The classification of naturally graded Zinbiel algebras with characteristic sequence equal to $(n - p,\, p) $
UDC 512.5 This work is a continuation of the description of some classes of nilpotent Zinbiel algebras. We focus on the study of Zinbiel algebras with restrictions imposed on gradation and characteristic sequence. Namely, we obtain the classification of naturally graded Zinbiel algebras with cha...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507266326724608 |
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| author | Adashev, J. K. Ladra, M. Omirov, B. A. Адашев, Й. К. Ладра, М. Оміров, Б. А. |
| author_facet | Adashev, J. K. Ladra, M. Omirov, B. A. Адашев, Й. К. Ладра, М. Оміров, Б. А. |
| author_sort | Adashev, J. K. |
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| description | UDC 512.5
This work is a continuation of the description of some classes of nilpotent Zinbiel algebras.
We focus on the study of Zinbiel algebras with restrictions imposed on gradation and characteristic sequence.
Namely, we obtain the classification of naturally graded Zinbiel algebras with characteristic sequence equal to $ (n-p, p)$. |
| first_indexed | 2026-03-24T02:06:35Z |
| format | Article |
| fulltext |
UDC 512.5
J. K. Adashev (Inst. Math., Nat. Univ. Uzbekistan, Tashkent),
M. Ladra (Univ. Santiago de Compostela, Spain),
B. A. Omirov (Inst. Math., Nat. Univ. Uzbekistan, Tashkent)
THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS
WITH CHARACTERISTIC SEQUENCE EQUAL TO (\bfitn - \bfitp , \bfitp ) *
КЛАСИФIКАЦIЯ ПРИРОДНО ГРАДУЙОВАНИХ АЛГЕБР ЗIНБЄЛЯ
З ХАРАКТЕРИСТИЧНОЮ ПОСЛIДОВНIСТЮ (\bfitn - \bfitp , \bfitp )
This work is a continuation of the description of some classes of nilpotent Zinbiel algebras. We focus on the study of
Zinbiel algebras with restrictions imposed on gradation and characteristic sequence. Namely, we obtain the classification
of naturally graded Zinbiel algebras with characteristic sequence equal to (n - p, p).
Продовжено опис деяких класiв нiльпотентних алгебр Зiнбєля. Основну увагу зосереджено на вивченнi алгебр
Зiнбєля з обмеженнями на градацiю та характеристичну послiдовнiсть, а саме, отримано класифiкацiю природно
градуйованих алгебр Зiнбєля з характеристичною послiдовнiстю (n - p, p).
1. Introduction. This paper is devoted to investigation of algebras, which are Koszul dual to Leibniz
algebras. These algebras were introduced in the middle of 90th of the last century by the French
mathematician J.-L. Loday [15] and they are called Zinbiel algebras (Leibniz written in reverse
order).
A crucial fact of the theory of finite dimensional Zinbiel algebras is the nilpotency of such
algebras over a field of zero characteristic [12]. Since the description of finite-dimensional complex
Zinbiel algebras is a boundless problem (even if they are nilpotent), their study should be carried out
by adding some additional restrictions (on index of nilpotency, gradation, characteristic sequence,
etc.).
In general, investigation of Zinbiel algebras goes parallel to the study of nilpotent Leibniz alge-
bras. For instance, n-dimensional Leibniz algebras of nilindices n + 1 and n (which is equivalent
to admit characteristic sequences equal to (n) and (n - 1, 1), respectively) were described in papers
[5] and [14]. Similar description for Zinbiel algebras were obtained in the paper [3].
In the study of n-dimensional Leibniz algebras of nilindex n - 1 (see [10]) it was noted that
characteristic sequences of such algebras are equal to either (n - 2, 1, 1) or (n - 2, 2). Descrip-
tion of Leibniz (Zinbiel) algebras with such characteristic sequence were obtained in [9] and [10]
(respectively, [4]).
Later on, naturally graded Leibniz algebras of nilindex n - 2 that admit the following character-
istic sequences:
(n - 3, 3), (n - 3, 2, 1), (n - 3, 1, 1, 1)
* The work was partially supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European
FEDER support included, UE), and by Xunta de Galicia, grant ED431C2019/10 (European FEDER support included).
c\bigcirc J. K. ADASHEV, M. LADRA, B. A. OMIROV, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7 867
868 J. K. ADASHEV, M. LADRA, B. A. OMIROV
were investigated in a series of papers [6, 7, 11], respectively. Description of naturally graded Zinbiel
algebras with these properties was given in [1] and [2].
Finally, the latest progress in the description of the structure of nilpotent Leibniz algebras was
obtained in papers [11] and [16]. In particular, naturally graded nilpotent n-dimensional Leibniz
algebras with characteristic sequences equal to (n - p, p) and (n - p, 1, . . . , 1) were described. Since
the description of p-filiform Zinbiel algebras (that are Zinbiel algebras with characteristic sequence
equal to (n - p, 1, . . . , 1)) was obtained in [8], in order to complete the description similar to [11], in
this paper we present the description (up to isomorphism) of naturally graded Zinbiel algebras with
characteristic sequence equal to (n - p, p).
All considered algebras and vector spaces in this work are assumed to be finite dimensional and
complex. In order to keep tables of multiplications of algebras short, we will omit zero products.
2. Preliminaries. In this section we give definitions and known results necessary to proceed
further to the main part of the work.
Definition 2.1. An algebra A over a field F is called a Zinbiel algebra if for any x, y, z \in A
the following identity holds:
(x \circ y) \circ z = x \circ (y \circ z) + x \circ (z \circ y),
where \circ is the multiplication of the algebra A.
For an arbitrary Zinbiel algebra we define the lower series as follows:
A1 = A, Ak+1 = A \circ Ak, k \geq 1.
Definition 2.2. A Zinbiel algebra A is called nilpotent if there exists s \in \BbbN such that As = 0.
The minimal such number is called the nilindex of A.
Definition 2.3. An n-dimensional Zinbiel algebra A is called null-filiform if \mathrm{d}\mathrm{i}\mathrm{m}Ai = (n+1) - i
for 1 \leq i \leq n+ 1.
It is clear by definition that an algebra A being null-filiform is equivalent to admitting the
maximal possible nilindex.
Let x be an element of the set A \setminus A2. For an operator of a left multiplication Lx (defined as
Lx(y) = x \circ y) we define a descending sequence C(x) = (n1, n2, . . . , nk), where n = n1+n2+ . . .
. . .+nk, which consists of the sizes of Jordan blocks of the operator Lx. On the set of such sequences
we consider the lexicographical order, that is, (n1, n2, . . . , nk) \leq (m1,m2, . . . ,ms) if there exists
i \in N such that nj = mj for all j < i and ni < mi.
Definition 2.4. The sequence C(A) = \mathrm{m}\mathrm{a}\mathrm{x}x\in A\setminus A2 C(x) is called the characteristic sequence of
the algebra A.
Example 2.1. Let C(A) = (1, 1, . . . , 1). Then the algebra A is Abelian.
Example 2.2. An n-dimensional Zinbiel algebra A is null-filiform if and only if C(A) = (n).
Let A be a finite-dimensional Zinbiel algebra of nilindex s. We set Ai := Ai/Ai+1, 1 \leq i \leq s - 1,
and \mathrm{g}\mathrm{r}A := A1 \oplus A2 \oplus \cdot \cdot \cdot \oplus As - 1. From the condition Ai \circ Aj \subseteq Ai+j we derive a graded algebra
\mathrm{g}\mathrm{r}A. The graduation constructed in a such way is called the natural graduation. If a Zinbiel algebra
A is isomorphic to the algebra \mathrm{g}\mathrm{r}A, then the algebra A is called a naturally graded Zinbiel algebra.
Further we need the following lemmas.
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THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 869
Lemma 2.1 [13]. For any n, a \in \BbbN the following equality holds:
n\sum
k=0
( - 1)kCk
aC
n - k
a+n - k - 1 = 0.
Lemma 2.2 [12]. Let A be a Zinbiel algebra with the following products to be known:
e1 \circ ei = ei+1, 1 \leq i \leq k - 1.
Then
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq k,
where Cb
a =
\bigl(
a
b
\bigr)
denotes the binomial coefficient.
3. Main results. Let A be a Zinbiel algebra and C(A) = (n1, n2, . . . , nk) its characteristic
sequence. Then there exists a basis \{ e1, e2, . . . , en\} such that the matrix of an operator of the left
multiplication by the element e1 has the form
Le1,\sigma =
\left(
Jn\sigma (1)
0 . . . 0
0 Jn\sigma (2)
. . . 0
. . . . . . . . . . . .
0 0 . . . Jn\sigma (s)
\right) ,
where \sigma (i) belongs to \{ 1, 2, . . . , s\} .
By a suitable permutation of basis elements we can assume that n\sigma (2) \geq n\sigma (3) \geq . . . \geq n\sigma (s).
Let A be a naturally graded Zinbiel algebra with characteristic sequence equal to (n1, n2, . . . , nk).
Proposition 3.1. There is no naturally graded Zinbiel algebra with n\sigma (1) = 1 and n\sigma (2) \geq 4.
Proof. From the condition of the proposition we have the products
e1 \circ e1 = 0, e1 \circ ei = ei+1, 2 \leq i \leq 4,
e1 \circ ei = ei+1,
t\sum
k=1
n\sigma (k) + 1 \leq i \leq
t+1\sum
k=1
n\sigma (k) - 1, 3 \leq t \leq s - 1,
e1 \circ ei = 0, i =
t\sum
k=1
n\sigma (k), 3 \leq t \leq s.
By using the property of Zinbiel algebras
(a \circ b) \circ c = (a \circ c) \circ b,
we obtain
e3 \circ e1 = (e1 \circ e2) \circ e1 = (e1 \circ e1) \circ e2 = 0 \Rightarrow e3 \circ e1 = 0.
The chain of equalities
0 = (e1 \circ e1) \circ e3 = e1 \circ (e1 \circ e3) + e1 \circ (e3 \circ e1) = e1 \circ e4 = e5
implies e5 = 0, that is, we get a contradiction with the condition n\sigma (2) \geq 4 which completes the
proof of the proposition.
The next example shows that the condition n\sigma (2) \geq 4 of Proposition 3.1 is essential.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
870 J. K. ADASHEV, M. LADRA, B. A. OMIROV
Example 3.1. Let A be a four-dimensional Zinbiel algebra with the multiplication table
e1 \circ e2 = e3, e1 \circ e3 = e4, e2 \circ e1 = - e3.
Then C(A) = (3, 1) and the matrix of the operator of the left multiplication on e1 has the
form
\biggl(
J1 0
0 J3
\biggr)
.
Let A be an arbitrary Zinbiel algebra with characteristic sequence equal to (n - p, p). Then the
matrix of the operator of the left multiplication by e1 admits one of the following forms:
\mathrm{I}.
\biggl(
Jn - p 0
0 Jp
\biggr)
; \mathrm{I}\mathrm{I}.
\biggl(
Jp 0
0 Jn - p
\biggr)
, n \geq 2p.
Definition 3.1. A Zinbiel algebra is called an algebra of the first type (of type I) if the operator
Le1 has the form
\biggl(
Jn - p 0
0 Jp
\biggr)
; otherwise it is called an algebra of the second type (of type II).
Taking into account results of papers [4] and [3], we will consider only n-dimensional naturally
graded Zinbiel algebras with C(A) = (n - p, p), p \geq 3.
3.1. Classification of Zinbiel algebras of type \bfI . Let A be a Zinbiel algebra of type I. Then we
have the existence of a basis \{ e1, e2, . . . , en - p, f1, f2, . . . , fp\} such that the products containing an
element e1 on the left are as follows:
e1 \circ ei = ei+1, 1 \leq i \leq n - p - 1.
From Lemma 2.2 we obtain
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq n - p, e1 \circ ep = 0,
e1 \circ fi = fi+1, 1 \leq i \leq p - 1, e1 \circ fp = 0.
(1)
It is easy to see that
A1 = \langle e1, f1\rangle , A2 = \langle e2, f2\rangle , . . . , Ap = \langle ep, fp\rangle , Ap+1 = \langle ep+1\rangle , . . . , An - p = \langle en - p\rangle .
Let
f1 \circ ei = \alpha iei+1 + \beta ifi+1, 1 \leq i \leq p - 1,
f1 \circ ei = \alpha iei+1, p \leq i \leq n - p - 1,
(2)
f1 \circ fi = \gamma iei+1 + \delta ifi+1, 1 \leq i \leq p - 1,
f1 \circ fp = \gamma pep+1.
Proposition 3.2. Let A be a Zinbiel algebra of type I. Then, for the structural constants \alpha i, \beta i,
\gamma i and \delta i, we have the following restrictions:
\alpha i+1 = 0, 1 \leq i \leq n - p - 2,
\beta i+1 =
i\prod
k=0
k + \beta 1
k + 1
, 1 \leq i \leq p - 2,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 871
(i+ 1)\gamma i = \beta 1
\Biggl(
2\gamma 1 +
i\sum
k=2
\gamma k
\Biggr)
, 1 \leq i \leq n - p - 2,
(i+ \beta 1)\delta i = \beta 1
\Biggl(
2\delta 1 +
i\sum
k=2
\delta k
\Biggr)
, 1 \leq i \leq p - 2.
Proof. First, we calculate the products fi \circ e1 and f2 \circ ei.
Consider
f2 \circ e1 = e1 \circ (f1 \circ e1) + e1 \circ (e1 \circ f1) = \alpha 1e3 + (1 + \beta 1)f3.
Using the chain of equalities
fi \circ e1 = (e1 \circ fi - 1) \circ e1 = e1 \circ (fi - 1 \circ e1) + e1 \circ (e1 \circ fi - 1),
we deduce fi \circ e1 = \alpha 1ei+1 + (i - 1 + \beta 1)fi+1 for 1 \leq i \leq p - 1 and fi \circ e1 = \alpha 1ei+1 for
p \leq i \leq n - p - 1.
From the equality
ei \circ f1 = (e1 \circ ei - 1) \circ f1 = e1 \circ (ei - 1 \circ f1) + e1 \circ (f1 \circ ei - 1),
we obtain ei \circ f1 =
\sum i - 1
k=1
\alpha kei+1 +
\sum i - 1
k=0
\beta kfi+1 for 1 \leq i \leq p - 1 and ei \circ f1 =
\sum i - 1
k=1
\alpha kei+1
for p \leq i \leq n - p - 1.
f2 \circ e2 = e1 \circ (f1 \circ e2) + e1 \circ (e2 \circ f1) = (\alpha 1 + \alpha 2)e4 + (1 + \beta 1 + \beta 2)f4.
From f2 \circ ei = (e1 \circ f1) \circ ei = e1 \circ (f1 \circ ei) + e1 \circ (ei \circ f1), we have
f2 \circ ei =
i\sum
k=1
\alpha kei+2 +
i\sum
k=0
\beta kfi+2, 1 \leq i \leq p - 2,
f2 \circ ei =
i\sum
k=1
\alpha kei+2, p - 1 \leq i \leq n - p - 2.
Now we calculate the products fi \circ f1 and f2 \circ fi. We get
f2 \circ f1 = (e1 \circ f1) \circ f1 = 2\gamma 1e3 + 2\delta 1f3,
f3 \circ f1 = (e1 \circ f2) \circ f1 = (2\gamma 1 + \gamma 2)e4 + (2\delta 1 + \delta 2)f4.
By induction we obtain
fi \circ f1 =
\Biggl(
2\gamma 1 +
i - 1\sum
k=2
\gamma k
\Biggr)
ei+1 +
\Biggl(
2\delta 1 +
i - 1\sum
k=2
\delta k
\Biggr)
fi+1, 2 \leq i \leq p - 1,
fp \circ f1 =
\Biggl(
2\gamma 1 +
p - 1\sum
k=2
\gamma k
\Biggr)
ep+1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
872 J. K. ADASHEV, M. LADRA, B. A. OMIROV
Similarly, from f2 \circ fi = (e1 \circ f1) \circ fi = e1 \circ (f1 \circ fi) + e1 \circ (fi \circ f1) we derive
f2 \circ fi =
\Biggl(
2\gamma 1 +
i\sum
k=2
\gamma k
\Biggr)
ei+2 +
\Biggl(
2\delta 1 +
i\sum
k=2
\delta k
\Biggr)
fi+2, 2 \leq i \leq p - 2,
f2 \circ fi =
\Biggl(
2\gamma 1 +
i\sum
k=2
\gamma k
\Biggr)
ei+2, p - 1 \leq i \leq p.
If \beta 1 = 1, then from the equality (f1 \circ f1) \circ e1 = (f1 \circ e1) \circ f1 we get
2\gamma 1(1 - \beta 1) = \alpha 2
1 - \delta 1\alpha 1, (1 - \beta 1)\delta 1 = \alpha 1(1 + \beta 1).
Consequently, \alpha 1 = 0.
Let \beta 1 \not = 1. Then taking the following change:
e\prime 1 = e1, f \prime
1 =
\alpha 1
\beta 1 - 1
e1 + f1,
we obtain \alpha \prime
1 = 0.
Form the equalities
f1 \circ ei+1 = f1 \circ (e1 \circ ei) = (f1 \circ e1) \circ ei - if1 \circ ei+1 = \beta 1f2 \circ ei - if1 \circ ei+1,
we derive (i+ 1)f1 \circ ei+1 = \beta 1f2 \circ ei. Therefore,
(i+ 1)\alpha i+1ei+2 + (i+ 1)\beta i+1fi+2 = \beta 1
\Biggl(
i\sum
k=1
\alpha kei+2 +
i\sum
k=0
\beta kfi+2
\Biggr)
.
Comparing coefficients at the basis elements and applying induction, we deduce
\alpha i+1 = 0, 1 \leq i \leq n - p - 2,
\beta i+1 =
i\prod
k=0
k + \beta 1
k + 1
, 1 \leq i \leq p - 2.
Considering the equality (f1 \circ fi) \circ e1 = (f1 \circ e1) \circ fi leads to the rest of the restrictions of the
proposition.
In the next proposition we calculate the products ei \circ fj and fj \circ ei.
Proposition 3.3. Let A be a Zinbiel algebra of type I. Then the following expressions are true:
ei \circ fj =
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j for 2 \leq i+ j \leq p, (3)
fi \circ ej =
j\sum
k=0
Ci - 2
i+j - 2 - k\beta kfi+j for 2 \leq i+ j \leq p, (4)
where \beta 0 = 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 873
Proof. We shall prove (3), (4) by induction. From (1) and (3.1) we get the correctness of (3) and
(4) for i = 1.
Consider (4) for j = 1. Since f1\circ e1 = \beta 1f2 and f2\circ e1 = e1\circ (f1\circ e1)+e1\circ (e1\circ f1) = (1+\beta 1)f3,
then using the equalities fi \circ e1 = (e1 \circ fi - 1) \circ e1 = e1 \circ (fi - 1 \circ e1) + e1 \circ (e1 \circ fi - 1) and induction,
we deduce fi \circ e1 = (i - 1 + \beta 1)fi+1 for 1 \leq i \leq p - 1. From
ei \circ f1 = (e1 \circ ei - 1) \circ f1 = e1 \circ (ei - 1 \circ f1) + e1 \circ (f1 \circ ei - 1),
it implies
ei \circ f1 =
i - 1\sum
k=0
\beta kfi+1 for 1 \leq i \leq p - 1.
Therefore, the equalities (3) are true for j = 1 and arbitrary i.
Let us suppose that expressions (3), (4) are true for i and any value of j. The proof of the
expressions for i+ 1 is obtained by the following chain of equalities:
ei+1 \circ fj = e1 \circ (ei \circ fj) + e1 \circ (fj \circ ei) =
= e1 \circ
\Biggl(
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta kfi+j
\Biggr)
=
=
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j+1 +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta kfi+j+1 =
=
\Biggl(
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta k +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta k
\Biggr)
fi+j+1 =
i\sum
k=0
Cj - 1
i+j - 1 - k\beta kfi+j+1.
Here we used the well-known formula Cm - 1
n + Cm
n = Cm
n+1.
The proof of expressions (4) is analogous.
Below, we clarify the restrictions on structural constants of the algebra with relation to the
dimension and the parameter \beta 1.
Proposition 3.4. Let A be a Zinbiel algebra of type I. Then the following restrictions are true:
(1) Case \mathrm{d}\mathrm{i}\mathrm{m}A \geq 2p+ 1. If \beta 1 \not = 1, then
\gamma i = 0, 1 \leq i \leq p - 1,
\delta i = 0, 1 \leq i \leq p - 2,
(p - 1 + \beta 1)\gamma p = 0,
(p - 2 + \beta 1)\delta p - 1 = 0.
If \beta 1 = 1, then
\beta i = 1, 1 \leq i \leq p - 1,
\gamma i = \gamma 1, 1 \leq i \leq p,
\delta i = \delta 1, 1 \leq i \leq p - 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
874 J. K. ADASHEV, M. LADRA, B. A. OMIROV
(2) Case \mathrm{d}\mathrm{i}\mathrm{m}A = 2p. If \beta 1 \not = 1, then
\gamma i = 0, 1 \leq i \leq p - 2,
\delta i = 0, 1 \leq i \leq p - 2,
(p - 2 + \beta 1)\gamma p - 1 = 0,
(p - 2 + \beta 1)\delta p - 1 = 0.
If \beta 1 = 1, then
\beta i = 1, 1 \leq i \leq p - 1,
\gamma i = \gamma 1, 1 \leq i \leq p - 1,
\delta i = \delta 1, 1 \leq i \leq p - 1.
Proof. Let \mathrm{d}\mathrm{i}\mathrm{m}A \geq 2p+ 1. Then from Proposition 3.2 we have
(i+ 1)\gamma i = \beta 1
\Biggl(
2\gamma 1 +
k\sum
k=2
\gamma k
\Biggr)
, 1 \leq i \leq p - 1,
(i+ \beta 1)\delta i = \beta 1
\Biggl(
2\delta 1 +
k\sum
k=2
\delta k
\Biggr)
, 1 \leq i \leq p - 2.
(5)
Consider
(f1 \circ fi) \circ e1 = f1 \circ (fi \circ e1) + f1 \circ (e1 \circ fi) = (i+ \beta 1)(\gamma i+1ei+2 + \delta i+1fi+2).
On the other hand,
(f1 \circ fi) \circ e1 = (\gamma iei+1 + \delta ifi+1) \circ e1 = (i+ 1)\gamma iei+2 + (i+ \beta 1)\delta ifi+2.
Hence,
(i+ 1)\gamma i = (i+ \beta 1)\gamma i+1, 1 \leq i \leq p - 1,
(i+ \beta 1)\delta i = (i+ \beta 1)\delta i+1, 1 \leq i \leq p - 2.
(6)
Considering the cases \beta 1 \not = 1 and \beta 1 = 1 together with the expressions (5) and (6) leads to the
restrictions of the case \mathrm{d}\mathrm{i}\mathrm{m}A \geq 2p + 1. The proof of the remaining case is carried out in a similar
fashion.
Consider a general change of basis of the algebra A. It is known that for naturally graded Zinbiel
algebras it is sufficient to take the change of basis in the form
e\prime 1 = Ae1 +Bf1, f \prime
1 = Ce1 +Df1,
where AD - BC \not = 0.
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THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 875
Proposition 3.5. Let A be a Zinbiel algebra of type I and let \beta 1 \not = 1. Then
e\prime i+1 = e\prime 1 \circ e\prime i, 1 \leq i \leq n - p - 1,
f \prime
i+1 = e\prime 1 \circ f \prime
i , 1 \leq i \leq p - 1,
e\prime i = Aiei +Ai - 1B
i - 1\sum
k=0
\beta kfi, 1 \leq i \leq p - 1,
f \prime
i = Ai - 1Dfi, 1 \leq i \leq p - 1,
C = 0.
Proof. From f \prime
1\circ f \prime
1 = 0 we get C = 0. The proof of the proposition is completed by considering
products e\prime 1 \circ e\prime i = e\prime i+1 and e\prime 1 \circ f \prime
i = f \prime
i+1.
Theorem 3.1. Let A be an n-dimensional (n \geq 2p + 2) Zinbiel algebra of type I and with
characteristic sequence equal to (n - p, p). Then it is isomorphic to one of the following non-
isomorphic algebras:
A1 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq n - p,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 2 and \beta 1 \in \BbbC ;
A2 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq n - p, f1 \circ fp - 1 = fp,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta i = ( - 1)iCi
p - 2 for 1 \leq i \leq p - 2;
A3 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq n - p,
ei \circ fj = fi \circ ej = fi \circ fj = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p.
Proof. From Proposition 3.4 for \beta 1 \not = 1 we obtain a multiplication table of the algebra:
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq n - p,
f1 \circ fi = \delta ifi+1, 1 \leq i \leq p - 1, fi \circ fj = \varphi (\delta 1, \delta 2, . . . , \delta s)fi+j , 2 \leq i+ j \leq p,
ei \circ fj =
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
j\sum
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where (p - 1 + \beta 1)\gamma p = 0, (p - 2 + \beta 1)\delta p - 1 = 0.
Consider
(f1 \circ fp) \circ e1 = f1 \circ (fp \circ e1) + f1 \circ (e1 \circ fp) = 0.
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876 J. K. ADASHEV, M. LADRA, B. A. OMIROV
On the other hand,
(f1 \circ fp) \circ e1 = \gamma pep+1 \circ e1 = (p+ 1)\gamma pep+2.
Therefore, we deduce \gamma p = 0.
Applying Proposition 3.5 to the general change of basis, from the equalities
f \prime
1 \circ f \prime
p - 1 = (Df1) \circ
\bigl(
Ap - 2Dfp - 1
\bigr)
= Ap - 2D2\delta p - 1fp =
= \delta \prime p - 1f
\prime
p = \delta \prime p - 1
\bigl(
Ap - 1D +Ap - 2BD\delta p - 1
\bigr)
fp,
we get \delta \prime p - 1 =
D\delta p - 1
A+B\delta p - 1
.
If \delta p - 1 = 0, then \delta \prime p - 1 = 0, and we obtain the algebra A1.
If \delta p - 1 \not = 0, then, by choosing D =
A+B\delta p - 1
\delta p - 1
and from (p - 2 + \beta 1)\delta p - 1 = 0, we have
\delta \prime p - 1 = 1, \beta 1 = 2 - p, that is, we have the algebra A2.
In case of \beta 1 = 1 we have \delta i = \delta 1, 1 \leq i \leq p - 1. Putting D =
A+B\delta 1
\delta 1
we obtain \delta \prime 1 = 1.
Consequently, f1 \circ fi = fi+1 for 1 \leq i \leq p - 1. From Lemma 2.2 we deduce fi \circ fj = Cj
i+j - 1fi+j ,
2 \leq i+ j \leq p. Thus, we get the algebra A3.
In the following theorem the classification for n = 2p+ 1 is presented.
Theorem 3.2. Let A be a Zinbiel algebra of type I and with characteristic sequence equal to
(p+ 1, p). Then it is isomorphic to one of the following non-isomorphic algebras:
A4 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p+ 1, f1 \circ fp = ep+1,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, \beta i = ( - 1)iCi
p - 1 for 1 \leq i \leq p - 1;
A5 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p+ 1,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 2 and \beta 1 \in \BbbC ;
A6 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p+ 1, f1 \circ fp - 1 = fp,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, \beta i = ( - 1)iCi
p - 2 for 1 \leq i \leq p - 2 and \beta p - 1 = 0;
A7 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p+ 1,
ei \circ fj = fi \circ ej = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p,
fi \circ fj = \gamma 1C
j
i+j - 1ei+j + \delta 1C
j
i+j - 1fi+j , 2 \leq i+ j \leq p,
fi \circ fj = \gamma 1C
j
i+j - 1ep+1, i+ j = p+ 1,
where \gamma 1, \delta 1 \in \BbbC .
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THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 877
Proof. From Proposition 3.4 for \beta 1 \not = 1 we obtain a multiplication table of A
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p+ 1,
f1 \circ fp - 1 = \delta p - 1fp, f1 \circ fp = \gamma pep+1,
ei \circ fj =
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
j\sum
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, (p - 1 + \beta 1)\gamma p = 0 and (p - 2 + \beta 1)\delta p - 1 = 0.
Consider the general change of basis as above. Then from Proposition 3.5 we have
e\prime p = Apep +
\Biggl(
Ap - 1B
p - 1\sum
k=0
\beta k +Ap - 2B2
p - 2\sum
k=0
\beta k\delta p - 1
\Biggr)
fp,
e\prime p+1 =
\Biggl(
Ap+1 +
\Biggl(
Ap - 1B2
p - 1\sum
k=0
\beta k +Ap - 2B3
p - 2\sum
k=0
\beta k\delta p - 1
\Biggr)
\gamma p
\Biggr)
ep+1,
f \prime
p =
\bigl(
Ap - 1D +Ap - 2BD\delta p - 1
\bigr)
fp.
The equality e\prime 1 \circ f \prime
p = 0 in the new basis implies B\gamma p = 0.
Case 1. Let \gamma p \not = 0. Then B = 0 and \beta 1 = 1 - p, \delta p - 1 = 0.
Considering the equality f \prime
1 \circ f \prime
p = \gamma \prime pe
\prime
p+1, we derive A2\gamma \prime p = D2\gamma p.
Setting D =
A
\surd
\gamma p
, we obtain \gamma \prime p = 1. Thus, we get the algebra A4.
Case 2. Let \gamma p = 0. Then, considering the equality f \prime
1 \circ f \prime
p - 1 = \delta \prime p - 1f
\prime
p, we deduce \delta \prime p - 1 =
=
D\delta p - 1
A+B\delta p - 1
.
If \delta p - 1 = 0, then \delta \prime p - 1 = 0, that is, we obtain the algebra A5.
If \delta p - 1 \not = 0, then \beta 1 = 2 - p and putting D =
A+B\delta p - 1
\delta p - 1
, we get \delta \prime p - 1 = 1 and the algebra A6.
Now we consider case \beta 1 = 1. Using Proposition 3.4, we obtain the algebra A7.
Below, we present the classification of Zinbiel algebras with characteristic sequence equal to
C(A) = (p, p).
Theorem 3.3. Let A be a Zinbiel algebra with characteristic sequence (p, p). Then it is isomor-
phic to one of the following non-isomorphic algebras:
A8 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 2 and \beta 1 \in \BbbC ;
A9 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp - 1 = fp,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
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878 J. K. ADASHEV, M. LADRA, B. A. OMIROV
where \beta 0 = 1, \beta i = ( - 1)iCi
p - 2 for 1 \leq i \leq p - 2 and \beta p - 1 = 0;
A10 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp - 1 = ep + \delta p - 1fp,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p,
where \beta 0 = 1, \beta i = ( - 1)iCi
p - 2 for 1 \leq i \leq p - 2, \beta p - 1 = 0 and \delta p - 1 \in \BbbC ;
A11 : ei \circ ej = Cj
i+j - 1ei+j , ei \circ fj = fi \circ ej = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p,
A12 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj = fi \circ ej = fi \circ fj = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p.
Proof. The proof of this theorem is carried out by applying the methods and arguments as in the
proof of Theorems 3.1 and 3.2.
3.2. Classification of Zinbiel algebras of type \bfI \bfI . Consider a Zinbiel algebra of type II. From
the condition on the operator Le1 we have the existence of a basis \{ e1, e2, . . . , ep, f1, f2, . . . , fn - p\}
such that the products involving e1 on the left-hand side have the form
e1 \circ ei = ei+1, 1 \leq i \leq p - 1.
Applying Lemma 2.2, we get
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, e1 \circ ep = 0,
e1 \circ fi = fi+1, 1 \leq i \leq n - p - 1, e1 \circ fn - p = 0.
It is easy to see that
A1 = \langle e1, f1\rangle , A2 = \langle e2, f2\rangle , . . . , Ap = \langle ep, fp\rangle , Ap+1 = \langle fp+1\rangle , . . . , An - p = \langle fn - p\rangle .
Let us introduce notations:
f1 \circ ei = \alpha iei+1 + \beta ifi+1, 1 \leq i \leq p - 1, f1 \circ ep = \beta pfp+1,
f1 \circ fi = \gamma iei+1 + \delta ifi+1, 1 \leq i \leq p - 1,
f1 \circ fi = \delta ifi+1, p \leq i \leq n - p - 1, f1 \circ fn - p = 0.
The following proposition can be proved similar to Proposition 3.2.
Proposition 3.6. Let A be a Zinbiel algebra of type II. Then for structural constants \alpha i, \beta i, \gamma i
and \delta i the following restrictions hold:
\alpha i+1 = 0, 1 \leq i \leq p - 2,
\beta i+1 =
i\prod
k=0
k + \beta 1
k + 1
, 1 \leq i \leq p - 1,
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THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 879
(i+ 1)\gamma i = \beta 1
\Biggl(
2\gamma 1 +
i\sum
k=2
\gamma k
\Biggr)
, 1 \leq i \leq p - 2,
(i+ \beta 1)\delta i = \beta 1
\Biggl(
2\delta 1 +
i\sum
k=2
\delta k
\Biggr)
, 1 \leq i \leq n - p - 2.
Proposition 3.7. Let A be a Zinbiel algebra of type II. Then the following expressions hold:
ei \circ fj =
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, p+ 1 \leq i+ j \leq n - p, (7)
fi \circ ej =
j\sum
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, p+ 1 \leq i+ j \leq n - p, (8)
where \beta 0 = 1.
Proof. We shall prove the assertion of the proposition be induction. Clearly, the relation (7) is
true for i = 1.
We have
f2 \circ e1 = e1 \circ (f1 \circ e1) + e1 \circ (e1 \circ f1) = (1 + \beta 1)f3.
Using the chain of equalities
fi \circ e1 = (e1 \circ fi - 1) \circ e1 = e1 \circ (fi - 1 \circ e1) + e1 \circ (e1 \circ fi - 1),
and induction, we derive fi \circ e1 = (i - 1+\beta 1)fi+1 for 1 \leq i \leq p - 1 and fi \circ e1 = (i - 1+\beta 1)fi+1
for p \leq i \leq n - p - 1. Therefore, the relation (8) is true for j = 1.
Let us assume that the relations (7), (8) are true for i and any value of j. The proof of these
relations for i+ 1 follows from the following chain of equalities:
ei+1 \circ fj = e1 \circ (ei \circ fj) + e1 \circ (fj \circ ei) =
= e1 \circ
\Biggl(
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta kfi+j
\Biggr)
=
=
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j+1 +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta kfi+j+1 =
=
\Biggl(
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta k +
i\sum
k=0
Cj - 2
i+j - 2 - k\beta k
\Biggr)
fi+j+1 =
i\sum
k=0
Cj - 1
i+j - 1 - k\beta kfi+j+1.
Checking the correctness of the remaining relations of the proposition is analogous.
Similar to the case of Zinbiel algebra of type I for algebras of type II we obtain the restrictions
on structure constants with relation to parameter \beta 1.
Proposition 3.8. Let A be a Zinbiel algebra of type II.
If \beta 1 \not = 1, then
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880 J. K. ADASHEV, M. LADRA, B. A. OMIROV
\gamma i = 0, 1 \leq i \leq p - 2,
\delta i = 0, 1 \leq i \leq n - p - 2,
(p - 2 + \beta 1)\gamma p - 1 = 0,
(n - p - 2 + \beta 1)\delta n - p - 1 = 0;
if \beta 1 = 1, then
\beta i = 1, 1 \leq i \leq p,
\gamma i = \gamma 1, 1 \leq i \leq p - 1,
\delta i = \delta 1, 1 \leq i \leq n - p - 1.
In the next theorem we prove that there is no n-dimensional Zinbiel algebras of type II with
n \geq 3p+ 2.
Theorem 3.4. There is no Zinbiel algebras of type II with characteristic sequence equal to
(n - p, p) for n \geq 3p+ 2.
Proof. Consider for 1 \leq i \leq p+ 1 equalities
0 = (e1 \circ ep) \circ fi = e1 \circ (ep \circ fi) + e1 \circ (fi \circ ep).
Applying the relations (7), (8) and arguments similar to the ones that are used in the proof of
Proposition 3.1, we derive the relation
p\sum
k=0
Ci - 1
p+i - 1 - k\beta k = 0, (9)
where \beta 0 = 1 and 1 \leq i \leq n - p - 1.
Now we consider the determinant of the matrix of order p+ 1:
M =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 1 1 . . . 1 1
C1
p+1 C1
p C1
p - 1 . . . C1
2 1
C2
p+2 C2
p+1 C2
p . . . C2
3 1
. . . . . . . . . . . . . . . . . .
Cp - 1
2p - 1 Cp - 1
2p - 2 Cp - 1
2p - 3 . . . Cp - 1
p 1
Cp
2p Cp
2p - 1 Cp
2p - 2 . . . Cp
p+1 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
.
Taking into account identity Cm - 1
n + Cm
n = Cm
n+1 and subtracting from each row the previous
one we obtain
M =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
1 1 1 . . . 1 1
C1
p C1
p - 1 C1
p - 2 . . . 1 0
C2
p C2
p - 1 C2
p - 2 . . . 0 0
. . . . . . . . . . . . . . . . . .
Cp - 1
p 1 0 . . . 0 0
1 0 0 . . . 0 0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
= - 1.
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THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 881
Since M = - 1, the system of equations (9) for i = p+1 has only trivial solution with respect to
unknown variables \beta i. In particular, \beta 0 = 0. However, \beta 0 = 1, that is, we get a contradiction to the
condition i = p+ 1 \leq n - p - 1, which implies the non existence of an algebra under the condition
n \geq 3p+ 2.
Let A be an n-dimensional algebra with a basis \{ e1, e2, . . . , ep, f1, f2, . . . , fn - p\} and the multi-
plication table
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, e1 \circ ep = 0,
e1 \circ fn - p = 0, fi \circ fj = 0, 1 \leq i, j \leq n - p,
ei \circ fj =
i - 1\sum
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
j\sum
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq n - p,
where \beta i = ( - 1)iCi
p, 0 \leq i \leq p.
It is easy to check that this algebra is a Zinbiel algebra.
The correctness of the relation (9) for parameters \beta i for n = 3p + 1 follows from Lemma 2.1.
Thus, the condition n \geq 3p+ 2 is necessary.
We list the next theorems on the description of Zinbiel algebras of type II without proofs. It can
be carried out by applying similar arguments that were used above.
Theorem 3.5. A Zinbiel algebra of type II with characteristic sequence equal to (p + 1, p) is
isomorphic to one of the following non-isomorphic algebras:
\widetilde A1 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p+ 1,
where \beta 0 = 1, \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 1, \beta 1 \in \{ - p, - (p - 1), . . . , - 2, - 1\} ;
\widetilde A2 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp - 1 = ep,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p+ 1,
where \beta i = ( - 1)iCi
p - 2 for 0 \leq i \leq p - 2 \beta p - 1 = \beta p = 0;
\widetilde A3 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp = fp+1,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 2 \leq i+ j \leq p+ 1,
where \beta i = ( - 1)iCi
p - 1 for 0 \leq i \leq p - 1 and \beta p = 0;
\widetilde A4 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj = fi \circ ej = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p+ 1,
\widetilde A5 :
\left\{ ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj = fi \circ ej = Cj
i+j - 1fi+j , fi \circ fj = Cj
i+j - 1fi+j , 2 \leq i+ j \leq p+ 1.
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882 J. K. ADASHEV, M. LADRA, B. A. OMIROV
Theorem 3.6. A Zinbiel algebra of type II with characteristic sequence equal to (p + 2, p) is
isomorphic to one of the following non-isomorphic algebras:
\widetilde A6 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, 2 \leq i+ j \leq p+ 2,
fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, 2 \leq i+ j \leq p+ 2,
where \beta 0 = 1, \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 1, \beta 1 \in \{ - p, - (p - 1), . . . , - 2, - 1\} ;
\widetilde A7 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp - 1 = ep,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, 2 \leq i+ j \leq p+ 2,
fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, 2 \leq i+ j \leq p+ 2,
where \beta i = ( - 1)iCi
p - 2 for 0 \leq i \leq p - 2, \beta p - 1 = \beta p = 0;
\widetilde A8 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp+1 = fp+2,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, 2 \leq i+ j \leq p+ 2,
fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, 2 \leq i+ j \leq p+ 2,
where \beta i = ( - 1)iCi
p for 0 \leq i \leq p.
Theorem 3.7. A Zinbiel algebra of type II with characteristic sequence equal to (p + t, p), for
3 \leq t \leq p+ 1, is isomorphic to one of the following non-isomorphic algebras:
\widetilde A9 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, 2 \leq i+ j \leq p+ t,
fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, 2 \leq i+ j \leq p+ t,
where \beta 0 = 1, \beta i+1 =
\prod i
k=0
k + \beta 1
k + 1
for 1 \leq i \leq p - 1, \beta 1 \in \{ - p, - (p - 1), . . . , - (t - 1)\} ;
\widetilde A10 :
\left\{
ei \circ ej = Cj
i+j - 1ei+j , 2 \leq i+ j \leq p, f1 \circ fp - 1 = ep,
ei \circ fj =
\sum i - 1
k=0
Cj - 1
i+j - 2 - k\beta kfi+j , 1 \leq i \leq p, 2 \leq i+ j \leq p+ t,
fi \circ ej =
\sum j
k=0
Ci - 2
i+j - 2 - k\beta kfi+j , 1 \leq j \leq p, 2 \leq i+ j \leq p+ t,
where \beta i = ( - 1)iCi
p - 2 for 0 \leq i \leq p - 2 and \beta p - 1 = \beta p = 0.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
THE CLASSIFICATION OF NATURALLY GRADED ZINBIEL ALGEBRAS . . . 883
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Received 15.07.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
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| id | umjimathkievua-article-1482 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:35Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/71/cbe5246aa26450917adbcbee6cb07471.pdf |
| spelling | umjimathkievua-article-14822019-12-05T08:57:08Z The classification of naturally graded Zinbiel algebras with characteristic sequence equal to $(n - p,\, p) $ Класифiкацiя природно градуйованих алгебр Зiнбєля з характеристичною послiдовнiстю $(n - p,\, p) $ Adashev, J. K. Ladra, M. Omirov, B. A. Адашев, Й. К. Ладра, М. Оміров, Б. А. UDC 512.5 This work is a continuation of the description of some classes of nilpotent Zinbiel algebras. We focus on the study of Zinbiel algebras with restrictions imposed on gradation and characteristic sequence. Namely, we obtain the classification of naturally graded Zinbiel algebras with characteristic sequence equal to $ (n-p, p)$. УДК 512.5 Продовжено опис деяких класів нільпотентних алгебр Зінбєля. Основну увагу зосереджено на вивченні алгебр Зінбєля з обмеженнями на градацію та характеристичну послідовність, а саме, отримано класифікацію природно градуйованих алгебр Зінбєля з характеристичною послідовністю $(n-p,p).$ Institute of Mathematics, NAS of Ukraine 2019-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1482 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 7 (2019); 867-883 Український математичний журнал; Том 71 № 7 (2019); 867-883 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1482/466 Copyright (c) 2019 Adashev J. K.; Ladra M.; Omirov B. A. |
| spellingShingle | Adashev, J. K. Ladra, M. Omirov, B. A. Адашев, Й. К. Ладра, М. Оміров, Б. А. The classification of naturally graded Zinbiel algebras with characteristic sequence equal to $(n - p,\, p) $ |
| title | The classification of naturally graded Zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| title_alt | Класифiкацiя природно градуйованих алгебр Зiнбєля
з характеристичною послiдовнiстю $(n - p,\, p) $ |
| title_full | The classification of naturally graded Zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| title_fullStr | The classification of naturally graded Zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| title_full_unstemmed | The classification of naturally graded Zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| title_short | The classification of naturally graded Zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| title_sort | classification of naturally graded zinbiel algebras
with characteristic sequence equal to $(n - p,\, p) $ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1482 |
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