Metabelian Non-Lie Leibniz Algebras

• Metabelian non-Lie Leibniz algebras are nonassociative systems with a bilinear bracket that satisfies the left Leibniz identity, and their derived algebra is abelian.
• They generalize metabelian Lie algebras by omitting bracket anti-symmetry, enabling rich classification in low dimensions with explicit structural models.
• These algebras exhibit specialized polynomial identities, distinct automorphism groups, and integrated cohomological aspects that drive further algebraic research.

Metabelian non-Lie Leibniz algebras are nonassociative algebras—possessing a bilinear bracket that satisfies the (left) Leibniz identity—whose derived algebra is abelian but for which the bracket is not skew-symmetric. These algebras generalize the concept of metabelian Lie algebras by omitting the anti-symmetry of the bracket, thus enabling a wider and more intricate landscape of algebraic structures. They are central in the study of solvable non-Lie nonassociative symmetries and exhibit specialized polynomial identities, automorphism group structures, and cohomological features.

1. Structure of Metabelian Non-Lie Leibniz Algebras

A Leibniz algebra over a field $k$ is a $k$-vector space $\mathfrak{g}$ with a bilinear bracket $$[-,-]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$$ obeying the left Leibniz identity: $$[x,[y,z]] = [[x,y],z] - [[x,z],y],\quad \forall x,y,z\in\mathfrak{g}.$$ The derived series is defined by $\mathfrak{g}^{(0)} = \mathfrak{g}$, $\mathfrak{g}^{(1)} = [\mathfrak{g},\mathfrak{g}]$, $$\mathfrak{g}^{(n+1)} = [\mathfrak{g}^{(n)},\mathfrak{g}^{(n)}]$$. The algebra is metabelian if $$[\,[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]\,]=0$$, so that $\mathfrak{g}' = [\mathfrak{g},\mathfrak{g}]$ is abelian as a two-sided ideal and the derived length is at most $2$ (Agore et al., 2014).

A pivotal result states that every metabelian Leibniz algebra admits a structure as a “unified product” $V\oplus P$, where $V$, $P$ are abelian Leibniz algebras (i.e., vector spaces with trivial bracket), and the bracket is defined for $x,y\in V$, $p,q\in P$ by

$$\{(x,p),(y,q)\} = (x\cdot q + p\cdot y + f(p,q),0),$$

where $\cdot:V\otimes P \to V$, $\cdot:P\otimes V\to V$ are bilinear “actions” and $f:P\otimes P\to V$ is a bilinear “cocycle” subject to two families of associativity–Leibniz compatibilities (Agore et al., 2014). This description encompasses all metabelian Leibniz algebras and gives rise to a wide moduli of non-Lie structures.

When $\dim \mathfrak{g}' = 1$, the classification becomes explicit: any such algebra is isomorphic to one of the types $\mathfrak{g}(P,\alpha,\beta,f)$, where $\alpha,\beta\in P^*$ and $f\in \mathrm{Bil}(P\times P, k)$, distinguished by the vanishing or nonvanishing of $\alpha$, $\beta$, $f$. Non-Lie behavior emerges as soon as either $\alpha+\beta\neq 0$ or $f(p,p)\neq 0$ for some $p$ (Agore et al., 2014).

2. Classification in Small Dimensions

In dimension $2$ over an arbitrary field, there are exactly three nontrivial isomorphism classes of metabelian non-Lie Leibniz algebras (Fertunani et al., 13 Dec 2025):

• $L_2$: Non-abelian Lie algebra, $e_1e_2=-e_2e_1=e_1$.
• $L_3$: Commutative non-Lie algebra, $e_2e_2=e_1$.
• $L_4$: Non-Lie, non-commutative, $e_1e_2=e_2e_2=e_1$.

In dimension $3$, ten of the eleven classical non-Lie Leibniz algebras are metabelian, with canonical representatives $RR_2(\alpha)$ to $RR_{11}$. All these classes are defined by their bracket structure and satisfy the metabelian identity $(x_1x_2)(x_3x_4)=0$ (Fertunani et al., 13 Dec 2025).

For algebras with one-dimensional derived subalgebra and which are non-nilpotent and non-Lie, a full classification is available: any such $n$-dimensional algebra over a field of characteristic $\neq 2$ is isomorphic to $L_n \cong S_2\oplus\mathbb{F}^{n-2}$, with $S_2$ generated by $[e_2,e_1]=e_1$ (Bartolo et al., 2023).

3. Polynomial Identities and Codimensions

Metabelian non-Lie Leibniz algebras satisfy specific polynomial identities: $(x_1x_2)(x_3x_4)=0$ for all elements, characterizing the metabelian property in the language of varieties.

Finite bases for the $T$-ideals of polynomial identities are constructed for all 2- and 3-dimensional classes. For example, $T(L_2) = \langle x_1^2, (x_1x_2)(x_3x_4)\rangle_T$ over infinite fields, while $T(L_4) = \langle x_1(x_2x_3)\rangle_T$ (Fertunani et al., 13 Dec 2025). Minimality of these bases arises by specialized evaluations in canonical bases.

The $n$-th codimension $c_n(A)$ of these algebras—$\dim_K(P_n/(P_n\cap T(A)))$ where $P_n$ is the component of degree $n$—ranges from linear to zero growth, with $\exp(L_2)=1$, $\exp(L_3)=0$, $\exp(L_4)=1$, and explicit formulas for all values (Fertunani et al., 13 Dec 2025).

Explicit monomial bases for the relatively free algebras are constructed by normal forms arising from repeated application of the Leibniz identity and, for finite fields, by reductions controlling exponents.

4. Automorphism Groups and Derivations

Let $$G=\operatorname{Aut}_{Lbz}(\mathfrak{g}(P,\alpha,\beta,f))$$. Every automorphism has the form $\varphi(a,p) = (u a + \gamma(p), \psi(p))$ for $u\in k^*$, $\gamma\in P^*$, $\psi\in \operatorname{Aut}_k(P)$, with compatibility conditions:

• $u\alpha(p)=\alpha\circ\psi(p)$,
• $u\beta(p)=\beta\circ\psi(p)$,
• $u f(p,q) = f(\psi(p),\psi(q))$.

Thus $G$ sits inside a semidirect product $P^* \rtimes (k^* \times \operatorname{Aut}_k(P))$ (Agore et al., 2014).

For $L_n$ (dimension $n$, derived algebra one-dimensional), every derivation $d$ is given by a block matrix with arbitrary scalars and matrices, and the automorphism group is $$\mathbb{F}^\times \times \operatorname{GL}_{n-2}(\mathbb{F}) \ltimes \mathbb{F}^{n-2}$$ (Bartolo et al., 2023). The structure of biderivations is also given by paired block matrices with explicit compatibility constraints.

5. Normal Forms and Images of Polynomial Maps

In all cases, the explicit computation of normal forms of left-normed monomials allows the determination of codimension sequences, monomial bases, and the images of multihomogeneous polynomials. On $L_2$ and $L_4$, every nonzero multihomogeneous polynomial attains as its set of values the one-dimensional subspace $Ke_1$, while on $L_3$ the possibilities for the image are $\{0\}$, $Ke_1$, a coset, or the full algebra (Fertunani et al., 13 Dec 2025). The image of any multilinear polynomial on any 2-dimensional Leibniz algebra is always a vector subspace.

The reduction methods rely crucially on systematic use of the Leibniz identity for monomial rewrites, multilinearity for variable elimination, and field-specific exponent reductions (Fertunani et al., 13 Dec 2025).

6. Integration and Cohomological Aspects

Non-nilpotent, one-derived metabelian non-Lie Leibniz algebras $L_n$ can be integrated into a global Lie rack structure, with an explicit operation on $\mathbb{R}^n$: $$x = (x_1,\dots,x_n) \rhd y = (y_1, e^{x_1} y_2, y_3, \dots, y_n)$$ where the tangent bracket recovers the original Leibniz bracket $[e_2,e_1]=e_1$. The integration respects left-autodistributivity and smooth manifold structure (Bartolo et al., 2023).

Biderivations and cohomological interpretations play a critical role in extension theory and in demonstrating the rigidity of the splitting for certain extensions, as all cocycles can be eliminated by change of basis in $L_n$ (Bartolo et al., 2023).

7. Further Directions and Open Problems

While the structure of metabelian non-Lie Leibniz algebras with one-dimensional derived algebra is fully classified and explicit in arbitrary dimension, the general case for higher-dimensional derived ideals rapidly becomes wild. Classification in dimensions higher than $3$ (beyond low-dimensional results) requires intricate analysis of bilinear maps up to isomorphism, intractable without employing major theory such as Kronecker–Williamson for bilinear pairs (Agore et al., 2014). This suggests a rich, yet unconquered, landscape for further exploration of such non-Lie structures, particularly in deformation theory, invariants, and applications to generalized symmetry.