Restricted post-Lie structures induced by tridendriform algebras in prime characteristic

A tridendriform algebra $(T,\succ,\prec,\cdot)$ over a field of characteristic $p>0$ carries a post-Lie algebra structure with bracket $[a,b]_T=a\cdot b-b\cdot a$ and product $a\triangleright_T b=a\succ b-b\prec a$ (Bai–Guo–Ni), whose sub-adjacent Lie bracket is the commutator of the associative product $\star=\succ+\prec+\cdot$ ; the $p$ -fold $\cdot$ -power makes $(T,[-,-]_T)$ a restricted Lie algebra in Jacobson's sense. Li and Ou (arXiv:2606.19244, Remark 4.16) expect that $(T,[-,-]_T,(-)^{\cdot p},\triangleright_T)$ is always a restricted post-Lie algebra in their sense — a trivially restricted post-Lie algebra in the terminology of Ehret–Gilliers, meaning the sub-adjacent $p$ -map is canonically determined by a Jacobson-type formula. Ehret–Gilliers (arXiv:2504.14540, Proposition 3.4.1 on quasi-shuffle algebras) had already proved the statement under the weaker, general definition of restricted post-Lie algebra, with the $\star$ -power as the sub-adjacent $p$ -map, including the restricted-derivation axiom. The genuinely open core is whether the canonical Jacobson-formula $p$ -map coincides, up to elements annihilated by every $\triangleright_T$ -action, with the $\star$ -power in each tridendriform algebra — equivalently, whether the second defining axiom holds for the canonical map. The problem asks for a proof valid for all tridendriform algebras and all primes, or an explicit tridendriform algebra in some characteristic $p$ where the canonical map fails the axiom.

Background

Over a field $\mathbf{k}$ of prime characteristic $p$ , a restricted Lie algebra (introduced by N. Jacobson in 1937) is a Lie algebra $\mathfrak{g}$ together with a map $x\mapsto x^{[p]}$ , called the $p$ -map, which abstracts the Frobenius power: it is $p$ -semilinear in $x$ , satisfies $\operatorname{ad}(x^{[p]})=(\operatorname{ad}x)^p$ , and obeys Jacobson's expansion formula for $(x+y)^{[p]}$ . The basic example is the commutator Lie algebra $A^-$ of any associative $\mathbf{k}$ -algebra $A$ , with $x^{[p]}=x^p$ the $p$ -fold associative product. Restricted analogues of related algebra types have been developed similarly: Dzhumadil'daev initiated restricted pre-Lie algebras in characteristic $p$ (a pre-Lie algebra, also called left-symmetric, Vinberg, or Gerstenhaber algebra, is a vector space with a product $*$ satisfying $(x*y)*z-x*(y*z)=(y*x)*z-y*(x*z)$ , so that the commutator is a Lie bracket), and Dokas proved that dendriform algebras (the two-operation splitting structure) in characteristic $p$ carry restricted pre-Lie structures.

A post-Lie algebra (introduced by B. Vallette via the Koszul duality of operads; the notion also arises in geometric numerical integration on manifolds and in the theory of regularity structures for stochastic PDEs) is a Lie algebra $(\mathfrak{g},[-,-])$ equipped with a bilinear product $\triangleright$ such that $x\triangleright[y,z]=[x\triangleright y,z]+[y,x\triangleright z]$ and $([x,y]+x\triangleright y-y\triangleright x)\triangleright z=x\triangleright(y\triangleright z)-y\triangleright(x\triangleright z)$ . The sub-adjacent bracket $\llbracket x,y\rrbracket:=[x,y]+x\triangleright y-y\triangleright x$ is then again a Lie bracket; with an abelian $[-,-]$ one recovers the pre-Lie notion. Ehret and Gilliers recently introduced restricted post-Lie algebras in two strengths. In the general version, a post-Lie algebra over a restricted Lie algebra $(\mathfrak{g},[-,-],(-)^{[p]})$ carries a second, independently specified $p$ -map $(-)^{\llbracket p\rrbracket}$ making the sub-adjacent Lie algebra restricted and compatible with $\triangleright$ . In the stronger, trivially restricted version — which is Li–Ou's Definition 3.1 and the one their whole framework of restricted Rota–Baxter Lie algebras is built on — the sub-adjacent $p$ -map is not free data but is canonically determined by the Jacobson-type formula

$x^{\llbracket p\rrbracket}=x^{[p]}+\sum_{r=1}^{p-1}\frac{1}{r}\sum_{\alpha\vDash p,\ \ell(\alpha)=r}C_\alpha\,[x^{\triangleright\alpha}],$

where $\alpha=(\alpha_1,\dots,\alpha_r)$ runs over compositions of $p$ with $r$ parts, $x^{\triangleright k}:=x\triangleright(x\triangleright\cdots(x\triangleright x)\cdots)$ is the $k$ -fold left-nested $\triangleright$ -self-product of $x$ , $[x^{\triangleright\alpha}]:=[x^{\triangleright\alpha_1},[x^{\triangleright\alpha_2},[\cdots[x^{\triangleright\alpha_{r-1}},x^{\triangleright\alpha_r}]\cdots]]]$ , $C_\alpha=\prod_{k=1}^{r}\binom{\alpha_1+\cdots+\alpha_k-1}{\alpha_k-1}$ counts the set partitions of $\{1,\dots,p\}$ into blocks ordered by increasing maxima, and $1/r$ is the inverse of $r$ modulo $p$ . A trivially restricted post-Lie algebra must satisfy two compatibility axioms: first, $y\triangleright x^{[p]}=[x,[x,\cdots[x,y\triangleright x]\cdots]]$ with $p-1$ copies of $x$ (each left $\triangleright$ -multiplication is a restricted derivation of the restricted Lie algebra); second, $x^{\llbracket p\rrbracket}\triangleright y=x\triangleright(x\triangleright\cdots(x\triangleright y)\cdots)$ with $p$ copies of $x$ .

A tridendriform algebra $(T,\succ,\prec,\cdot)$ (Loday–Ronco, from the study of polytopes and Koszul duality; also called a dendriform trialgebra or quasi-shuffle algebra) is a vector space with three binary operations satisfying the seven identities $(x\star y)\succ z=x\succ(y\succ z)$ , $(x\succ y)\prec z=x\succ(y\prec z)$ , $(x\prec y)\prec z=x\prec(y\star z)$ , $x\succ(y\cdot z)=(x\succ y)\cdot z$ , $x\cdot(y\succ z)=(x\prec y)\cdot z$ , $(x\cdot y)\prec z=x\cdot(y\prec z)$ and $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ , where $x\star y:=x\succ y+x\prec y+x\cdot y$ ; in particular $\star$ and $\cdot$ are associative. The paradigmatic source is a Rota–Baxter algebra: an associative algebra $A$ with a linear operator $P$ of weight $\lambda\in\mathbf{k}$ satisfying $P(x)P(y)=P(P(x)y+xP(y)+\lambda xy)$ ; Ebrahimi-Fard showed that such a pair yields a tridendriform algebra with $x\succ_P y=P(x)y$ , $x\prec_P y=xP(y)$ , $x\cdot_\lambda y=\lambda xy$ . Bai–Guo–Ni proved that every tridendriform algebra induces a post-Lie algebra via $[a,b]_T:=a\cdot b-b\cdot a$ and $a\triangleright_T b:=a\succ b-b\prec a$ , and a direct two-line computation shows that the sub-adjacent bracket $\llbracket a,b\rrbracket$ equals the $\star$ -commutator $a\star b-b\star a$ . Consequently $(T,[-,-]_T,(-)^{\cdot p})$ and $(T,\llbracket-,-\rrbracket,(-)^{\star p})$ are both automatically restricted Lie algebras, so the entire content of the question below lies in the compatibility of $\triangleright_T$ with these $p$ -maps.

Two results delimit the current state. Li–Ou (Corollary 4.15, via their theory of restricted Rota–Baxter Lie algebras) proved that a Rota–Baxter algebra of weight $\lambda$ in characteristic $p$ induces a trivially restricted post-Lie algebra $(A,\lambda[-,-],\lambda^{p-1}(-)^p,\triangleright_P)$ , where the $p$ -map $\lambda^{p-1}(-)^p$ is exactly the $p$ -fold $\cdot_\lambda$ -power — so the expectation holds for every tridendriform algebra arising from a Rota–Baxter operator. Ehret–Gilliers (Proposition 3.4.1) proved that every tridendriform algebra in characteristic $p$ yields a restricted post-Lie algebra in the general sense, with $p$ -map the $\cdot$ -power and sub-adjacent $p$ -map the $\star$ -power; in particular the restricted-derivation axiom always holds, and the second axiom holds with $x^{\star p}$ in place of the canonical map. No source establishes the second axiom for the canonical Jacobson-formula $p$ -map, which is the constraint distinguishing the trivially restricted notion that Li–Ou's Remark 4.16 expectation is about.

Problem Statement

Let $\mathbf{k}$ be a field of characteristic $p>0$ and let $(T,\succ,\prec,\cdot)$ be a tridendriform algebra over $\mathbf{k}$ (the seven defining identities are recalled in the background; in particular the operations $\cdot$ and $\star:=\succ+\prec+\cdot$ are associative). Set $[a,b]_T:=a\cdot b-b\cdot a$ , $a\triangleright_T b:=a\succ b-b\prec a$ , and $x^{[p]}:=x\cdot x\cdots x$ (the $p$ -fold product under the associative operation $\cdot$ ). Then $(T,[-,-]_T,\triangleright_T)$ is a post-Lie algebra and $(T,[-,-]_T,(-)^{[p]})$ is a restricted Lie algebra. Determine whether $(T,[-,-]_T,(-)^{[p]},\triangleright_T)$ is a restricted post-Lie algebra in the sense of Li–Ou's Definition 3.1 (a trivially restricted post-Lie algebra in the sense of Ehret–Gilliers) for every tridendriform algebra $T$ : that is, whether for all $x,y\in T$ the following two identities hold,

$y\triangleright_T x^{[p]}=[x,[x,\cdots[x,y\triangleright_T x]\cdots]]\quad(p-1\text{ copies of }x,\ \text{all brackets }[-,-]_T),$

$x^{\llbracket p\rrbracket}\triangleright_T y=x\triangleright_T(x\triangleright_T\cdots(x\triangleright_T y)\cdots)\quad(p\text{ copies of }x),$

where the canonical sub-adjacent $p$ -map $x^{\llbracket p\rrbracket}$ is defined by the Jacobson-type formula

$x^{\llbracket p\rrbracket}=x^{[p]}+\sum_{r=1}^{p-1}\frac{1}{r}\sum_{\alpha\vDash p,\ \ell(\alpha)=r}C_\alpha\,[x^{\triangleright_T\alpha}],$

in which: $\alpha=(\alpha_1,\dots,\alpha_r)$ runs over the compositions of $p$ with $r$ parts; $1/r$ denotes the inverse of the integer $r$ modulo $p$ ; $x^{\triangleright_T k}:=x\triangleright_T(x\triangleright_T\cdots(x\triangleright_T x)\cdots)$ is the $k$ -fold left-nested $\triangleright_T$ -self-product of $x$ ( $k\ge 1$ ); $[x^{\triangleright_T\alpha}]:=[x^{\triangleright_T\alpha_1},[x^{\triangleright_T\alpha_2},[\cdots[x^{\triangleright_T\alpha_{r-1}},x^{\triangleright_T\alpha_r}]\cdots]]]$ with all brackets $[-,-]_T$ ; and $C_\alpha:=\prod_{k=1}^{r}\binom{\alpha_1+\cdots+\alpha_k-1}{\alpha_k-1}$ , the number of set partitions of $\{1,\dots,p\}$ into $r$ blocks with block sizes $\alpha_1,\dots,\alpha_r$ ordered by increasing maxima. An affirmative answer is a proof valid for all tridendriform algebras over all fields of prime characteristic; a negative answer is an explicit tridendriform algebra over a field of some characteristic $p$ , together with elements $x,y$ , for which one of the two displayed identities fails. Two delimiting facts: the first displayed identity for this situation already follows from Ehret–Gilliers' Proposition 3.4.1 (arXiv:2504.14540) after translating their opposite sign convention $a\blacktriangleright b=b\prec a-a\succ b=-a\triangleright_T b$ , and the same proposition establishes the second identity with $x^{\llbracket p\rrbracket}$ replaced by the $\star$ -power $x^{\star p}$ (the $p$ -fold product under $\star$ ); hence the load-bearing open part is the second identity for the canonical $p$ -map, equivalently whether $(x^{\llbracket p\rrbracket}-x^{\star p})\triangleright_T y=0$ for all $x,y\in T$ (in particular whether $x^{\llbracket p\rrbracket}=x^{\star p}$ holds identically in every tridendriform algebra). Li–Ou's Corollary 4.15 (arXiv:2606.19244) gives the affirmative answer for all tridendriform algebras induced by Rota–Baxter operators of arbitrary weight, so any counterexample must come from outside the Rota–Baxter-induced class.

References

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