Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras

Abstract: Over an arbitrary field, we conduct a comprehensive study of the polynomial identities and codimensions of two- and three-dimensional metabelian non-Lie Leibniz algebras. In addition, we compute the images of multihomogeneous polynomials on two-dimensional Leibniz algebras and, as a consequence, prove that the image of any multilinear polynomial evaluated on such algebras is always a vector space. Our analysis includes the three nontrivial isomorphism classes in dimension two and the ten isomorphism classes in dimension three, all of which are metabelian. In particular, we determine finite bases for their corresponding $T$-ideals and provide explicit bases for the associated relatively free graded algebras.

• The paper provides complete and explicit bases for the T-ideals of polynomial identities, enabling systematic classification of low-dimensional metabelian non-Lie Leibniz algebras.
• It uses detailed reduction algorithms and evaluations of multihomogeneous polynomials to establish vector space images and linear codimension growth with PI-exponent equal to 1.
• The study distinguishes PI-equivalence from isomorphism in three-dimensional cases, offering computational tools for invariant theory and further cohomological investigations.

Polynomial Identities and Codimensions in Low-Dimensional Metabelian Non-Lie Leibniz Algebras

Introduction and Scope

This paper presents a systematic study of the polynomial identities (PIs), codimension sequences, and images of multihomogeneous polynomials for all two- and three-dimensional metabelian non-Lie Leibniz algebras over arbitrary fields. By analyzing three isomorphism classes in dimension two and ten in dimension three, explicit and finite bases are obtained for their $T$-ideals of identities, together with thorough descriptions of the relatively free algebras and the codimension growth. The results unify the treatment across fields of arbitrary characteristic, offering explicit computations crucial for the PI-theory in nonassociative algebraic structures.

Background: PI-Theory and Metabelian Leibniz Algebras

Leibniz algebras extend Lie algebras by relaxing antisymmetry but maintaining the derivation property (Leibniz identity). Metabelian algebras, in this context, are characterized by the identity $(x_1x_2)(x_3x_4) = 0$. The investigation of explicit bases of $T$-ideals for polynomial identities in such algebras, especially in low dimension, remains a challenging and central task, with implications for structural and representation theory.

There is an established connection between the PI equivalence of finite-dimensional simple algebras and their isomorphism over algebraically closed fields, but such a direct correspondence generally fails for non-simple or non-Lie settings. The scarcity of explicit PI-bases in small nonassociative algebras motivates the detailed, case-by-case study in this work.

Polynomial Identities in Two-Dimensional Algebras

A complete isomorphism classification for two-dimensional Leibniz algebras yields three cases: $L_2$, $L_3$, and $L_4$. For $L_2$ (Lie), the $T$-ideal of identities is generated by $x_1^2$ and the metabelian identity $(x_1x_2)(x_3x_4)$, with a characteristic-dependent generator added for finite fields. For $L_4$ (non-Lie), $x_1(x_2x_3)$ always generates the ideal of identities, with $x_1 x_2^{(q)} - x_1 x_2$ required in the finite field case.

The explicit reduction algorithms detailed in the paper allow expression of any PI as a combinatorial linear combination of left-normed or symmetrized monomials, modulo the generators.

Multihomogeneous Images and the L’vov–Kaplansky Conjecture

The image sets of multihomogeneous polynomials evaluated on these algebras are shown to always be vector spaces, affirming a version of the L’vov–Kaplansky conjecture for all two-dimensional Leibniz algebras. Explicitly, for $L_2$ and $L_4$ any non-identity multihomogeneous polynomial $f$ satisfies $f(L_i) = K e_1$. For $L_3$ (commutative), images correspond to $K e_1$ or $\lambda K^2 e_1$, depending on the structure of $f$.

Codimension Sequence and Exponent

For both $L_2$ and $L_4$, the codimension sequence satisfies $c_1 = 1$ and $c_n = n-1$ (for $L_2$) or $n$ (for $L_4$, $n \ge 2$). This establishes linear boundedness and PI-exponent equal to $1$. Notably, in all two-dimensional cases, algebras are PI-equivalent if and only if they are isomorphic—a nontrivial structural result.

Identities and Codimensions in Three-Dimensional Cases

For three-dimensional metabelian non-Lie Leibniz algebras, ten isomorphism classes are analyzed (excluding those isomorphic to Lie or abelian cases). For each, generators of the $T$-ideal are produced, as well as bases for the relatively free algebra and explicit codimension sequences. For instance:

• $RR_2$: Generated by the metabelian identity and standard Lie-type alternator $s_3(x_1,x_2,x_3)$; in finite characteristic, additional field-dependent identities involving powers appear. Codimensions are $c_1 = 1$, $c_2 = 2$, $c_n = 2n-1$ ($n \geq 3$).
• $RR_3$: Depending on the characteristic, generated by various symmetrizing polynomials (e.g., $x_1 x_2 x_3 + x_2 x_1 x_3$) and the metabelian identity; in characteristic $2$, the commutator vanishes entirely. Codimensions are $c_1 = 1$, $c_2 = 2$, $c_n = n-1$ for $n>2$.
• $RR_4$, $RR_{10}$: Nilpotency imposes $x_1 x_2 x_3$ as a generator.
• $RR_6$, $RR_7$, $RR_9$, $RR_{11}$: Each is governed by combinations of two-variable power identities and the associator vanishing, with field-specific additional generators if the field is finite.

A striking phenomenon is observed: for $n \geq 3$, PI-equivalent does not imply isomorphism, in contrast with two-dimensional cases.

Explicit Bases for Relatively Free Algebras

For each isomorphism class, explicit combinatorial bases are constructed for the relatively free algebra modulo the $T$-ideal of identities. These are indexed by restricted compositions of degrees, governed by the generating identities and powers forced by field characteristic.

The methodology leverages evaluations on canonical bases, exploiting symmetries from the underlying Leibniz identity and, when necessary, diagonalizability properties of structure-related matrices (notably in $RR_7$ and $RR_9$).

Theoretical and Practical Implications

The comprehensive computation of PI generators, codimension sequences, and images delivers a toolkit for invariant-theoretic and representation-theoretic studies in nonassociative algebra. The negative result concerning PI-equivalence and isomorphism in dimension three provides a strong constraint for attempts at PI-based classification of Leibniz algebras. Furthermore, linear codimension growth across all computed cases, with exponent one, strongly constrains the possible complexity for related varieties.

From a practical standpoint, the explicit forms provided also facilitate computational approaches to automorphism, derivation, and cohomology problems on such small-dimension Leibniz algebras, which have applications in homological algebra and related combinatorial algebraic structures.

Potential future directions include extension to higher-dimensional metabelian Leibniz algebras, and more general classes beyond the metabelian constraint, as well as computational enumeration of relatively free algebra growth in these nonassociative settings.

Conclusion

This paper provides a comprehensive and explicit characterization of the polynomial identities, codimensions, and polynomial images for all two- and three-dimensional metabelian non-Lie Leibniz algebras over arbitrary fields. The main contributions are:

• Complete, explicit PI-bases for all isomorphism classes,
• Proof that the multilinear images are always vector spaces,
• Linear codimension growth with PI-exponent equal to one,
• Identification of PI-equivalence versus isomorphism dichotomy across dimensions.

These results significantly advance the explicit PI-theory for nonassociative structures and form a basis for algebraic and computational developments in Leibniz and related non-Lie algebraic systems.

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Citation:

Polynomial Identities and Codimensions of Two- and Three-Dimensional Metabelian Non-Lie Leibniz Algebras (2512.12282)