A new properties of varieties of Leibnitz algebras

The paper is devoted to two new results concerning varieties of Leibnitz algebras over a field of the zero characteristic. Here is proved the sufficient condition for finiteness colength of variety of Leibnitz algebras. Here is also defined the basis of identities of variety V3 of Leibnitz algebras and the basis of its multilinear part.


Introduction
A linear algebra with bilinear multiplication, which is satisfies to the Leibnitz identity (xy)z ≡ (xz)y + x(yz), is called a Leibnitz algebra. Perhaps for the first time this concept was discussed in the article [2] as a generalization of Lie algebras. The Leibnitz identity allows any element expressed as a linear combination of elements in which the brackets are arranged from left to right. Therefore further agree omit brackets in left-normed products, i.e. (((ab)c) . . . d) = abc . . . d. A variety V of linear algebras over a field Φ is a set of algebras over this field that satisfy a fixed set of identities. Note, that the system of identities can be given implicitly. In this case the variety V is usually defined generating algebra given constructively.
Let F (X, V) be a relatively free algebra of variety V with countable set of free generators X = {x 1 , x 2 , . . . }. Consider the space of the multilinear elements of algebra F (X, V). This space we will denote P n (V) and call multilinear part of variety V. On this space naturally introduce the action of permutations that we can consider it as a ΦS n -module, where S n is a symmetric group. Since the field Φ has zero characteristic, then the space P n (V) is the direct sum of irreducible submodules. Denote by χ λ the character of the irreducible representations of the symmetric group, which corresponds to the partition λ of the number n. Then the character of module P n (V) is expressed by the formula where m λ are the multiplicities of irreducible submodules in this sum. An important numerical characteristic of variety V of linear algebras is the colength l n (V), which is defined as the number of terms in the decomposition of character in the sum of irreducible characters: We say that the colength V of variety is finite if there exists a constant C independent of n such that for any n is true the inequality l n (V) ≤ C. The right multiplication operator, for example, on element z, we denote by Z, assuming that xz = xZ. This designation allows the element xy...y n to write in the form xY n . Recall that the standard polynomial of degree n has the form: St n (x 1 , x 2 , . . . , x n ) = q∈Sn (−1) q x q(1) x q(2) . . . x q(n) , where the summation is carried out by elements of the symmetric group, and (−1) q is equal to +1 or −1 depending on the parity of permutation q. Agree variables in standard polynomial denote with special characters above (below, wave and so on). For example the standard polynomial of degree n in the variables x 1 , x 2 , . . . , x n we will write as follows: St n = x 1 x 2 . . . x n . It is clear that the standard polynomial is skew symmetric. Variables in different skew symmetric sets will be denoted by different symbols, for example:

Sufficient condition for the finiteness colength of varieties of Leibnitz algebras
Previously, in the article [7] are identified the necessary conditions for the finiteness colength of varieties of Leibnitz algebras. Further, we consider it sufficient conditions. Following article [6] denote the variety of all Leibnitz algebras (Lie algebras), that satisfy the identity ( . Let in addition V 1 = N 2 A is a variety of all Lie algebras, commutator of which is nilpotent of class not more then two, and V 1 is a variety of Leibnitz algebras defined by the identity Theorem 1. Let V be a subvariety of variety N s A, which for any natural numbers k, m, k ≤ m, and α 1 , . . . , α k ∈ K satisfies the identity Then the variety V has the final colength. Proof. Because identity (3) is not satisfied in the varieties V 1 and V 1 , from conditions of the theorem follows that V 1 , V 1 ⊂ V ⊂ N s A. Then by theorem 1 of article [4], there exists a constant C independent of n for such in the sum (1) is true the condition (n − λ 1 ) < C. In this case in the sum (2) the number of the non-zero terms is bounded by a constant independent of n. Thus, to prove the result, it suffices to establish that all multiplicities m λ are bounded by a constant, which also is independent of n.
The article [5] is proved that the multiplicity m λ (V) is equal to the number of linearly independent polyhomogeneous elements of special form. We will show that the dimension of the space of polyhomogeneous elements is bounded by a constant independent of n, which will complete the proof of the theorem.
Consider λ, for which m λ = 0. For such partition is true the condition (n − λ 1 ) < C and to it correspond monomials of the form where s < C. Denote by Q λ 1 the space generated by elements g s . We prove that the number of linearly independent monomials g s bounded by a constant. The proof is by induction on the number s of generators x i and lexicographic order on lines of the form (α 1 , α 2 , . . . , α s+1 ).
Consider the case s = 1. Then generating monomials of the space Q λ 1 have the form: Y α 1 x 1 Y α 2 . If for these elements are true the conditions α 1 ≥ m and α 2 ≥ m, then by the identity (3) they can be represented as a linear combination of the elements, in which α 1 < m. Thus any monomial will be expressed through such monomials in which either only α 1 ≥ m or α 2 ≥ m. The number of such monomials is bounded by the constant 2m independent of n.
In the general case the space Q λ 1 will generate by elements, in which only one α i is not less then tm. Note, that the general number of such elements is bounded by a constant independent of n.
Let i will be a smallest index for which α i ≥ tm. consider the corresponding element: If α i+1 ≥ tm, then the identity (3) allows to bring the element g s to a linear combination of words, that are lexicographically less.lexicographically smaller. If α i+1 < tm, then modulo words, lexicographically smaller, the element g s can be written as The Leibnitz identity allows to bring the last element to the sum of terms, that are lexicographically smaller, and term We will contain element with fewer generators x ir covered by the induction assumption. The theorem is proved.

The basis of multilinear part of variety V 3 of Leibnitz algebras
The variety V 3 of Leibnitz algebras is an equivalent to the well-known variety V 3 of Lie algebras. Previously, in the article [1] the growth of this variety was designated, and in the article [8] -its multiplicities and colength.
Let T = Φ[t] be a ring of polynomial in the variable t. Consider threedimensional Heisenberg algebra H with the besis {a, b, c} and multiplication ba = −ab = c, the product of the remaining basis elements is zero. Well known and easy to verify that the algebra H is nilpotent of the class two Lie algebra. Transform the polynomial ring T in the right module of algebra H, in which the basis elements of algebra H act on the right on the polynomial f from T follows: Consider the direct sum of vector spaces H and T with multiplication by the rule: (x + f )(y + g) = xy + f y, where x, y are from H; f, g are from T . Denote it by the symbol H. Direct verification shows that H is an algebra of Leibnitz. The algebra H is the Leibnitz algebra, satisfies to the identity x(y(zt)) ≡ 0 and generates the variety V 3 of Leibnitz algebras.
Lemma. The variety V 3 satisfies to the identities: (4) x(y(zt)) ≡ 0, where A, B, C, D are some words from generators. Proof. The truth of identities (4) and (5) verified by arbitrary replacement generators by elements of algebra H and was showed in the paper [1]. Consider the following special form of the second identity: Presenting it as a sum and using the identity xyz − xzy ≡ x(yz), we obtain: Dividing this identity by 2 and moving the second term to the right, we obtain the identity (6). The lemma is proved.
Proof. Consider an arbitrary element of the space P n ( V 3 ). Using corollary xy(zt) ≡ x(zt)y from the Leibnitz identity and identity (4), move the all pairs as far right as possible.
We order the elements obtained using the lexicographic ordering of lines (k 1 , k 2 , . . . , k n−2m−1 ). Let the considering element has a form: and k s > k s+1 . Using the identity xyz ≡ xzy + x(yz) we can write this element as a sum where the first term is lexicographically less, than parent element, and the second term has fewer number of single elements. Applying the same method to the resulting term, we eventually present our original element as a sum of terms, in that k 1 < k 2 < ... < k n−2m−1 .
Consider an arbitrary element, in which indexes of single elements are ordered. We choose the two lowest index in the considered element and redenote them through 1 ′ and 2 ′ relatively. We introduce the lexicographic order on lines (j 1 , j 2 , . . . , j m ). Using also the induction on the number of brackets, we will prove, that all received elements can be represented as a linear combination of elements θ(i, i 1 , ..., i m , j 1 , ..., j m ). The corollary x(yz) ≡ −x(zy) of Leibnitz identity allows to order the indexes of elements in couples, and the identity xy(zt) ≡ x(zt)y allows to order the brackets by the indexes of first elements. According to these identities, the element can be written either in the form ..x k n−2m−1 , either in the form ..x k n−2m−1 . In the first case we can consider the ordering on m−1 brackets that runs by induction. In the second case we apply the identity (6) and obtain: ..x k n−2m−1 , where for the first term can be again apply the induction hypothesis, and the second term is lexicographically less. Therefore, any element of the space P n ( V 3 ) can be written as a linear combination of elements θ(i, i 1 , ..., i m , j 1 , ..., j m ) modulo Id( V 3 ).