monomial of degree $n$, i.e., this is the monomial $x_{i_{1}}*(x_{i_{2}}*(\ldots(x_{i_{n-1}}*x_{i_{n}})\ldots))$. The black multiplication $\bullet$ corresponds to the Lie bracket $[\cdot,\cdot]$ and white multiplication $\circ$ corresponds to $\{\cdot,\cdot\}$. We denote by $\mathcal{T}$ a set of trees that satisfy conditions (1), (2), (3) and (4), and we denote by $\mathcal{M}$ the set of monomials which correspond to the trees from $\mathcal{T}$.

Calculating the dimension of operad $\operatorname{{M}}\operatorname{Lie}$ by means of the package [6], we get the following result:

We see that the dimension of this operad is growing at a high rate, and this sequence does not coincide with any sequence from OEIS. In [22] was given the dimension of the Lie-admissible operad up to degree $7$, which is

It will be interesting to find a general formula for the dimension of the metabelian Lie-admissible and Lie-admissible operads.

For $n$, let us define the sequence $(*_{x_{i_{1}}},*_{x_{i_{2}}},\ldots,*_{x_{i_{n-1}}},x_{i_{n}})$, where $*$ can be $\circ$ or $\bullet$. For each sequence, we define a space $(*_{x_{i_{1}}},*_{x_{i_{2}}},\ldots,*_{x_{i_{n-1}}},x_{i_{n}})_{(*,*,\ldots,*)}$ as follows:

For each space $(*_{x_{i_{1}}},*_{x_{i_{2}}},\ldots,*_{x_{i_{n-1}}},x_{i_{n}})_{(*,*,\ldots,*)}$, $n\geq 3$, we define a polynomial $p_{(*,*,\ldots,*,n)}$ as follows:

1) We divide this sequence into consecutive vertices of the same colour;

2) For $k_{i}$ consecutive white vertices, we select $k_{i}$ generators and write all possible permutations for them;

3) For $k_{i}$ consecutive black vertices, we select $k_{i}$ generating ones and write them in descending order;

4) If the sequence ends with $k_{i}$ vertices of white colour, then for the selected generator we write all possible permutations of $k_{i}$ generators so that the last two generators are always ordered;

5) If the sequence ends with $k_{i}$ vertices of black colour, then for the selected generator we write symmetric polynomials of the free metabelian Lie algebra on $k_{i+1}$ generators.

The sum of such monomials gives polynomial $p_{(*,*,\ldots,*,n)}$. For $n=1,2$, we set