Unique addition in centerless metabelian Lie algebras

Henry Zweiman

September 8, 2026

Abstract

A Lie ring has unique addition if every bracket-preserving bijection from it onto an arbitrary Lie ring is additive. We prove that every finite-dimensional centerless metabelian Lie algebra over an infinite field has unique addition, with no restriction on the characteristic. The underlying criterion applies to a faithful semidirect product , where is an arbitrary abelian group and the acting Lie ring contains a central element acting invertibly on . Applications include the nine-dimensional algebra in a question of Arzhantsev and families for which the least number of elements with zero common centralizer is arbitrarily large. We explain the relation with earlier additivity results for triangular associative algebras.

1. Introduction

A unique-addition Lie ring, or a UA-Lie ring, is a Lie ring such that, for every Lie ring , each bijection satisfying

is additive. Neither additivity nor linearity is included in the hypothesis on . In particular, when is a Lie algebra over a field, the target need not be a Lie algebra over that field.

Arzhantsev’s recent account [1] develops structural criteria for this property and poses several open questions. An important sufficient condition, over an infinite field, is the existence of two elements whose common centralizer is zero. His Example 2 is a nine-dimensional centerless solvable Lie algebra for which this condition fails; Problem 4 asks whether that algebra nevertheless has unique addition. Our main result concerns a full class of centerless solvable Lie algebras.

Theorem 1.

Every finite-dimensional centerless metabelian Lie algebra over an infinite field is a UA-Lie ring.

Here metabelian means that the derived algebra is abelian. The characteristic in Theorem 1 is arbitrary. The proof combines a faithful-action criterion with an elementary fact about commuting linear operators. Its first ingredient is valid without a field or a finiteness assumption.

Theorem 2.

Let be an abelian group, and let be a Lie subring of , with the commutator bracket. Suppose that there is which commutes with every element of and is bijective on . Then

is a UA-Lie ring.

In particular, it is enough that contain the identity endomorphism. The ring is not required to be closed under associative multiplication; its inclusion in expresses faithfulness of the action. The inverse of need not belong to .

There is a relevant earlier approach. Qi and Hou [3] proved additivity modulo a central error for Lie-multiplicative bijections from triangular associative algebras. Their theorem is reproduced in [2, Theorem 1.1], which also extends it. The target calculations in the proof of [2, Theorem 4.1, ] use addition and brackets and adapt to arbitrary Lie-ring targets. Consequently the associative unital special case of Theorem 2, including Arzhantsev’s particular matrix example, is obtainable by a central-extension argument from that earlier method. We give the argument in Section 5. The results here are formulated for arbitrary acting Lie subrings and yield Theorem 1 without an associative-closure assumption.

For comparison, Sosnov [4] recently classifies the UA property for and over fields and settles another question from [1]. Those results concern the full matrix Lie rings and their special linear subrings. The present argument concerns faithful affine actions and metabelian Lie algebras.

2. Recovering addition from a faithful action

Proof of Theorem 2.

Fix a bracket-preserving bijection . We identify and with their summands in . Since , we have . Set

The map is an additive endomorphism of the given additive group of . On we have

Centrality of in and bijectivity of on therefore give

Thus and are additive subgroups of . The subgroup has zero bracket, and is bijective, since it is conjugate as a set map to .

We first prove

Indeed, , so lies in . Write this difference as with . For every , we have

The last equality uses . Injectivity implies for all , and faithfulness gives . This proves (2).

For , the identity

and (2) give

Both arguments of belong to the additive subgroup . Its restriction to is injective, whence

It follows that is additive.

Now take . Since is an additive subgroup, write

For every , additivity on yields

Again injectivity and faithfulness give . Thus is additive. Equation (2) now proves additivity on .

Corollary 3.

Let be a Lie ring with an abelian ideal satisfying . Suppose that some satisfies and that is bijective. Then is a UA-Lie ring.

Proof.

Put . For , choose the unique with . Then . Also , so . The action of on is faithful because

The element is central in and acts bijectively on . Apply Theorem 2.

Remark 4.

Surjectivity of is used in (1): it identifies its images of the two source summands with an image and a kernel in the target additive group. Theorem 2 makes no assertion about arbitrary bracket-preserving injections. All subtractions in its proof occur in the already-given Lie ring ; no transported addition is assumed in advance.

3. Centerless metabelian Lie algebras

Lemma 5.

Let be commuting endomorphisms of a nonzero finite-dimensional vector space over an infinite field . If

then some -linear combination of the is invertible.

Proof.

Extend scalars to an algebraic closure of . The common kernel stays zero, since it is the kernel of the scalar extension of the map .

Successive decompositions into generalized eigenspaces give a direct-sum decomposition of into nonzero simultaneous generalized eigenspaces . Commutativity ensures invariance at each step. On we have

where the commute and are nilpotent. The tuple cannot be zero. Otherwise the commuting nilpotents would have a nonzero common kernel: successively take kernels, each time restricting the next nilpotent to the nonzero invariant subspace already obtained.

Choose so that for every one of the finitely many nonzero weight tuples. This is possible over the infinite field . The operator is invertible on every , since a sum of commuting nilpotents is nilpotent. For the latter fact, expand a sufficiently high power; each monomial contains a power at least the nilpotence exponent of one of the operators.

It follows that

is a nonzero polynomial. Over the infinite field it has a nonzero value at some -rational tuple, as required.

Proof of Theorem 1.

Let be centerless and metabelian, and set . If , then centerlessness forces , and the conclusion is immediate. Otherwise the endomorphisms of induced by the adjoint action of commute, because is abelian. Their common kernel is

Apply Lemma 5 to a basis of the space of acting endomorphisms. There is such that is invertible.

Set . Since , for every there is a unique with . Thus as vector spaces. For , the bracket belongs to and commutes with , so it is zero. Therefore is abelian. If an element of annihilates , it commutes with both and and hence is central in . The action of on is consequently faithful.

We have expressed as , with central in and invertible on . Theorem 2 applies.

Remark 6.

The infinite-field assumption enters only when choosing a rational point at which the determinant polynomial is nonzero. The affine criterion itself applies over finite fields. The preceding argument does not assert that every centerless metabelian Lie algebra over a finite field contains an element acting invertibly on its derived algebra.

4. Common centralizers and the matrix example

For a finite-dimensional centerless Lie algebra , denote by the least size of a tuple with zero common centralizer. Thus the C-condition of [1] is . We use this notation only to quantify the following separation.

Let be any field, , , and , where . Extend each element of by zero on , and put

Two elements of have zero product, so is an abelian Lie algebra. In block form,

Proposition 7.

Over every field, is a centerless UA-Lie ring and

Proof.

The UA assertion follows from Theorem 2. Take any elements, and write

An element centralizes all of them precisely when

The pairs identify with maps by . Thus their solution space has dimension

For this is positive.

Conversely, let be a basis of . The common centralizer of and the translation vectors is zero. Commuting with removes the translation component. An element annihilating every has by its -component and then . This proves the upper bound and also centerlessness.

For , formula (3) is exactly the algebra in [1, Example 2, Problem 4, p. 184]. Proposition 7 gives an affirmative answer to that question. The family also shows that there is no universal bound on among UA-Lie algebras.

The same phenomenon occurs when the acting Lie algebra is not closed under associative multiplication. Let , let be a nilpotent Jordan block of size three on , and extend it by zero on . On , set

The operators in commute, but . Theorem 2 applies to .

Moreover . The lower-bound calculation above is unchanged, since acts trivially on . For the upper bound, choose a Jordan basis with , and use and the translation vectors

An element annihilating these vectors has successively , , and . Thus these examples simultaneously lack associative closure of the acting algebra and, when , fail the two-centralizer condition.

5. Relation with triangular associative algebras

Suppose for this paragraph that is an associative unital -algebra acting faithfully on . The Lie algebra underlying is isomorphic to

where the last summand is central: the isomorphism sends a triangular block to .

Given a bracket-preserving bijection , extend it by the identity on the central copy of . The target is the Lie ring . The argument for triangular algebras in [3], reproduced and extended in [2], yields additivity modulo the center after adapting its bracket calculations to this target. Since is centerless, so is . Comparing the extra -coordinates eliminates the remaining central error. This recovers the associative unital affine case.

Accordingly, the particular matrix answer should be understood together with this prior proof-level route. The proof in Section 2 is a direct structural argument requiring only a faithful Lie action and a central invertible acting element. It permits acting Lie subrings without associative closure, arbitrary abelian translation groups, and the metabelian consequence of Section 3. No inheritance of the UA property by arbitrary Lie subalgebras is used or asserted.

References

[1] I. Arzhantsev, Uniqueness of addition in Lie algebras revisited, Math. Commun. 30 (2025), 179–189. doi:10.64785/mc.30.2.3.

[2] B. L. M. Ferreira and H. Guzzo Jr., Lie -multiplicative mappings on triangular -matrix rings, Rev. Un. Mat. Argentina 60 (2019), no. 1, 9–20. doi:10.33044/revuma.v60n1a02.

[3] X. Qi and J. Hou, Additivity of Lie multiplicative maps on triangular algebras, Linear Multilinear Algebra 59 (2011), no. 4, 391–397. doi:10.1080/03081080903582094.

[4] G. Sosnov, Uniqueness of addition in Lie rings and , preprint (2026). arXiv:2606.08201v1.