Three-Dimensional Leibniz Algebras

• Three-dimensional Leibniz algebras are vector spaces with a bilinear bracket that satisfies the left Leibniz identity, splitting into Lie and proper non-Lie cases.
• They are classified by the dimension of their Leibniz kernel and derived series, with detailed examples over fields such as ℂ and ℝ showcasing distinct automorphism groups.
• The topic encompasses degeneration theory, polynomial invariants, and Nijenhuis operators, serving as a concrete laboratory for exploring algebraic and geometric properties.

Searching arXiv for recent and foundational papers on three-dimensional Leibniz algebras. Search query: all:("three-dimensional Leibniz algebras" OR "3-dimensional Leibniz algebras" classification automorphism degeneration) A three-dimensional Leibniz algebra is a three-dimensional algebra $L$ with a bilinear bracket $[\,,\,]$ satisfying the left Leibniz identity

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

without the antisymmetry required for Lie algebras. In dimension three, the subject is unusually explicit: classification by isomorphism can be carried out over several base fields, the proper non-Lie and Lie cases can be separated by the Leibniz kernel $\Leib(L)$, and structural features such as nilpotency, solvability, automorphism groups, degenerations, polynomial invariants, and Nijenhuis operators admit concrete descriptions. Over an arbitrary field, there are four Lie families and forty proper Leibniz families; over $\mathbb{C}$, the geometric classification of the affine variety of three-dimensional Leibniz structures yields four Lie algebras and nine non-Lie Leibniz algebras; over $\mathbb{R}$, one classification used in the study of Nijenhuis operators consists of thirteen pairwise non-isomorphic types (Kurdachenko et al., 2022, Ismailov et al., 2018, Ma et al., 16 Jul 2025).

1. Definition and basic structural dichotomies

A left Leibniz algebra is an algebra $L$ with bracket $[\,,\,]$ satisfying

$[[a,b],c]=[a,[b,c]]-[b,[a,c]]$

for all $a,b,c\in L$ (Kurdachenko et al., 2022). A Leibniz algebra is Lie exactly when $[\,,\,]$0, where $[\,,\,]$1 denotes the Leibniz kernel; in the three-dimensional classification over an arbitrary field, this yields the primary split into Lie algebras and proper (non-Lie) Leibniz algebras (Kurdachenko et al., 2022).

For three-dimensional proper Leibniz algebras over an arbitrary field $[\,,\,]$2, one has

$[\,,\,]$3

and the classification organizes accordingly (Kurdachenko et al., 2022). The same source records several general constraints. Nilpotent algebras occur only in the block $[\,,\,]$4 with $[\,,\,]$5 and $[\,,\,]$6 abelian, where there are eight $[\,,\,]$7-step nilpotent algebras, and in the block $[\,,\,]$8, where there is one $[\,,\,]$9-step nilpotent algebra. All non-nilpotent Leibniz algebras are solvable with derived length $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$0, and the left center

$[[x,y],z]=[x,[y,z]]-[y,[x,z]],$1

always contains $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$2 (Kurdachenko et al., 2022).

Over an algebraically closed field of characteristic zero, one particularly tractable subproblem is the classification of non-Lie solvable left Leibniz algebras of dimension three. In the formulation revisited in the solvable classification paper, every such algebra satisfies

$[[x,y],z]=[x,[y,z]]-[y,[x,z]],$3

hence is solvable of derived length $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$4 but not nilpotent (Demir et al., 2015). This sharp distinction between nilpotent and non-nilpotent behavior is a recurrent organizing principle throughout the dimension-three theory.

2. Classification over arbitrary fields

The classification over an arbitrary field $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$5 consists of four Lie families and forty proper Leibniz families, exhausting all three-dimensional left Leibniz algebras up to isomorphism (Kurdachenko et al., 2022). The Lie part contains the abelian algebra, the Heisenberg algebra, a one-parameter solvable non-nilpotent family $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$6 with $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$7, and split $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$8 when $[[x,y],z]=[x,[y,z]]-[y,[x,z]],$9 (Kurdachenko et al., 2022).

For proper Leibniz algebras, the classification splits by $\Leib(L)$0.

When $\Leib(L)$1, one writes $\Leib(L)$2. The next dichotomies are whether $\Leib(L)$3, and whether $\Leib(L)$4 is abelian. In the central-quotient-abelian case there are eight nilpotent extensions of the cyclic $\Leib(L)$5-algebra, including $\Leib(L)$6, $\Leib(L)$7, $\Leib(L)$8, and $\Leib(L)$9, as well as six further types subject to irreducibility conditions on a quadratic polynomial (Kurdachenko et al., 2022). In the central-quotient-nonabelian case there are six non-nilpotent extensions of the two-dimensional non-abelian Lie algebra, including $\mathbb{C}$0 and the one-parameter family $\mathbb{C}$1, plus four further mixed cyclic-ideal sums with a quadratic condition (Kurdachenko et al., 2022). If $\mathbb{C}$2, there are six non-nilpotent types with $\mathbb{C}$3 abelian and thirteen non-nilpotent mixed cyclic sums with $\mathbb{C}$4 non-abelian, appearing as $\mathbb{C}$5 with explicit linear-quadratic relations among structure constants and field constraints (Kurdachenko et al., 2022).

When $\mathbb{C}$6, one writes $\mathbb{C}$7, so $\mathbb{C}$8 is one-dimensional and the multiplication is determined by the linear map

$\mathbb{C}$9

Up to basis choice in $\mathbb{R}$0 and normalization of $\mathbb{R}$1, there are exactly six isomorphism types (Kurdachenko et al., 2022). Among these are the nilpotent cyclic algebra $\mathbb{R}$2 with

$\mathbb{R}$3

the non-nilpotent cyclic algebra $\mathbb{R}$4 with

$\mathbb{R}$5

and the one-parameter-type families $\mathbb{R}$6 and $\mathbb{R}$7, where $\mathbb{R}$8 on $\mathbb{R}$9 has either a nonzero eigenvalue or a single Jordan block, together with two further cyclic Jordan-block types (Kurdachenko et al., 2022).

This classification also identifies the field-dependent phenomena. They occur precisely in families involving $L$0, irreducibility conditions for a polynomial $L$1, or exclusion of values such as $L$2 in certain one-parameter families (Kurdachenko et al., 2022). A plausible implication is that dimension three is already large enough for arithmetic properties of the field to influence isomorphism classes in an essential way.

3. Solvable non-Lie algebras with two-dimensional derived algebra

Over an algebraically closed field of characteristic zero, every three-dimensional non-Lie solvable left Leibniz algebra is, up to isomorphism, one of six types: four discrete algebras and two one-parameter families (Demir et al., 2015). In all cases,

$L$3

and $L$4, so the algebra is solvable but not nilpotent (Demir et al., 2015).

The six types are summarized below.

Type	Nonzero products	Remarks
I	$L$5	discrete
II$L$6	$L$7	$L$8
III	$L$9	discrete
IV	$[\,,\,]$0	quotient by $[\,,\,]$1 is non-abelian Lie
V$[\,,\,]$2	$[\,,\,]$3	$[\,,\,]$4
VI	$[\,,\,]$5	discrete

Type II$[\,,\,]$6 is described as semidirect type, with left multiplication $[\,,\,]$7 diagonalizable with eigenvalues $[\,,\,]$8; Type III is explicitly noted as not of semidirect diagonal form; Type IV is “Lie-like” in the $[\,,\,]$9-plane and has quotient $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$0 equal to the two-dimensional non-abelian Lie algebra; Type V$[[a,b],c]=[a,[b,c]]-[b,[a,c]]$1 combines the patterns of II and IV; Type VI has a two-step flag with successive one-dimensional brackets (Demir et al., 2015).

The only parameters occur in Types II and V, with the identification $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$2 and no further identifications (Demir et al., 2015). The source states that one may take representatives of $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$3 if desired (Demir et al., 2015). This suggests that, in this subclassification, the moduli are extremely small: apart from inversion symmetry, the parameter is rigidly retained.

A separate 2024 study isolates two non-nilpotent families over an arbitrary field with $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$4 and a specified “one-generator” subalgebra pattern (Kurdachenko et al., 2024). The first is

$[[a,b],c]=[a,[b,c]]-[b,[a,c]]$5

and the second is the one-parameter family

$[[a,b],c]=[a,[b,c]]-[b,[a,c]]$6

The paper states that, under its hypotheses, these are exactly the two non-nilpotent types, and that over an algebraically closed field one may rescale $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$7 to make $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$8 (Kurdachenko et al., 2024). This does not purport to replace the broader classification of all three-dimensional non-nilpotent Leibniz algebras, but rather identifies two distinguished families within a more restricted construction.

4. Nilpotent algebras and automorphism groups

A 2023 study of automorphism groups considers three nilpotent, three-dimensional non-Lie Leibniz algebras that it describes as exhausting the non-Lie nilpotent types in dimension $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$9 (Kurdachenko et al., 2023). In the basis $a,b,c\in L$0, these are: $a,b,c\in L$1

$a,b,c\in L$2

$a,b,c\in L$3

Their nilpotency data are recorded there as class $a,b,c\in L$4 for $a,b,c\in L$5, and class $a,b,c\in L$6 for $a,b,c\in L$7 and $a,b,c\in L$8, with $a,b,c\in L$9 and $[\,,\,]$00 (Kurdachenko et al., 2023).

The automorphism groups admit explicit upper-triangular matrix realizations. For $[\,,\,]$01,

$[\,,\,]$02

with $[\,,\,]$03, $[\,,\,]$04 (Kurdachenko et al., 2023). For $[\,,\,]$05,

$[\,,\,]$06

with $[\,,\,]$07 and $[\,,\,]$08 (Kurdachenko et al., 2023). For $[\,,\,]$09, the automorphism group is

$[\,,\,]$10

and it decomposes as

$[\,,\,]$11

hence

$[\,,\,]$12

with $[\,,\,]$13, $[\,,\,]$14, $[\,,\,]$15 (Kurdachenko et al., 2023).

The same paper records the dimensions

$[\,,\,]$16

and emphasizes that the size and decomposition of $[\,,\,]$17 reflect the central and derived series of $[\,,\,]$18 (Kurdachenko et al., 2023). More specifically, invariance of $[\,,\,]$19 under automorphisms and the induced scalings on successive quotients account for the toral factors in the group structure (Kurdachenko et al., 2023).

For non-nilpotent examples, the 2024 automorphism study computes two explicit families. For

$[\,,\,]$20

one obtains $[\,,\,]$21 (Kurdachenko et al., 2024). For

$[\,,\,]$22

one obtains an abelian group

$[\,,\,]$23

with representatives

$[\,,\,]$24

(Kurdachenko et al., 2024). These formulas provide concrete instances of how small changes in the bracket can alter the automorphism group from a non-abelian linear group to a direct product.

5. Degenerations, irreducible components, and rigidity over $[\,,\,]$25

The geometric study of the affine variety $[\,,\,]$26 of all complex three-dimensional Leibniz algebras identifies four Lie algebras and nine non-Lie Leibniz algebras up to isomorphism (Ismailov et al., 2018). The non-Lie algebras are denoted $[\,,\,]$27, with explicit multiplication tables (Ismailov et al., 2018). In each of the one-parameter families, distinct parameters outside finitely many resonant values give non-isomorphic algebras; among the Lie families, the only further identifications are $[\,,\,]$28 (Ismailov et al., 2018).

Primary degenerations are given by one-parameter changes of basis $[\,,\,]$29 with polynomial entries in $[\,,\,]$30, such that the structure constants converge to those of the target algebra as $[\,,\,]$31 (Ismailov et al., 2018). The list of primary degenerations includes, among others,

$[\,,\,]$32

$[\,,\,]$33

$[\,,\,]$34

$[\,,\,]$35

and every algebra degenerates to the zero algebra (Ismailov et al., 2018).

The variety $[\,,\,]$36 has exactly five irreducible components (Ismailov et al., 2018). They are the closures of the families or rigid points: $[\,,\,]$37

$[\,,\,]$38

with respective dimensions $[\,,\,]$39 (Ismailov et al., 2018). The rigid algebras are exactly $[\,,\,]$40 and $[\,,\,]$41 (Ismailov et al., 2018). Since a rigid algebra has a Zariski-open $[\,,\,]$42-orbit, this gives an explicit geometric distinction between structurally stable and non-stable multiplication laws in dimension three.

This degeneration picture complements the isomorphism classification. The classification enumerates orbit representatives; the degeneration poset describes orbit closures. A plausible implication is that, in low dimension, deformation-theoretic and geometric invariant-theoretic viewpoints are unusually close to explicit computation.

6. Polynomial invariants and Nijenhuis operators

Over $[\,,\,]$43, the invariant-theoretic study of the nine non-Lie isomorphism classes gives explicit automorphism groups, rings of polynomial invariants, and low-degree trace invariants (Kaygorodov et al., 11 May 2025). The classification used there consists of the algebras $[\,,\,]$44, with exactly three nilpotent algebras: $[\,,\,]$45 while all others have infinite nilpotency class (Kaygorodov et al., 11 May 2025).

The polynomial invariant rings $[\,,\,]$46 are computed explicitly. Representative cases include

$[\,,\,]$47

$[\,,\,]$48

$[\,,\,]$49

$[\,,\,]$50

$[\,,\,]$51

$[\,,\,]$52

(Kaygorodov et al., 11 May 2025). In each non-trivial case these generators are algebraically independent (Kaygorodov et al., 11 May 2025).

The same paper states that among the six non-nilpotent algebras

$[\,,\,]$53

the first- and second-degree traces together with $[\,,\,]$54 form a complete invariant (Kaygorodov et al., 11 May 2025). Using

$[\,,\,]$55

it gives criteria such as

$[\,,\,]$56

$[\,,\,]$57

$[\,,\,]$58

(Kaygorodov et al., 11 May 2025). This yields an explicitly computable separation scheme for the non-nilpotent complex non-Lie cases.

A different but related direction is the study of Nijenhuis operators on real three-dimensional Leibniz algebras. One 2025 classification adopts thirteen non-isomorphic real types $[\,,\,]$59, each given by a bracket table in a fixed basis $[\,,\,]$60, and classifies all Nijenhuis operators

$[\,,\,]$61

satisfying

$[\,,\,]$62

for all $[\,,\,]$63 (Ma et al., 16 Jul 2025). For each type, the set of Nijenhuis operators is described as a finite union of affine families in $[\,,\,]$64; for example, Type $[\,,\,]$65 admits the single two-parameter family

$[\,,\,]$66

and Type $[\,,\,]$67 admits the one three-parameter family

$[\,,\,]$68

(Ma et al., 16 Jul 2025). The paper concludes that for each of the thirteen real isomorphism classes, the Nijenhuis operators form a finite union of affine algebraic subvarieties in $[\,,\,]$69 (Ma et al., 16 Jul 2025). This places integrable operator deformations alongside classification and invariant theory as a mature component of the three-dimensional subject.

7. Scope, correspondences, and recurrent themes

Several parallel classifications coexist because they work over different base fields, emphasize different subclasses, or encode different equivalence relations. The arbitrary-field classification gives four Lie cases and forty proper Leibniz families (Kurdachenko et al., 2022). The solvable non-Lie classification over an algebraically closed field of characteristic zero isolates six isomorphism types with $[\,,\,]$70 (Demir et al., 2015). The degeneration study over $[\,,\,]$71 describes four Lie algebras and nine non-Lie Leibniz algebras as points of the variety $[\,,\,]$72 (Ismailov et al., 2018). The real Nijenhuis-operator study uses thirteen real isomorphism classes (Ma et al., 16 Jul 2025). These are not contradictory counts; they reflect distinct classification problems.

Two recurrent invariants organize the literature. The first is the Leibniz kernel $[\,,\,]$73, whose dimension in the non-Lie three-dimensional case is always $[\,,\,]$74 or $[\,,\,]$75 over an arbitrary field (Kurdachenko et al., 2022). The second is the derived algebra $[\,,\,]$76, whose dimension often controls whether the algebra is nilpotent, which automorphism-group patterns can occur, and how the algebra sits in degeneration diagrams (Demir et al., 2015, Kurdachenko et al., 2023, Kurdachenko et al., 2024).

Another recurrent theme is the coexistence of algebraic and geometric classification. Invariant data such as centers, derived series, nilpotency class, automorphism groups, trace polynomials, and Nijenhuis families are all explicit in dimension three (Kurdachenko et al., 2022, Kurdachenko et al., 2023, Kaygorodov et al., 11 May 2025, Ma et al., 16 Jul 2025). At the same time, degeneration theory and irreducible components show how these isomorphism classes assemble into a global moduli picture over $[\,,\,]$77 (Ismailov et al., 2018).

A common misconception is that three-dimensional Leibniz algebras are merely a small perturbation of the three-dimensional Lie classification. The dimension-three results show otherwise. Proper Leibniz algebras admit central and non-central Leibniz kernels, cyclic and mixed cyclic constructions, field-sensitive irreducibility constraints, distinct automorphism-group decompositions, and nontrivial invariant rings that have no analogue in the purely Lie three-dimensional case (Kurdachenko et al., 2022, Kurdachenko et al., 2023, Kaygorodov et al., 11 May 2025). The low-dimensional setting is therefore not only a test case but also a complete laboratory in which classification, geometry, and operator theory can all be carried out explicitly.