arxiv: 2604.17664 · v1 · submitted 2026-04-19 · 🧮 math.RA

Recognition: unknown

The Jordan multiplication semigroup of matrix algebras is the full endomorphism semigroup

Ilja Gogi\'c, Mateo Toma\v{s}evi\'c, Matija Kazalicki

Pith reviewed 2026-05-10 04:37 UTC · model grok-4.3

classification 🧮 math.RA

keywords Jordan algebramatrix algebramultiplication operatorsendomorphism semigrouptransvectionsspecial linear group

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The pith

Every linear endomorphism of the space of n by n matrices equals a product of Jordan multiplication operators.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the semigroup generated by all operators of the form X maps to A circle X, where circle is the symmetrized matrix product, equals the full set of linear maps on the vector space of matrices. A sympathetic reader cares because this shows the Jordan structure alone can produce every possible linear transformation, rather than only a proper sub-semigroup. The argument proceeds by first embedding the special linear group into the unit group of the semigroup via explicit transvections, then using a rank-n squared minus one element together with determinant surjectivity to reach all endomorphisms. For finite fields the determinant step relies on Jacobi-sum estimates to guarantee every nonzero scalar appears.

Core claim

For the Jordan algebra A equal to M_n(K) with the symmetrized product, the semigroup JMS(A) generated by all multiplication operators L_A equals the full endomorphism algebra End_K(A_v). Equivalently, every K-linear map on the underlying vector space arises as a finite composition of these multiplication operators.

What carries the argument

The Jordan multiplication semigroup JMS(A), the semigroup generated inside End_K(A_v) by the operators L_A(X) = (A X + X A)/2.

If this is right

The special linear group SL(n squared, K) sits inside the unit group of JMS(A).
Every element of End_K(A_v) is reachable once one singular element of rank n squared minus 1 is known to lie in JMS(A).
Determinant surjectivity of the unit group of JMS(A) is the remaining step that completes the proof.
In the finite-field setting, Jacobi-sum estimates suffice to produce every nonzero determinant value.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same generation statement may hold for other simple Jordan algebras whose underlying vector spaces admit enough transvections.
One could ask whether the result extends to infinite-dimensional or topological versions of matrix algebras.
The construction supplies an explicit set of generators for the full matrix monoid over any field of characteristic not 2.

Load-bearing premise

The base field has characteristic different from 2, which is required both to define the symmetrized product and to construct the elementary transvections used in the proof.

What would settle it

An explicit linear map on the n by n matrices that cannot be expressed as any finite product of the operators L_A would falsify the equality.

read the original abstract

Let $\mathbb{K}$ be a field of characteristic different from $2$, and let $M_n(\mathbb{K})$ be the algebra of all $n\times n$ matrices over $\mathbb{K}$. We consider the corresponding special Jordan algebra $\mathcal{A}:=M_n(\mathbb{K})^+$ with symmetrized product $A\circ B:=(AB+BA)/2$, and write $\mathcal{A}_{\mathrm v}:=M_n(\mathbb{K})$ for the underlying $\mathbb{K}$-vector space of $\mathcal{A}$. For $A\in\mathcal{A}$, let $\mathrm{L}_A(X):=A\circ X$ be the multiplication operator. We consider the Jordan multiplication semigroup generated by all multiplication operators, \[ \mathrm{JMS}(\mathcal{A}):=\langle \mathrm{L}_A:A\in\mathcal{A}\rangle\subseteq \mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v}). \] We prove that $\mathrm{JMS}(\mathcal{A})=\mathrm{End}_{\mathbb{K}}(\mathcal{A}_{\mathrm v})$. Equivalently, every $\mathbb{K}$-linear endomorphism of $\mathcal{A}_{\mathrm v}$ is a composition of multiplication operators. The proof is primarily linear-algebraic. The main step is to show that $\mathrm{SL}(\mathcal{A}_{\mathrm v})\subseteq \mathrm{JMS}(\mathcal{A})$ by constructing elementary transvections inside the semigroup. We then prove determinant surjectivity on the unit group of $\mathrm{JMS}(\mathcal{A})$ and combine it with the existence of a singular element of rank $n^2-1$ to obtain the full endomorphism semigroup. In the finite-field case, the determinant-surjectivity step is established via Jacobi-sum estimates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves JMS of the matrix Jordan algebra equals full End via explicit transvections plus det surjectivity, with Jacobi sums as the finite-field hinge.

read the letter

The result is that for a field K of characteristic not 2, the semigroup generated by all left multiplication operators L_A where A is in the special Jordan algebra M_n(K)^+ is equal to the full set of K-linear endomorphisms of the underlying vector space M_n(K). Equivalently, every linear map is a composition of these symmetrized multiplications. They show this by constructing transvections inside the semigroup to get SL, proving det surjectivity on the group of units, and then using a rank n²-1 element to reach the whole semigroup. The transvection construction is the clean new piece and works via standard linear algebra. The determinant surjectivity is elementary except in the finite case, where Jacobi-sum estimates are used. That is the least transparent step, and it would be good to see if there's a more elementary way or if the estimates apply uniformly. If those hold, the proof is fine and free of circularity. This is a niche but solid result in Jordan algebra theory. Readers working on multiplication semigroups or explicit generators for endomorphism rings will get value from the explicit constructions. It deserves peer review as the main argument is clear and the novelty is in the generation result.

Referee Report

2 major / 2 minor

Summary. The paper proves that for a field K of char ≠ 2, the Jordan multiplication semigroup JMS(A) generated by the operators L_A (A in the special Jordan algebra A = M_n(K)^+) equals the full End_K(A_v), where A_v is the underlying vector space. Equivalently, every K-linear endomorphism of A_v is a composition of Jordan multiplication operators. The argument first embeds SL(A_v) into JMS(A) by exhibiting elementary transvections as products of L_A's, then establishes that the unit group of JMS(A) has surjective determinant onto K^*, and finally adjoins a single rank-(n²-1) element to reach all of End_K(A_v). For finite K the determinant-surjectivity step uses Jacobi-sum estimates.

Significance. If the result holds, it gives a semigroup-theoretic generation result showing that Jordan multiplications alone suffice to produce every linear map on the matrix algebra viewed as a vector space. This strengthens the understanding of operator semigroups arising from Jordan structures and provides an explicit linear-algebraic route from multiplication operators to the full endomorphism monoid. The explicit transvection construction and the reduction to a single high-rank element are concrete strengths; the finite-field case adds a number-theoretic ingredient via Jacobi sums.

major comments (2)

[determinant-surjectivity step (finite K)] Determinant-surjectivity paragraph (finite-field case): the Jacobi-sum estimates invoked to prove that the determinant map on the unit group of JMS(A) is surjective onto K^* are load-bearing for the finite-field case, since without full surjectivity the generated group may be a proper subgroup of GL(A_v) and the subsequent adjunction of the rank-(n²-1) element would not reach all of End. The manuscript should supply the precise statement of the estimates used, the range of n for which they apply, and a self-contained verification or reference that makes the counting argument transparent.
[SL(A_v) embedding via transvections] Transvection construction section: while the outline (using char ≠ 2 to produce elementary transvections inside the semigroup generated by L_A) is linear-algebraic and appears direct, the explicit formulas for the transvections as products of multiplication operators should be written out for a general matrix unit basis so that the reader can confirm they lie in JMS(A) without additional assumptions.

minor comments (2)

[Introduction / notation] Notation: the distinction between A (the Jordan algebra) and A_v (its underlying vector space) is used throughout; a single sentence clarifying that End_K(A_v) is identified with the full matrix algebra over the n²-dimensional space would help readers unfamiliar with the Jordan setting.
[Abstract and finite-field subsection] The abstract states the result for arbitrary n; the finite-field Jacobi-sum argument should explicitly note any lower bound on n required for the estimates to hold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. The points raised identify opportunities to improve clarity in two key technical sections. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [determinant-surjectivity step (finite K)] Determinant-surjectivity paragraph (finite-field case): the Jacobi-sum estimates invoked to prove that the determinant map on the unit group of JMS(A) is surjective onto K^* are load-bearing for the finite-field case, since without full surjectivity the generated group may be a proper subgroup of GL(A_v) and the subsequent adjunction of the rank-(n²-1) element would not reach all of End. The manuscript should supply the precise statement of the estimates used, the range of n for which they apply, and a self-contained verification or reference that makes the counting argument transparent.

Authors: We agree that the finite-field determinant-surjectivity argument is load-bearing and that the current reference to Jacobi-sum estimates requires greater transparency. In the revision we will insert the precise statement of the estimates (including the relevant character-sum bounds), specify the range of n to which they apply (all n ≥ 2, with the n = 1 case handled separately by direct verification), and supply either a short self-contained counting argument or an explicit citation to the number-theoretic source. This will render the surjectivity step fully verifiable without external lookup. revision: yes
Referee: [SL(A_v) embedding via transvections] Transvection construction section: while the outline (using char ≠ 2 to produce elementary transvections inside the semigroup generated by L_A) is linear-algebraic and appears direct, the explicit formulas for the transvections as products of multiplication operators should be written out for a general matrix unit basis so that the reader can confirm they lie in JMS(A) without additional assumptions.

Authors: We accept the suggestion to make the transvection construction fully explicit. The revised manuscript will contain the explicit products of Jordan multiplication operators L_A that realize each elementary transvection with respect to the standard matrix-unit basis {E_{ij}}. The derivations will be written step-by-step, using only the definition of the symmetrized product and the hypothesis char K ≠ 2, so that membership in JMS(A) is immediate for any reader. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit constructions and external estimates

full rationale

The derivation proceeds by direct linear-algebraic construction of elementary transvections as finite products of multiplication operators L_A inside the Jordan semigroup, using only the symmetrized product and char ≠ 2. This embeds SL(A_v) into JMS(A). Determinant surjectivity on units is obtained for finite fields via Jacobi-sum estimates (standard character-sum techniques from finite-field theory, not derived inside the paper). A single rank-(n²-1) singular element is then adjoined to reach all of End. No equation or step is defined in terms of the target conclusion, no parameter is fitted to a subset and renamed as prediction, and no load-bearing premise reduces to a self-citation. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the definition of the Jordan product (which requires char K ≠ 2) and on standard facts from linear algebra and finite-field character sums; no free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption The base field K has characteristic different from 2.
Required for the symmetrized product A ∘ B = (AB + BA)/2 to behave as a Jordan algebra and for the transvection constructions to work.

pith-pipeline@v0.9.0 · 5627 in / 1258 out tokens · 26145 ms · 2026-05-10T04:37:17.965576+00:00 · methodology

discussion (0)

Reference graph

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