CC BY
11
DOI: 10.17516/1997-1397-2020-13-6-733-745 УДК 512.554.38
Almost Inner Derivations of Some Nilpotent Leibniz Algebras
Zhobir K. Adashev*
Institute of Mathematics of Uzbek Academy of Sciences
Tashkent, Uzbekistan
Tuuelbay K. KurbanbaeV
Karakalpak State University Nukus, Uzbekistan
Received 06.07.2020, received in revised form 08.08.2020, accepted 16.10.2020 Abstract. We investigate almost inner derivations of some finite-dimensional nilpotent Leibniz algebras. We show the existence of almost inner derivations of Leibniz filiform non-Lie algebras differing from inner derivations, we also show that the almost inner derivations of some filiform Leibniz algebras containing filiform Lie algebras do not coincide with inner derivations.
Keywords: Leibniz algebra, derivation, inner derivation, almost inner derivation.
Citation: J.K. Adashev, T.K. Kurbanbaev, Almost Inner Derivations of some Nilpotent Leibniz Algebras, J. Sib. Fed. Univ. Math. Phys., 2020, 13(6), 733-745. DOI: 10.17516/1997-1397-2020-13-6-733745.
Introduction
Lie algebra is an algebra satisfying the anticommutativity identity and the Jacobi identity. The derivations of finite-dimensional Lie algebras are a well-studied direction of the theory of Lie algebras. It should be noted that the space of all derivations of Lie algebras is also Lie algebra with respect to the commutator. In the set of derivations of Lie algebras, there exist subsets of the so-called inner derivations. Naturally, there is a question: in what classes of algebras do derivations exist? and which are not inner? For the semisimple Lie algebras the sets of inner derivations and derivations coincide [14].
Almost inner derivations of Lie algebras were introduced by C. S. Gordon and E. N. Wilson [13] in the study of isospectral deformations of compact manifolds. Gordon and Wilson wanted to construct not only finite families of isospectral nonisometric manifolds, but rather continuous families. They constructed isospectral but nonisometric compact Riemannian manifolds of the form G/r, with a simply connected exponential solvable Lie group G, and a discrete cocompact subgroup r of G. For this construction, almost inner automorphisms and almost inner derivations were crucial.
Gordon and Wilson considered not only almost-inner derivations, but they studied almost inner automorphisms of Lie groups. The concepts of "almost inner" automorphisms and derivations, almost homomorphisms or almost conjugate subgroups arise in many contexts in algebra, number theory and geometry. There are several other studies of related concepts, for example, local derivations, which are a generalization of almost inner derivations and automorphisms [2,3].
In [4] we initiated the study of derivation type maps on non-associative algebras, namely, we investigated so-called 2-local derivations on finite-dimensional Lie algebras, and showed an essential difference between semisimple and nilpotent Lie algebras is the behavior of their 2-local
* [email protected] https://orcid.org/0000-0002-4882-4098 t [email protected] https://orcid.org/0000-0002-9963-872X © Siberian Federal University. All rights reserved
derivations. The present paper is devoted to local derivation on finite-dimensional Lie algebra over an algebraically closed field of characteristic zero.
Local derivation first was considered in 1990, Kadison [16] and Larson and Sourour [18]. Let X be a Banach A-bimodule over a Banach algebra A, a linear mapping A : A ^ X is said to be a local derivation if for every x in A there exists a derivation Dx : A ^ X, depending on x, satisfying A(x) = Dx(x).
The main problems concerning this notion are to find conditions under which local derivations become derivations and to present examples of algebras with local derivations that are not derivations [8,16,18]. Kadison proves in [16, Theorem A] that each continuous local derivation of a von Neumann algebra M into a dual Banach M-bimodule is a derivation. This theorem gave rise to studies and several results on local derivations on C*-algebras, culminating with a definitive contribution due to Johnson, which asserts that every continuous local derivation of a C*-algebra A into a Banach A-bimodule is a derivation [15, Theorem 5.3]. Moreover in his paper, Johnson also gives an automatic continuity result by proving that local derivations of a C*-algebra A into a Banach A-bimodule X are continuous even if not assumed a priori to be so (cf. [15, Theorem 7.5]).
In the theory of Lie algebras, there is a theorem which says that in the finite-dimensional nilpotent Lie algebra there are not inner (i.e. outer) derivations [12]. We give an Example 2.1 to shows that that there exists 4-dimensional nilpotent Lie algebras, where any almost inner derivation is an outer derivation, and the converse is true also. But this question is still open for the general case. In [9] authors study almost inner derivations of some nilpotent Lie algebras. Prove the basic properties of almost inner derivations, calculate all almost inner derivations of Lie algebras for small dimensions. They also introduced the concept of fixed basis vectors for nilpotent Lie algebras defined by graphs and studied free nilpotent Lie algebras of the nilindex 2 and 3.
We recall that the study of almost-inner derivations of the Leibniz algebras is an open problem. Therefore in this paper we consider almost-inner derivations for some nilpotent Leibniz algebras. We prove the basic properties of almost inner derivations of the Leibniz algebras. We get almost all inner derivations of four-dimensional nilpotent Leibniz algebras. The study of the inner derivations of nilpotent Leibniz algebras is a very difficult problem. Therefore, we consider some subclasses of these nilpotent algebras. We study almost inner derivations of the null-filiform Leibniz algebras, and also consider almost inner derivations of the some filiform Leibniz algebras.
1. Preliminaries
Definition 1.1. An algebra L over a field F is called the Leibniz algebra if for all x,y,z € L the Leibniz identity holds:
[x, [y,z]] = [[x,y],z] - [[x z\,y],
where [ , ] is the multiplication in L.
For an arbitrary Leibniz algebra L, we define a sequence:
L1 = L, Lk+1 = [Lk,L1], k > 1.
The Leibniz algebra L is said to be nilpotent if there exists s € N such that Ls = 0. The minimal number s with this property is called the nilpotency index or nilindex of the algebra L.
We recall that the Leibniz algebra is called
null-filiform, if dimL1 = (n +1) — i, 1 ^ i ^ n + 1;
filiform, if dimL1 = n — i, 2 ^ i ^ n.
Let L be a nilpotent Leibniz algebra with nilindex s.
We consider Li = Li/Li+i, 1 < i < s — 1 and grL = Li©L2© • • ©Ls-i. Then [Li, Lj] C Li+j and we obtain the graded algebra grL.
Definition 1.2. If the Leibniz algebra L is isomorphic algebra grL, then L is called naturally graded Leibniz algebra.
For the Leibniz algebra L, we denote the right and left annihilators, respectively, as follows
Annr(L) = {x e L | [L,x] = 0}, Annl(L) = {x e L | [x,L] = 0}.
We denote the center of the algebra by Cent(L) = Annr(L) n Annl(L). A linear map d is called a derivation of the Leibniz algebra L, if
d( [x,y\) = [d(x),y] + [x,d(y)].
We denote the space of all derivations by Der(L).
For each x e L, the operator Rx : L ^ L which is called the right multiplication, such that Rx(y) = [y,x], y e L, is a derivation. This derivation is called an inner derivation of L, and we denote the space of all inner derivations by Inner(L).
Definition 1.3. The derivation D e Der(L) of the Leibniz algebra L is called almost inner derivation, if D(x) e [x,L] {[x,L] C L) holds for all x e L; in other words, there exists ax e L such that D(x) = [x, ax].
The space of all almost inner derivations of L is denoted by AID(L).
Definition 1.4. The derivation D e AID(L) of the Leibniz algebra L is called the right central almost inner derivation, if there exists x e L such that the map (D — Rx) : L ^ Annr(L).
The space of right central almost inner derivations of L is denoted by RCAID(L), respectively.
Definition 1.5. The derivation D e AID(L) of the Leibniz algebra L is called central almost inner derivation, if there exists x e L such that the map (D — Rx) : L ^ Cent(L).
The space of central almost inner derivations of L is denoted by CAID(L), respectively.
2. Main results
2.1. The properties of almost inner derivations of the Leibniz algebras
Thesubspaces Inner(L), CAID(L), RCAID(L), AID(L), Der(L) are Lie subalgebras with [D,D'] = DD' — D'D.
Proposition 2.1. We have the following inclusions of Lie subalgebras
Inner(L) C CAID(L) C RCAID(L) C AID(L) C Der(L).
Proof. Let Di,D2 e AID(L) and x e L. Then there exist yi,y2 e L such that D\(x) = [x,y\], D2(x) = [x,y2]. Using the property of the derivation and the Leibniz identity, we get the following
[Di,D2](x) = (DiD2)(x) — (D2Di)(x) = [Di(x)y] + [x, Di(y2)] — [D2(x),yi] — [x,D2(y)] = = [[x, yi], y2] — [[x, y2],yi] + [x, Di(y2)] — [x, D2 (yi)] = = [x, [yi,y2]] + [x,Di(y2)] — [x,D2(yi)] = [x, [yi,y2] + Di(y2 ) — D2 (yi)].
Therefore, [Di,D2](x) = [x, |yi,y2]+ Di(y2) — D2(yi)] € [x,L], we have [Di,D2] € AID(L). Let C1,C2 € CAID(L). Then there exist y1,y2 € L such that C1 — Ryi and C2 — Ry2 are maps from L to Cent(L). We consider [C, Rx] = RC(x) for C € Der(L) and obtain the following
[C1 — Ryi , C2 — Ry2] = [C1, C2] — [C1, Ry2] — [Ryi, C2] + [Ryi , Ry2] =
= [C1, C2] — RCi(y2) + RC2(yi) — R[y2,yi] = [C1, C2] — (RCi(y2) — RC2(yi) + R[y2,yi]).
Hence we have that the linear transformation [C1, C2] — (RCi(y2) — Rc2(yi) + R[y2,yi]) maps L to Cent(L). Hence [C1,C2] € CAID(L).
Let D1 ,D2 € RCAID(L). Then there exist y1,y2 € L such that D1 — Ryi and D2 — Ry2 are maps L to Annr(L). We consider [D,Rx] = RD(x) for D € Der(L) and obtain the following
[D1 — Ryi ,D2 — Ry2] = [D1, D2] — [D1, Ry2] — [Ryi, D2] + [Ryi ,Ry2] =
= [D1, D2] — RDi(y2) + RD2(yi) — R[y2,yi] = [D1, D2] — (RDi(y2) — RD2(yi) + R[y2,yi]).
Hence we have that the linear transformation [D1,D2] — (RDi(y2) — RD2(yi) + R[y2,yi]) maps L to Annr(L). Hence [D1,D2] € RCAID(L).
Now let us show that Inner(L) C CAID(L). Let Rx,Ry € Inner(L) and Rx — Ry : L ^ Cent(L). For every z € L, a € Cent(L) we consider the following
(Rx — Ry)(z) = [z, x] — [z, y] = [z, x] — [z, a + x] = [z, a] € Cent(L).
Therefore, Inner(L) C CAID(L). □
Proposition 2.2. The subalgebra RCAID(L) is a Lie ideal in AID(L).
Proof. Let C € RCAID(L) and D € AID(L). We must show [D, C] € RCAID(L). We already know [D, C] € AID(L). We fix an element x € L such that C' := C — Rx maps L to Annr(L). We denote D' := [D,C] — RD(x). Then from [D,Rx] = RD(x) we obtain
[D, C'] = [D, C — Rx] = [D, C] — [D, Rx] = [D, C] — Rd(x) = D'
and D' maps L to Annr (L). Hence for all y € L we have
D'(y) = [D, C'](y) = D(C'(y)) — C'(D(y)),
because C' maps L to Annr (L) and D maps Annr (L) to Annr (L). □
Proposition 2.3. Let L be the Leibniz algebra. Then the followings are true:
1) Let D € AID(L). Then D(L) C [L, L], D(Cent(L)) = 0 and D(I) C I for every ideal I of L.
2) For D € CAID(L), there exists an x € L such that D\l,l] = Rx\[l,l] .
3) If L has nilindex 3, then CAID(L) = AID(L).
4) If Cent(L) = 0, then CAID(L) = Inner(D).
5) If L is nilpotent, then AID(L) is nilpotent.
6) AID(L © L') = AID(L) © AID(L').
Proof. 1) By definition, almost inner derivations of L maps to [L, L] and Cent(L) to 0. Let x € I. Then we have D(x) € [x, L] C [I, L] C I.
2) For a given D € CAID(L), there exists x € L such that D' = D — Rx satisfies D'(L) C Cent(L). Hence D' is derivation and for all u,v € L we have
D'([u, v]) = [D'(u), v] + [u, D'(v)] = 0.
3) If L is nilpotent with nilindex 3, i.e. L3 = 0, then for each D e AID(L) we get D(L) C [L, L] C Cent(L) and get equality.
4) We suppose Cent(L) = 0 and D e CAID(L). Then there is x e L such that D — Rx = 0. Therefore D is inner.
5) Let D e AID(L) and x e L. Then Dk (x) e [[[..., [x,L],... L], L] (k times L). If k is higher than nilpotent class over L, then we have Dk(x) = 0, therefore D is nilpotent. By Engel's theorem for Leibniz algebras [5], AID(L) is nilpotent.
6) Let D e AID(L © L'). Then the constraints are again almost inner derivations, i.e. D l e AID(L) and D\l e AID(L'). It is obvious that the mapping D ^ D\l © D\l gives a one-to-one correspondence between AID(L © L') and AID(L) © AID(L').
□
2.2. Almost inner derivations of null-filiform Leibniz algebras
Firstly we consider a certain class of nilpotent Leibniz algebras, the so-called null-filiform Leibniz algebra [7].
In any n-dimensional null-filiform Leibniz algebra L there exists a basis {ei,e2,..., en} such that the multiplication in L has the form:
NFn : [ei,ei] = e^+i, 1 < i < n — 1 (1)
(the omitted of products are equal to zero). Let L be a null-filiform Leibniz algebra.
Proposition 2.4. For the n-dimensional null-filiform Leibniz algebra NFn the following equality holds:
AID(NFn) = Inner (NFn).
Proof. The null-filiform algebra L is a one-generated algebra, i.e. generated by ei. Let D e AID(NFn). Then, by the definition of almost inner derivation, there exists aei such that D(ei) = Raei. Let D' e AID(NFn) and let D' = D — Raei, then we get D'(ei) = 0. Then by multiplication (1) we have
D'(ei) = D'([ei-i,ei]) = [D'(e-i, ei)] + [e-i, D'(ei)] =0, 2 < i < n.
This means that
AID(NFn) = Inner (NFn).
□
2.3. Almost inner derivation of non-lie filiform Leibniz algebras
Now we consider filiform non-Lie Leibniz algebras Fi(a4, a5,..., an, 0) and F2(@5,..., ¡3n, y) from [7]:
[ei,ei] = e3,
[ei, ei] = ei+i, 2 ^ i ^ n — 1,
Fi (a.4, a5,.. ., an, 0) : <
n-i
[ei,e2] = J2 ases + 0en,
s=4 n-j+2
[ej,e2] = 12 ases+j-2, 2 < j < n — 2,
s=4
[e1,e1] = e3,
[ej, e1] = ej+1, 3 ^ i ^ n — 1,
n
F2(P4,p5,...,pn,j): { [e1,e2] = £^,
[e2,e2] = Yen,
n+2-i
[ej,e2]= J] 3kek+i-2, 3 < i < n — 2.
k=4
Let L be an algebra from F1(a4, a5,..., an, 0) or F2(34, 35,..., ¡3n, y). Let L be the Leibniz algebra and En,2 : L ^ L be a linear mapping such that
En,2(ej) = Sj,2en, 1 < i < n, (2)
1, i = 2
where 6i2 = S 1 1 — Kronecker I U, i = 2
symbol.
Theorem 2.1. Let L be a non-Lie filiform Leibniz algebra and let D € AID(L). Then there exist an element x € L and X € C such that
D — Rx = XEn,2.
Proof. We first consider the non-Lie filiform Leibniz algebra L = F1(a4, a5,... ,an,0).
Let D € AID(L). This algebra is a two-generated algebra, i.e. we have generators e1 and e2. Then, by the definition of almost inner derivation, there exists aei such that D(e1) = Raei. Let D' € AID(L) and D' = D — Raei, then we get D'(e1) = 0. Since D'(e1) = 0, then we have the following:
D'(e3) = D'([e1, e1]) = [D '(e1), e1] + [eb D'(e1)] = 0,
D' (ej) = D ' ([e-1, e1]) = [D ' (ej-1), e1] + [ej-1,D' (e1)] = [D ' (e-1), e1] = 0, 4 < i < n.
Let D'(e2) = bjej. we check the following:
j=1
D' (es) = D' ([e2,ei]) = [D' (e2),ei]
- n
j=i
(bi + b2)es + bse4 +-----+ bri-ieri.
On the other hand, D'(e3) = D([e1, e1]) = 0. So we get
b1 = —b2, bi = 0, 3 < i < n — 1.
Now we check the following:
0 = D'([e!,e2]) = [D'(e1), e2] + [e1, D'(e2)] = [e1, b1e1 — b1e2 + bnen] =
= b1e3 — b1(a4 e4 +-----+ an-1en-1 + 0en).
We have b1 =0 and D'(e2) = bnen. On the other hand, by definition of almost inner derivation
bnen = D'(e2)= [e2,aC2]= [e2, a2,1e1+a2,2e2+-----+a2,nen]= a2,1e3+a2,2(a4e4+a5e5+-----+anen).
We obtain {
{a2 1 =0, a2 2ai =0, 4 < i < n — 1,
b2 ,1 ,2 ,2 j (3)
bn = a2 ,2an.
Hence D'(e2) = a2,2anen. If a2 2an = 0, then AID(L) = Inner(L), so
a2,2an = 0,
therefore from (3) we get
ai = 0, 4 < i < n — 1. In the end we obtain D' = a2,2anEn,2 = AEn,2.
Let L = F2(34,35,...,3n,Y) and D' e AID(L). By definition AID for e2 there exists ae2 such that
D'(e2) = [e2, ae2] = [e2,a2,iei +-----+ a2,n] = a2,2Yen.
Conducting analogously reasoning in this algebra we obtain D'(ei) = 0, D'(ei) = 0, 3 < i < n and D' = a2,2iEn,2 = AEn,2, where A e C. Now we consider the following equality:
a2,2Yen = D'(ei) + D'(e2 = D'(ei + e2) = [ei + e2, Cei+e2] = [ei + e2,ciei + C2e2] =
= cie3 + C234 e4 + C2(34 + 3s)ee +-----+ C2(34 +-----+ 3n-i)en-i +
+ C2(34 +-----+ 3n-i + 3n + Y)en.
We get
Ci =0,
c23i =0, 4 < i < n — 1, C2 (3n + y) = a2,2Y.
If at least one of 3i0 =0 (4 ^ i0 ^ n — 1), then we have c2 = 0, hence AID(L) = Inner(L). Therefore 3i = 0, 4 < i < n — 1.
Thus, for filiform non-Lie algebras we obtain D — Ra = AEn,2, A e C. □
Remark 2.1. Let L be a filiform non-Lie Leibniz algebra. If at least one of ai0 = 0 and 3j0 = 0, io,jo e {4, 5,... ,n — 1}, then we get AID(L) = Inner(L).
Theorem 2.2. Let L be an n-dimensional filiform non-Lie Leibniz algebra Fi(0,..., 0, an, 0) or F2(0,..., 0, 3n,0). Then at run 0 = 0, an =0 and 3n = 0, y = 0 respectively we obtain
AID(L) = Inner(L) © (En2),
where En,2 is the matrix of the elements in which in the place (n, 2) we have 1, and other elements are 0.
Proof. Let L = Fi(0,..., 0,an,0). We have to show that En,2 is an almost inner derivation of
n
the algebra L. We take the element x = ^ xiei e L, then there is cx = ciei + c2e2 e L and we
i=i
check up the following
En,2(x) = [x,cx] =
Xiei, ci ex + C2e2
= ci(xi + x2)e3 + cix3 e4 + Cix4e5 +-----+ cixn-2en-i + (cixn-i + C2(xi0 + x2a.n))en.
If 0 = 0 and x3 = 0, then for xi = —the map En,2 is not almost inner derivation.
01
Therefore 0 = 0 and for any x e L choosing ci = 0, c2 = — we have En 2(x) = x2en. Hence
—n
En,2 e AID(L).
Let L = F2(0, 0,..., 0, 3n, y). Let Vx = E xiei € L, then 3cx = c1e1 + c2e2 € L and we
i=1
obtain the following
n
En,2(x) = [x,cx]= E xiei,C1e1 + C2e2 = i=1
= C1x1 e3 + C1x3e4 + C1x4e5 +-----+ C1xn-2en-1 + (c1xn-1 + C2(x13n + x2Y))en.
x2 y
If 3n =0 and x4 = 0, then for x1 = —— the derivation En 2 is not almost inner derivation.
3n
Therefore 3n = 0 and for any x € L choosing c1 =0, c2 = — we have En 2(x) = x2en. Hence
Y
En,2 € AID(L). □
Theorem 2.1 and 2.2 imply the following consequence:
Corollary 2.1. In filiform non-Lie Leibniz algebras, if all parameters are equal to zero, then these algebras turn into a graded algebra. Then the almost inner derivations of graded non-Lie Leibniz algebras coincide with the inner derivations.
2.4. Almost inner derivations of sme filiform Leibniz algebras
We consider filiform Leibniz algebra L = F3(01, 02,03), which contain filiform Lie algebra [10]:
[e1,e1] = 01en, [ebe2 ] = —e3 + 02en, [e2,e2] = 03en, [ei, e{] = ei+1, 2 ^ i ^ n — 1,
F3(01,02,03): ^ [e1,ei] = —ei+1, 3 ^ i ^ n — 1
n-i+3
[ei,e2] = — [e2,ei]= J2 3kek+i-3, 3 < i < n — 2,
k=5
. [ei,ej] = —[ej, ei] = 0, i,j > 3.
Theorem 2.3. Let L = F3(01, 02, 03) and let D € AID(L). Then there exist an element x € L and X C such that
Proof. Let L = F3(01,02, 03). Let D € AID(L). Then D induces an almost inner derivation of D by L/(en). By induction, we can assume that after changing D to inner derivation, we have D = (En-1,2 for some ( € C. This implies such that D(e1) = aen for some a € C. Now we replace D with D' = D + Raen-i. Then we have
D'(e1) = D(e1) + Raen-i (e1) = aen + [e1, aen-1] = 0, D'(ei) = D(ei) + [ei, aen-1] = D(eH), i > 2.
We get
D'(e2) = D(e2) = iJ.eri-1 + Xen, (J,, X € C. Hence, we have the following
D'(e3) = D'([e2, e1]) = [D '(e2), e1] = [(en-1 + Xen, e1] = (en, D'(e4) = D'([e3, e1]) = [D'(e3), e1] = [(en, e1] = 0,
moreover, D'(ei) =0, i ^ 5.
D — Rx = XEn, 2.
Since we have D'(e3) = ¡¡en and D' G AID(L), then there exists an element ae3 = a3,1e1 + +a3,2e2 G L such that D'(e3) = [e3,ae3] = ¡¡en. Therefore we get the following
We obtain
H^n = [e3, a3,1e1 + a3,2e2] = a3,ie4 + a3,2(ß5e5 + ß6e6 +----+ ßn^n)-
«3,1 = 0, a3 2ßi =0, 5 < i < n - 1, a3 2ßn
Since we assume j = 0, then we have
= 0, 5 < i < n - 1.
Now we consider the following
D'(e2) = [e2, ae2]
e2,J2a2,j ei ■ j=i
a2,ie3 + (a2,2^3 - a2ßßn)en
On the other hand D'(e2) = jen-1 + Xen. We have
a2,ie3 + (a2,2^3 - a,2,3ftn)en = jen-i + Xen.
Since we assume that j = 0, this equation does not have a solution, which is a contradiction. Hence indeed j = 0, and therefore D' = XE2,n. □
Proposition 2.5. Let L be an n-dimensional filiform Leibniz algebra F3(01, 02, 03). Then AID(F3(0i,02, 03)) = Inner(F3(0i,02, h)) © (En,2),
where En,2 is the matrix of the elements in which in the place (n, 2) we have 1, and other elements are 0.
Proof. The proof is analogous to Proposition 7.4 in [9]. □
2.5. Almost inner derivations of low dimensional nilpotent Leibniz algebras
N. Jacobson proved the following theorem [12]:
Theorem 2.4. Every nilpotent Lie algebra has a derivation D which is not inner.
There is a question: Are almost inner derivations of nilpotent Lie algebras outer derivations? And is the converse right? Generally this question is open. We give an example which answers in the positive on this question.
Example 2.1. We consider 5-dimensional nilpotent Lie algebra in which there exist almost inner derivations which are not inner [9].
1) 05,3 : [e1,e2] = e4, [e1,e4] = e5, [e2,e3] = e5, the omitted products are equal to zero. Derivations, inner derivations and almost inner derivations of this algebra have the following matrix forms respectively:
Der(g5,3)
/ ai,i 0 0 0 0 \
ai,2 a2,2 0 0 0
ai,3 a2,3 2ai,i 0 0
ai,4 a2,4 -a2,2 ai,i + 0'2,2 0
V ai,5 a2,5 a3,5 -ai,3 + a2,4 2ai,i + a2,2 /
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Fnner(g5,3) = 0 0 0 0 0 , AFD(q5,3) = 0 0 0 0 0
M2 -Mi 0 0 0 ai,4 a2,4 0 0 0
\ M4 M3 -M2 -Mi 0 /
\ «1,5 «2,5 a3j5 a2,4
/
/ «1,1 0 0 0 0 \
ai,2 a2,2 0 0 0
ai,3 «2,3 2ai,i 0 0
0 0 — «2,2 «1,1 + «2,2 0
V 0 0 «3,5 -«1,3 2«1,1 + «2,2 /
If a1j4 = a^5 = a2,4 = a2,5 = 0, then we obtain the matrix of outer derivation of algebra 05,3:
Outer(Q5i3)
Therefore, AID(g5,3) C Outer(g5,3) and any almost inner derivation of the algebra g5,3 is outer. If in Outer(g5,3) we have a1;1 = a1j2 = a1j3 = a2,2 = a2,3 = 0, then the space of all outer derivations coincides with the space of all almost inner derivations.
Now we give examples for low dimensional nilpotent Leibniz algebras. Example 2.2. Let L be the three-dimensional nilpotent Leibniz algebra:
L1( a) : [e2,e2] = e1 , [e3, e3] = ae1, [e2 ,e3] = e1, a G C,
L2 [e2,e2] = e1 , [e3, e2] = e1 , [e2, e3] = e1,
L3 [e2,e2] = e1 , [e3, e3] = e1 , [e3, e2] = e1, [e2, e3] =
L4 [e3,e3] = e1 ,
L5 [e2,e3] = e1 , [e3, e3] = e1 ,
Fe [e3,e3] = e1 , [e1, e3] = e2.
For three-dimensional nilpotent Leibniz algebras L, the following equality
AID(L) = Inner (L)
holds.
Example 2.3. Let L be four-dimensional nilpotent Leibniz algebra. Then from [1] there are 28 algebras and we give only those algebras which will be necessary to us:
L4 : [e1,e1] [e3,e1] = e3, = e4, [e1, e2] = ae4, a G {0, 1}; [e2,e1] = e3, [e2, e2] = e4,
F9 : [e1,e1] [e3,e1] = e4, = e4, [e2, e1] = e3, [e1,e3] = -e4, [e2,e2] = e4, [e1, e2] = -e3 + 2e4,
L10 : [e1,e1] [e1,e2] = e4, = -e3, [e2, e1] = e3, [e1, e3] = -e4; [e2,e2] = e4, [e3 ,e1] = e4,
L11 : [e1,e1] = e4, [e1,e2] = e3, [e2,e1] = -e3, [e2 ,e2] = -2e3 + e4;
L12 : [e1,e1] = e3, [e2,e1] = e4, [e2,e2] = -e3;
L13 : [e1,e1] = e3, [e1,e2] = e4, [e2,e1] = -ae3, [e2 ,e2] = -e4;
L20 : [e1,e2] = e4, 1 + a [e2,e1]= e4, 1a [e2,e2] = e3, a G C\{1}.
Let us show the calculation of the dimension of almost inner derivations and the inner derivations of these algebras.
0
• The algebra L4 is a filiform algebra from the class F1(0,..., 0, an, 0). Therefore, by Theorem 2.2 we have: if a = 0, then AID(L4) = Inner(L4), and if a = 1, then AID(L4) = = Inner(L4) © (E42).
• We consider the algebra L9. Let D G AID(L9), then by definition AID for 1 ^ i ^ 4 for
4
each ei there is aei = aijej and we have the following:
j=1
D(e1) = [e1 ,aei] = -a1,2e3 + (a1,1 + 2a1,2 - a1,3^4, D(e2) = [e2,ae2] = a2^e3 + 0,2, 2^4
D(e3) = [e3, a,e3] = a3,1e4, D(e4) = [e4, ae4] = 0.
Since D is derivation, we check the following:
a3,1e4 = D(e3) = D([e2, e1 ]) = [D(e2), e1] + [e2,D(e1)] = a2,1e4, from here we get a21 = a31. Therefore, the matrix AID of this algebra has the following form:
AID(Lg)
hence dimAID(L9) = 4.
Now we calculate the dimension of the space of inner derivations. To do this, we take the
4
element x = J2 xiei and consider Rx(ei), (1 ^ i ^ 4) :
i=1
Rx(e\) = [e1,x] = -X2e3 + (x1 +2x2 - X3)e4, Rx(e2) = [e2,x] = X1e3 + X2e4, Rx(e3) = [e3,x] = x1e4, The matrix of inner derivation of algebra L9:
0 0 0 0
0 0 0 0
-ai,2 a2,i 0 0
\ aii + 2ai,2 - a,i3 a2,2 a2,i 0
Rx(e4) = [e4, x] = 0.
/ 0 0 0 0
0 0 0 0
-x2 xi 0 0
V xi + 2x2 - x3 x2 xi 0
Inner (Lg) =
hence dimInner(L9) = 3.
From the matrices AID(L9) and Inner(L9) it is clear that AID(L9) = Inner(L9) © (E4,2).
4
Now let's calculate the dimension of RCAID(L9), for this we take every element of x = ^ xiei G
L9 and
(D - Rx)(x)
/ 0 0 0 0 0 0
-ai2 — X2 a2,i — xi 0
\
ai,3 — x3
a2,2 - X2 a2,i - Xi
0 \ { xi \
0
0 0
xi x2
x3 x4
0.
Then we have a1,2 = x2, a1 3 = x'3, a2}1 = x1, a222 = x2. Hence, dimRCAID(L9) = 3.
• For algebras L10, L11, L12, L20 similarly conducted reasoning and calculated dimension AID(L) and Inner(L).
• Now we consider L13 and get the following matrices:
AID(Li3) =
0 0
ai,i \ ai,2
0 0 0 \
0 0 0
-a2i 0 0 -a2,2 0 0
Inner (Li3) =
0 0 0 0
0 0 0 0
xi -xi 0 0
x2 - x2 0 0
This shows that dimAID(L13) = 4, dimRCAID(L13) = dimInner(L13) = 2, hence we obtain AID(L 13) = Inner(L13) © (£3,2 + £4,2}.
For other algebras, except those shown, almost inner derivations coincide with inner derivations.
Therefore, we have the following table:
Algebra dim Inner(L) dimRCAID(L) dimAID(L) dimDer(L) D
L4 2 2 3 4 E4,2
L9 3 3 4 4 E4,2
L10 3 3 4 4 E4,2
L11 2 2 3 5 E4,2
L12 2 2 3 5 E4,2
L13 2 2 4 5 E42 + E32
L20 2 2 3 7 E4,2
Example 2.4. Let L be a complex Leibniz algebra of dimension n ^ 2. Then we have
AID(L) = RCAID(L) = Inner(L). It is clear that for abelian Leibniz algebras Inner (L) = RCAID(L) = AID(L) = 0.
References
[1] S.Albeverio, B.A.Omirov, I.S.Rakhimov, Classification of 4-dimensional nilpotent complex Leibniz algebras, Extracta Mathematicae, , 21(2006), no. 3, 197-210.
[2] Sh.A.Ayupov, K.K.Kudaybergenov, Local derivation on finite-dimensional, Linear Algebra and its Applications, 493(2016), 381-398.
[3] Ayupov Sh.A., Kudaybergenov K.K., Local automorphisms on finite-dimensional Lie and Leibniz algebras, 2018, arxiv: 1803.03142v2.
[4] Sh.A.Ayupov, K.K.Kudaybergenov, I.S.Rakhimov, 2-Local derivations on finite-dimensional Lie algebras, Linear Algebra and its Applications, 474(2015), 1-11.
[5] Sh.A.Ayupov, B.A.Omirov, On Leibniz algebras, In: Algebra and Operator Theory (Tashkent, 1997), Kluwer Acad. Publ., Dordrecht, 1998, 1-12.
DOI: 10.1007/978-94-011-5072-9_1
[6] Sh.A.Ayupov, B.A.Omirov, On 3-dimensional Leibniz algebras, Uzbek Math. Journal, 1999, 9-14
[7] Sh.A.Ayupov, B.A.Omirov, On some classes of nilpotent Leibniz algebras, Siberian Math. J., 42(2001), no. 1, 15-24.
[8] M.Bresar, P.Semrl, Mapping which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math., 45(1993), 483-496. DOI: 10.4153/CJM-1993-025-4
[9] D.Burde, K.Dekimpe, B.Verbeke, Almost inner derivation of Lie algebras, Journal of Algebra and Its Applications, (2018), ID: 119142860. DOI: 10.1142/S0219498818502146.
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[11] N.Jacobson, Lie algebras, Interscience Publishers, Wiley, New York, 1962 .
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[14] J.E.Humphreys, Introduction to Lie algebras and Representation theory, Springer-Verlag, New York, 1972.
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Почти внутренние дифференцирования некоторых нильпотентных алгебр Лейбница
Жобир К. Адашев
Институт математики АН РУз Ташкент, Узбекистан
Тууелбай К. Курбанбаев
Каракалпакский государственный университет
Нукус, Узбекистан
Аннотация. В статье исследуется почти внутренние дифференцирования некоторых конечномерных нильпотентных алгебр Лейбница. Мы показываем существование почти внутренних дифференцирований филиформных нелиевых алгебр Лейбница, отличных от внутренних дифференцирований, а также показываем, что почти внутренние дифференцирования некоторых филиформных алгебр Лейбница, содержащих филиформные алгебры Ли, не совпадают с внутренними дифференцированиями.
Ключевые слова: алгебра Лейбница, дифференцирование, внутреннее дифференцирование, почти дифференцирование.
