Filiform Leibniz Algebras

• Filiform Leibniz algebras are nilpotent algebras with an extremal lower-central-series profile (dim Lᶦ = n−i) similar to filiform Lie algebras.
• They are classified via naturally graded models (F₁, F₂, and F₂(α)) and encompass variants like null-filiform, p-filiform, and quasi-filiform algebras with explicit normal forms.
• Research highlights include studies on derivations, central extensions, and cohomological rigidity, revealing insights into solvable extensions and infinite-dimensional analogues.

Filiform Leibniz algebras are nilpotent Leibniz algebras that realize an extremal lower-central-series profile and therefore occupy, in Leibniz theory, a role analogous to filiform Lie algebras in nilpotent Lie theory. In finite dimension, an $n$-dimensional Leibniz algebra $L$ is filiform when $\dim L^i=n-i$ for $2\le i\le n$, while broader $p$-filiform and quasi-filiform notions are organized by the characteristic sequence of right multiplication and by nilindex. The subject combines explicit normal forms, naturally graded models, derivation theory, central and solvable extensions, cohomology, rigidity, and, in recent work, infinite-dimensional pro-nilpotent and residually solvable analogues (Ladra et al., 2016, Camacho et al., 2013, Abdurasulov et al., 2021).

1. Definitional framework and extremal nilpotency

The literature uses both left and right Leibniz conventions. In the right convention, a Leibniz algebra is a vector space with bilinear bracket satisfying

$[x,[y,z]]=[[x,y],z]-[[x,z],y],$

whereas several finite-dimensional classification papers use the left Leibniz identity in equivalent structural roles. In either convention, the failure of skew-symmetry distinguishes Leibniz algebras from Lie algebras; when $[x,x]=0$ for all $x$, the bracket becomes skew-symmetric and the Jacobi identity is recovered. Standard structural objects include the lower central series $L^1=L$, $L^{k+1}=[L^k,L]$, the right annihilator, and the center (Abdurasulov et al., 2021).

For finite-dimensional nilpotent Leibniz algebras, filiformity is defined by

$L$0

This is the maximal-length lower-central-series pattern compatible with non-null-filiform nilpotency. The null-filiform condition is stronger: $L$1 and yields maximal nilindex $L$2. More generally, an $L$3-dimensional Leibniz algebra is $L$4-filiform when its characteristic sequence is

$L$5

with $L$6 copies of $L$7. Quasi-filiform Leibniz algebras are those of nilindex $L$8; in Leibniz theory this permits characteristic sequences $L$9 and $\dim L^i=n-i$0, a dichotomy absent in the Lie case described in the same source (Camacho et al., 2010, Adashev et al., 2015).

Naturally graded algebras are obtained from the associated graded algebra of the lower central series,

$\dim L^i=n-i$1

and satisfy $\dim L^i=n-i$2. These graded models are the organizing centers of the classification theory: both finite-dimensional filiform Leibniz algebras and their $\dim L^i=n-i$3-filiform variants are studied as deformations or extensions of a small set of naturally graded prototypes (Ladra et al., 2016).

2. Canonical graded models

A basic finite-dimensional classification result states that every complex naturally graded filiform Leibniz algebra is isomorphic to one of three pairwise nonisomorphic models: $\dim L^i=n-i$4

$\dim L^i=n-i$5

and

$\dim L^i=n-i$6

together with the additional top-degree brackets

$\dim L^i=n-i$7

Here $\dim L^i=n-i$8 and $\dim L^i=n-i$9 are non-Lie, while $2\le i\le n$0 is Lie. The algebra usually denoted $2\le i\le n$1 is the $2\le i\le n$2 model in dimension $2\le i\le n$3, with basis $2\le i\le n$4 and nonzero products $2\le i\le n$5 for $2\le i\le n$6; it is naturally graded, filiform, non-Lie, and indecomposable (Ladra et al., 2016).

Null-filiform Leibniz algebras admit an even more rigid model. Every complex $2\le i\le n$7-dimensional null-filiform Leibniz algebra is isomorphic to

$2\le i\le n$8

with all other products zero. Thus the entire algebra is generated by a single right-multiplication chain (Adashev et al., 2015).

The infinite-dimensional analogue used in the pro-nilpotent setting is the algebra $2\le i\le n$9 with basis $p$0 and multiplication

$p$1

all other products being zero. Its lower central series is

$p$2

so $p$3, every quotient $p$4 is finite-dimensional and nilpotent, and $p$5 is an $p$6-graded, two-generated, maximal-class pro-nilpotent Leibniz algebra. In the cited paper this algebra functions as the canonical infinite-dimensional filiform, or thin, Leibniz model (Abdurasulov et al., 2021).

3. Finite-dimensional classification programs

A global structural theorem due to Omirov, as presented in the cited study of derivations, states that all complex $p$7-dimensional filiform Leibniz algebras split into three disjoint families $p$8, $p$9, and $[x,[y,z]]=[[x,y],z]-[[x,z],y],$0, each written in an adapted basis $[x,[y,z]]=[[x,y],z]-[[x,z],y],$1 with explicit structure constants. The same work classifies non-characteristically nilpotent members of these families; for the first family, the non-characteristically nilpotent cases are governed by a Catalan-number pattern in the coefficients $[x,[y,z]]=[[x,y],z]-[[x,z],y],$2, and the paper corrects an earlier incorrect criterion for characteristic nilpotency (Khudoyberdiyev et al., 2012).

Low-dimensional subclasses have been classified in detail. For the subclass $[x,[y,z]]=[[x,y],z]-[[x,z],y],$3 arising from naturally graded filiform Lie algebras, the dimensions $[x,[y,z]]=[[x,y],z]-[[x,z],y],$4 and $[x,[y,z]]=[[x,y],z]-[[x,z],y],$5 were worked out via adapted bases, invariant functions, and explicit isomorphism criteria. The resulting counts are $[x,[y,z]]=[[x,y],z]-[[x,z],y],$6 isomorphism classes in $[x,[y,z]]=[[x,y],z]-[[x,z],y],$7 and $[x,[y,z]]=[[x,y],z]-[[x,z],y],$8 in $[x,[y,z]]=[[x,y],z]-[[x,z],y],$9, and the analysis also re-examines the corresponding filiform Lie algebras inside these Leibniz families (Abdulkareem et al., 2013).

The maximum-length program organizes another major branch of the subject. For quasi-filiform non-Lie Leibniz algebras of characteristic sequence $[x,x]=0$0, the maximum-length algebras are exactly the families $[x,x]=0$1, $[x,x]=0$2, and $[x,x]=0$3, with type I and type II distinguished by the Jordan form of a right multiplication operator (Camacho et al., 2010). For 3-filiform Leibniz algebras of maximum length, the classification is completed by the families $[x,x]=0$4 on the Lie side and $[x,x]=0$5, $[x,x]=0$6 on the non-Lie side, while several naturally graded candidates are shown not to admit gradations of maximum length (Camacho et al., 2013). For general $[x,x]=0$7, the cited classification shows that the even-$[x,x]=0$8 case yields only the pairwise non-isomorphic families $[x,x]=0$9 with $x$0 and $x$1, whereas for odd $x$2 no $x$3-filiform Leibniz algebra of maximum length exists under the stated dimensional assumptions (Camacho et al., 2013).

4. Central extensions, solvable envelopes, and infinite-dimensional analogues

Central extension theory is especially explicit for filiform-type Leibniz algebras. For the null-filiform algebra $x$4, every $x$5-dimensional central extension is isomorphic either to

$x$6

or to

$x$7

Thus the only nontrivial central phenomenon is the extension of the null-filiform chain by one further step. For the naturally graded non-Lie filiform algebras $x$8 and $x$9, the cited paper computes $L^1=L$0, classifies non-split central extensions of dimensions $L^1=L$1 through $L^1=L$2, and proves that every $L^1=L$3-dimensional central extension with $L^1=L$4 is split (Adashev et al., 2015).

A related finite-dimensional solvable-extension problem concerns Leibniz algebras whose nilradical is $L^1=L$5. An earlier classification had asserted that no $L^1=L$6-dimensional solvable Leibniz algebra with nilradical $L^1=L$7 exists; the correction paper shows that this statement is false and that there is a unique $L^1=L$8-dimensional solvable Leibniz algebra $L^1=L$9 with multiplication

$L^{k+1}=[L^k,L]$0

$L^{k+1}=[L^k,L]$1

all other basic products being zero. This algebra is unique up to isomorphism and has trivial second cohomology with coefficients in itself, hence it is rigid (Ladra et al., 2016).

In the infinite-dimensional setting, residually solvable extensions of the canonical filiform algebra $L^{k+1}=[L^k,L]$2 are classified when the maximal pro-nilpotent ideal is $L^{k+1}=[L^k,L]$3. The codimension of the complementary space is bounded by the maximal number of residually nil-independent derivations of $L^{k+1}=[L^k,L]$4, which is $L^{k+1}=[L^k,L]$5. Accordingly, the paper classifies the $L^{k+1}=[L^k,L]$6-dimensional families $L^{k+1}=[L^k,L]$7, $L^{k+1}=[L^k,L]$8, $L^{k+1}=[L^k,L]$9 and the maximal $L$00-dimensional family $L$01. For the maximal case,

$L$02

$L$03

with all other brackets between $L$04 and $L$05 equal to zero. These algebras are complete and have trivial second Leibniz cohomology (Abdurasulov et al., 2021).

5. Derivations, gradings, automorphisms, and module constructions

Derivations are central to the structure theory. For the infinite-dimensional algebra $L$06, every derivation has the form

$L$07

This explicit description underlies the classification of residually solvable extensions and the proof that the maximal complementary dimension is $L$08 (Abdurasulov et al., 2021).

Gradings exhibit a similarly strong rigidity. For null-filiform Leibniz algebras, every abelian-group grading is toral. By contrast, for one-parametric filiform Leibniz algebras there exist non-toral gradings, and the paper classifies all abelian-group gradings up to equivalence in both settings (Calderón et al., 2017). On the symmetry side, naturally graded indecomposable non-Lie $L$09-filiform Leibniz algebras $L$10 admit local automorphisms that are not automorphisms; the paper proves that $L$11-filiform Leibniz algebras, as a rule, have this local-global discrepancy (Yusupov, 2023).

Another line of work studies Leibniz algebras through a filiform Lie quotient. If $L$12, then $L$13 is Lie. For Leibniz algebras with corresponding Lie algebra $L$14, the cited paper introduces a Fock module over $L$15 and classifies algebras for which $L$16 is either the Fock module or a minimal faithful $L$17-module. This yields both infinite-dimensional Fock-type Leibniz algebras $L$18 and finite-dimensional families classified explicitly in the $L$19-dimensional case (Ayupov et al., 2014).

6. Cohomology, rigidity, and geometric perspectives

Cohomology enters filiform Leibniz theory through derivations, deformations, and extension problems. For the unique $L$20-dimensional solvable Leibniz algebra $L$21 with nilradical $L$22, the calculations

$L$23

lead to

$L$24

and therefore to rigidity in the Leibniz variety (Ladra et al., 2016). In the infinite-dimensional pro-solvable setting, the maximal extensions $L$25 satisfy

$L$26

and also

$L$27

so completeness and cohomological rigidity persist in the infinite-dimensional filiform context (Abdurasulov et al., 2021).

A plausible implication is that maximality of the complementary solvable part and maximality of the grading length tend to correlate with rigidity phenomena, whereas smaller extensions often retain outer derivations. This pattern is explicit in the cited infinite-dimensional paper, where the codimension-$L$28 extensions are not complete but the codimension-$L$29 extensions are complete (Abdurasulov et al., 2021).

The surrounding geometric language is developed most fully on the Lie side. Filiform Lie algebras in a fixed Vergne basis form algebraic varieties, and in odd dimension contact structures and in even dimension symplectic structures are characterized by nonvanishing top-level structure constants; this Lie-theoretic framework furnishes a natural comparison class for Leibniz investigations of graded varieties and deformation spaces (Remm, 2017). A second comparison point is the $L$30-dimensional filiform Lie algebra $L$31, which is strongly characteristically nilpotent yet admits a nontrivial filiform deformation, showing that extreme nilpotency and non-rigidity can coexist in filiform geometry (Herrera-Granada et al., 2018). This suggests analogous deformation-theoretic questions for filiform Leibniz varieties, especially beyond the graded and central-extension regimes already classified.