Classification of Filiform Lie Algebras

• Filiform Lie algebras are nilpotent structures with a maximally long lower central series, where each successive quotient is one-dimensional.
• Their classification involves analyzing isomorphism and isotopism classes, deformation families, and the geometry of moduli spaces in dimensions 2 to 8.
• Techniques such as adapted (Vergne) bases, invariant polynomials, and central extensions bridge the understanding from finite to infinite-dimensional cases.

A filiform Lie algebra is a finite-dimensional nilpotent Lie algebra whose lower central series attains maximal length: for an $n$-dimensional filiform Lie algebra $\mathfrak{g}$ over a field $K$, the lower central series $$C^1\mathfrak{g} = \mathfrak{g},\, C^{j+1}\mathfrak{g} = [C^j\mathfrak{g},\, \mathfrak{g}]$$ satisfies $\dim C^{k}\mathfrak{g} = n - k$ for all $2 \leq k \leq n$, equivalently, the nilpotency class is $n-1$ and each successive quotient in the central series is of dimension one. These algebras play a central role in the study of the varieties and deformation theory of nilpotent Lie algebras, both in finite and infinite dimensions. The classification of filiform Lie algebras involves understanding isomorphism and isotopism classes, deformation families, moduli space geometry, and invariant-theoretic distinctions in small dimensions.

1. Structure, Definitions, and Model Algebras

A filiform Lie algebra $\mathfrak{g}$ in dimension $n$ is defined via the structure of its lower central series and its adapted basis (often called the Vergne basis) $\{e_1, ..., e_n\}$, in which the model brackets are

$[e_1, e_{i+1}] = e_{i}, \quad i = 1, ..., n-1,$

with all other brackets either zero or determined by central extension parameters and deformations specific to the isomorphism class. The model (or "naturally graded") filiform algebra is the unique algebra in which only these brackets are nonzero. Non-model filiforms arise from one-dimensional central extensions applied successively to the model, and all such extensions can be understood through the parameterization of structure constants in adapted bases (see (Falcón et al., 2015, Remm, 2017, Rakhimov et al., 2010)).

In the infinite-dimensional context, $N$-graded Lie algebras of maximal class (residually nilpotent, $\dim L/L^2 = 2$, and $\dim L^i/L^{i+1} = 1$ for $i \geq 2$) admit only three isomorphism types when generated in degrees 1 and 2: $m_0$ (the shift algebra), $m_2$ (the extension of $m_0$), and $W$ (Witt algebra), with further restrictions for higher step generators (see (Barron et al., 2014)).

2. Classification in Low Dimensions: Isomorphism and Isotopism

The classification of filiform Lie algebras in dimensions up to 7 has been completed via various approaches:

Dimensions 2–4: There is only one isomorphism class (the abelian and model Heisenberg types) in each dimension. Isotopism partitions coincide with isomorphism classes.

Dimension 5: Two isomorphism classes exist: the model and a one-parameter deformation (all nonzero deformation parameters equivalent up to isomorphism). Isotopism invariants match the isomorphism classes.

Dimension 6: Five isomorphism classes occur, corresponding to different parameterizations for central extensions. Isotopism classes still equal isomorphism classes unless characteristic is 2, in which case some classes merge.

Dimension 7: The classification is distinguished by a subtler interplay of invariants. Over algebraically closed fields of characteristic $\neq 2$, there are eight isomorphism classes, given by explicit choices of parameters $(a,b,c,d)$ in the basis as

$$\begin{align*} [e_1, e_{i+1}] &= e_i\quad (i = 1,\ldots,6), \ [e_4, e_7] &= a e_2, \ [e_5, e_6] &= b e_2, \ [e_5, e_7] &= c e_2 + (a+b) e_3, \ [e_6, e_7] &= d e_2 + c e_3 + (a+b) e_4, \end{align*}$$

with allowed $(a,b,c,d)$ tuples detailed in (Falcón et al., 2015). Over finite fields $\mathbb{F}_p$ ($p \neq 2$), the number of distinct isomorphism classes increases to $p + 8$ due to square-class invariants ($\Delta = 4ad - 5c^2$). Isotopism coarsens these classes, giving always 8 isotopism classes.

Table: Isomorphism and Isotopism Classes for Dimensions $2 \leq n \leq 7$ (Falcón et al., 2015)

$n$	# Isomorphism Classes (char $\neq 2$)	# Isotopism Classes (char $\neq 2$)
2	1	1
3	1	1
4	1	1
5	2	2
6	5	5
7	$p + 8$ ($\mathbb{F}_p$)	8

This pattern generalizes: for $n \leq 6$ over $\mathbb{F}_p$ or algebraically closed fields, isotopism classes coincide with isomorphism classes; for $n=7$, isotopism classes fuse the "square class" families, producing a strictly coarser partition. See (Falcón et al., 2015, Abdulkareem et al., 2013, Rakhimov et al., 2010).

3. Varieties, Deformation Theory, and Geometric Components

The moduli of filiform Lie algebra structures on a fixed vector space can be realized as algebraic varieties $\mathcal{F}_n$ parameterized via adapted basis coordinates. The defining equations arise from the Jacobi identities and the maximal nilpotency condition; for small $n$, these are Zariski-closed subsets of affine space. For example, in $n=8$, $\mathcal{F}_8$ splits into two irreducible 6-dimensional components classified by constraints on structure constants (e.g., $a_1 = 0$ and $5a_4 + 2a_2 = 0$ in (Remm, 2017)). Components are typically understood as orbit closures under the group action of $GL(n)$ on structure constants.

Rigidity is rare: in dimensions $n \geq 8$, the variety is never zero-dimensional, so no single filiform is rigid; instead, "rigid families" with positive-dimensional open orbits appear. The cohomology group $H^2_{filiform}$ parameterizes genuine deformations that remain within the filiform class (Remm, 2017).

In the infinite-dimensional case, the associated projective moduli are cut out by weighted quadratic systems—see (Barron et al., 2014) for explicit equations and geometric structure.

4. Invariant-Based Classification: $(z_1, z_2)$ and Hilbert Polynomials

Recent advances use the bracket-ideal bifiltration

$F(k,\ell) = [C^k\mathfrak{g}, C^\ell\mathfrak{g}]$

and associated symmetric bivariate Hilbert polynomials

$$\mathrm{HP}_\mathfrak{g}(t,s) = \sum_{k,\ell\geq1} (\dim F(k,\ell))\, t^k s^\ell$$

as refined invariants. Two classical combinatorial invariants, $z_1$ (centralizer depth) and $z_2$ (maximal abelian stair), partition filiforms into finitely many types, but these invariants may fail to distinguish all isomorphism classes.

The Hilbert polynomial $\mathrm{HP}_\mathfrak{g}(t,s)$ encodes all bracket dimensions, is symmetric, and in low-dimensional cases can be computed directly from structure constants in an adapted basis. Its support and "arrow-shape" allow for the recovery of $z_1$ and $z_2$; within each $(z_1, z_2)$ family, distinct $\mathrm{HP}_\mathfrak{g}$ often separate otherwise undistinguished isomorphism classes. Explicit examples in $n=8,9,10$ demonstrate this phenomenon (Castro-Jiménez et al., 2 May 2025).

5. Filiform Algebras in Dimensions 7 and 8: Canonical Lists

The dimension 7 case has five rigid single orbit isomorphism classes, each given by explicit normal forms in an adapted basis and historically confirmed in lists by Ancochea-Bermúdez, Goze, Gómez, Jiménez-Merchán, and Khakimdjanov. In dimension 8, there are eleven classes, two of which are one-parameter families; all others are rigid. Explicit structure constants, orbit invariants, and adapted normal forms are given by Rakhimov–Hassan (Rakhimov et al., 2010), and comparison of different classification methods confirms all cases. See table:

Dimension	Number of Classes	Parameterization
7	5	Single orbits
8	11	2 families, 9 rigid

These lists include the contribution from central extension and the action of the adapted-change group (see (Abdulkareem et al., 2013, Rakhimov et al., 2010, Falcón et al., 2015)).

6. Contact, Symplectic, and Further Geometric Structures

Filiform Lie algebras exhibit geometric features related to contact and symplectic forms. The presence of a contact form in odd dimension requires specific nonvanishing conditions on structure constants in the Vergne basis, while symplectic structures in even dimension are equivalent to the existence of a contact structure on a one-dimensional central extension of the algebra. Explicit polynomial criteria for existence are available for $n=8,10$, and these conditions partition the variety into components correlating with geometric structure (see (Remm, 2017)).

7. Infinite-Dimensional and Graded Filiform Algebras

In the infinite-dimensional case, the $N$-graded Lie algebras of maximal class have a much simpler landscape: up to isomorphism, only finitely many families (namely, shifts and Witt-type) are possible, with all finite-dimensional filiforms arising as central truncations and deformations thereof (Barron et al., 2014). The passage between finite and infinite cases explains the rigid skeleton upon which all filiform deformations are built.

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The classification of filiform Lie algebras thus uses a hierarchy of invariants—from combinatorial ($z_1, z_2$) and Hilbert polynomial profiles to geometric and cohomological criteria and moduli-variety decomposition. Explicit isomorphism criteria, parameter spaces, and orbit invariants have been constructed for all cases up to dimension 8, with higher-dimensional cases organized by families within varieties. The connection with infinite-dimensional graded algebras elucidates the theoretical foundation for the moduli geometry of filiform Lie algebra structures on vector spaces (Falcón et al., 2015, Castro-Jiménez et al., 2 May 2025, Barron et al., 2014, Remm, 2017, Rakhimov et al., 2010, Abdulkareem et al., 2013).