Rota-Baxter indices of upper-triangular and semisimple matrix algebras and the derived-algebra nilpotency bound for $M_n(F)$

The paper "Spectrum of Rota-Baxter operators" (Gubarev, 2020; published Int. J. Algebra Comput. 2025) proved that the spectrum of every Rota-Baxter operator of weight $\lambda$ on a finite-dimensional unital algebra lies in $\{0,-\lambda\}$ and, building on this, introduced the Rota-Baxter $\lambda$ -index $\operatorname{rb}_\lambda(A)$ — the least $n$ such that every weight- $\lambda$ operator $R$ on $A$ satisfies $R^k(R+\lambda\,\mathrm{id})^{n-k}=0$ for some $k\le n$ — computing $\operatorname{rb}_\lambda(M_n(F))=2n-1$ . The paper explicitly poses three unresolved targets that remain open: (1) $\operatorname{rb}_0(t_n(F))=n$ for the algebra $t_n(F)$ of upper-triangular $n\times n$ matrices over a characteristic-zero field; (2) for every semisimple finite-dimensional associative algebra $A=\bigoplus_{i=1}^k M_{n_i}(F)$ , the formulas $\operatorname{rb}_0(A)=2\max\{n_i\}-1$ and $\operatorname{rb}_\lambda(A)=\sum_i n_i+\max\{n_i\}-1$ for $\lambda\ne 0$ ; and (3) the minimal integer $m$ depending only on $n$ such that the $m$ -th derived Rota-Baxter algebra of $M_n(F)$ under any weight-zero operator has trivial product. Later work has confirmed item (1) at $n=3$ and verified degenerate cases of item (2), but the general statements and the exact value of $m(n)$ are unsettled.

Background

A Rota-Baxter operator (RB-operator) of weight $\lambda$ on an algebra $A$ over a field $F$ is a linear map $R:A\to A$ satisfying $R(x)R(y)=R(R(x)y+xR(y)+\lambda xy)$ for all $x,y\in A$ — an abstract form of the integration-by-parts identity, with deep ties to the classical and associative Yang–Baxter equations (compatibility conditions on a tensor $r\in A\otimes A$ originating in integrable-systems theory; over matrix algebras, weight-zero RB-operators correspond to solutions of the associative one), to dendriform and pre-Lie algebras (algebras whose product splits into two or three partially associative operations, axiomatizing how an integral turns a product into a sum of products), and to double Lie algebras (Lie algebras carrying a second compatible coalgebra-type structure, the infinitesimal counterpart of Lie bialgebras). These ties motivate classifying RB-operators on matrix algebras and measuring their complexity.

Two structural results anchor the source paper. First, every RB-operator of weight $\lambda$ on a finite-dimensional unital algebra (any characteristic; algebraic algebras in characteristic zero) has spectrum contained in $\{0,-\lambda\}$ , so $R^k(R+\lambda\,\mathrm{id})^l=0$ for some $k,l$ . Second, every weight-zero RB-operator on a unital associative algebraic algebra of characteristic zero is nilpotent (with $R(1)$ nilpotent). These facts suggest a numerical invariant: the Rota-Baxter $\lambda$ -index $\operatorname{rb}_\lambda(A)=\min\{n\mid \text{for all }R\in\operatorname{RB}_\lambda(A)\text{ there is }k\le n\text{ with }R^k(R+\lambda\,\mathrm{id})^{n-k}=0\}$ (and $\operatorname{rb}_\lambda(A)=\infty$ if no such $n$ exists). For weight zero this is $\min\{n\mid R^n=0\text{ for every weight-zero }R\text{ on }A\}$ , the RB-index $\operatorname{rb}(A)$ introduced earlier in Gubarev's "Rota-Baxter operators on unital algebras" (Mosc. Math. J. 2021).

The known computations are: $\operatorname{rb}_\lambda(M_n(F))=2n-1$ for all $\lambda$ over a characteristic-zero field (weight zero from the Mosc. Math. J. paper, nonzero weight in the source paper); $\operatorname{rb}_\lambda(t_n(F))=2n-1$ for $\lambda\ne 0$ , where $t_n(F)$ is the subalgebra of upper-triangular $n\times n$ matrices (the source paper's phrase is "not strictly upper-triangular", i.e. the diagonal is included); $\operatorname{rb}_0(F^k)=1$ and $\operatorname{rb}_\lambda(F^k)=k$ for $\lambda\ne0$ for a direct sum of $k$ copies of the field; and $\operatorname{rb}(J)$ values for simple Jordan algebras $J$ of a bilinear form (a Jordan algebra is a commutative algebra satisfying $(x\circ x)\circ(x\circ y)=x\circ(x\circ(x\circ y))$ , e.g. the symmetrized product of an associative algebra). For an algebra of the form $A=\bigoplus_{i=1}^k M_{n_i}(F)$ over $F$ (by the Wedderburn–Artin theorem, every finite-dimensional semisimple associative algebra over an algebraically closed field takes exactly this form; the source paper states its conjecture for such direct sums over any characteristic-zero field) at weight zero, the sandwich $\max_i\{\operatorname{rb}(A_i)\}=2\max\{n_i\}-1\le\operatorname{rb}(A)\le 1+\max_i\{\operatorname{rb}(A_i)\}$ is proved, and the equality $\operatorname{rb}(A)=\max_i\{\operatorname{rb}(A_i)\}$ (Conjecture 5.24 of the Mosc. Math. J. paper) is verified only for $A=M_2(F)\oplus M_2(F)$ and $A=F\oplus\cdots\oplus F$ . At nonzero weight the source paper constructs a triangular-splitting operator on $\bigoplus_i M_{n_i}(F)$ whose minimal polynomial is $x^{\sum_i n_i}(x+\lambda)^{\max\{n_i\}-1}$ , attaining $\sum_i n_i+\max\{n_i\}-1$ as a lower bound for $\operatorname{rb}_\lambda(A)$ ; the available general upper bounds (e.g. $\operatorname{rb}_\lambda(A)\le 2\cdot\operatorname{capacity}(A)=2\sum_i n_i$ over an algebraically closed field, where capacity is the maximal number of pairwise orthogonal idempotents in $A$ ) are weaker.

The source paper also studies derived Rota-Baxter algebras: given an RB-operator $R$ of weight $\lambda$ on $A$ , define algebras $A_i$ on the same vector space by $x\circ_0 y=xy$ and $x\circ_{i+1}y=R(x)\circ_i y+x\circ_i R(y)+\lambda\,x\circ_i y$ ; $R$ is again an RB-operator on each $A_i$ . For weight zero, $x\circ_k y=\sum_{i=0}^k\binom{k}{i}R^i(x)R^{k-i}(y)$ , so if $R^N=0$ then all products of $A_k$ vanish for $k\ge 2N-1$ ; in particular $A_\infty$ (the eventual limiting algebra) has trivial product. Since $\operatorname{rb}_0(M_n(F))=2n-1$ , a uniform bound $m\le 4n-3$ exists for $M_n(F)$ , but its sharpness is unknown.

Problem Statement

Over an arbitrary field $F$ of characteristic zero, determine the following three values posed as open problems in Gubarev's "Spectrum of Rota-Baxter operators":

(1) The exact weight-zero Rota-Baxter index of the upper-triangular matrix algebra: prove or disprove that $\operatorname{rb}_0(t_n(F))=n$ for every $n\ge 1$ , where $t_n(F)$ is the algebra of $n\times n$ upper-triangular matrices (including the diagonal) and $\operatorname{rb}_0(t_n(F))=\min\{m\mid R^m=0\text{ for every RB-operator }R\text{ of weight zero on }t_n(F)\}$ .

(2) The exact indices of every finite direct sum $A=\bigoplus_{i=1}^k M_{n_i}(F)$ of full matrix algebras over $F$ (the form in which the source paper states its conjecture for semisimple finite-dimensional associative algebras): prove or disprove that $\operatorname{rb}_0(A)=2\max_{i}\{n_i\}-1$ , and that $\operatorname{rb}_\lambda(A)=\sum_{i=1}^k n_i+\max_i\{n_i\}-1$ for every nonzero $\lambda\in F$ , where $\operatorname{rb}_\lambda(A)=\min\{n\mid\text{for every RB-operator }R\text{ of weight }\lambda\text{ on }A\text{ there is }k\le n\text{ with }R^k(R+\lambda\,\mathrm{id})^{n-k}=0\}$ .

(3) The exact stabilization bound for the derived Rota-Baxter algebra tower of the full matrix algebra at weight zero: find the minimal integer $m=m(n)$ , depending only on $n$ , such that for every RB-operator $R$ of weight zero on $M_n(F)$ the $m$ -th derived algebra $A_m$ (products defined recursively by $x\circ_0 y=xy$ , $x\circ_{i+1}y=R(x)\circ_i y+x\circ_i R(y)+\lambda\,x\circ_i y$ with $\lambda=0$ ) has trivial product.

The three items share one objective — completing the computation of the RB-index invariants for the matrix-algebra families adjacent to $M_n(F)$ , whose index $\operatorname{rb}_\lambda(M_n(F))=2n-1$ is already known.

References

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