Gao, Guo and Zhang introduced matching Rota–Baxter algebras and their Lie, dendriform, pre-Lie and PostLie counterparts, and proved that in the associative case solutions of the polarized associative Yang–Baxter equation (PAYBE) satisfying a tensor commutation condition induce matching Rota–Baxter operators. For the Lie level they explicitly deferred the analogous question, writing that there should be a close relationship between the weight-zero matching Rota–Baxter Lie equation and a polarized classical Yang–Baxter equation. The open problem is to formulate that Lie-level tensor equation and to establish the operator–tensor correspondence: tensor solutions should induce matching Rota–Baxter Lie operators of weight zero, with tensor conditions playing the role that PAYBE together with r12s23=s12r23r_{12}s_{23}=s_{12}r_{23} plays in the associative case.

Background

A Rota–Baxter operator of weight λ\lambda on an associative algebra RR is a linear map P:R→RP:R\to R satisfying P(x)P(y)=P(P(x)y)+P(xP(y))+λP(xy)P(x)P(y)=P(P(x)y)+P(xP(y))+\lambda P(xy); on a Lie algebra g\mathfrak{g} the weight-λ\lambda identity, in the normalization used by the source paper, is [P(x),P(y)]=P([P(x),y])+P([x,P(y)])+λP([x,y])[P(x),P(y)]=P([P(x),y])+P([x,P(y)])+\lambda P([x,y]). The Lie identity with weight zero is the operator form of the classical Yang–Baxter equation (CYBE), the quadratic tensor equation [[r,r]]=0[[r,r]]=0 for r∈g⊗gr\in\mathfrak{g}\otimes\mathfrak{g}, where [[⋅,⋅]][[\cdot,\cdot]] is the Schouten bracket, the natural extension of the Lie bracket to tensor powers acting componentwise on the tensor legs, and tensor legs are indexed by the standard notation r12=∑iui⊗vi⊗1r_{12}=\sum_i u_i\otimes v_i\otimes 1, r13=∑iui⊗1⊗vir_{13}=\sum_i u_i\otimes 1\otimes v_i, r23=∑i1⊗ui⊗vir_{23}=\sum_i 1\otimes u_i\otimes v_i acting in the three tensor factors. Concretely, an O-operator (a weight-zero Rota–Baxter operator twisted by a representation on a module VV) corresponds to a skew-symmetric solution of the CYBE in the semidirect product Lie algebra g⋉V\mathfrak{g}\ltimes V, a dictionary developed by Aguiar and by Bai–Guo–Ni building on Golubchik and Sokolov's observation that weight-zero Rota–Baxter Lie operators lead to pre-Lie algebras (algebras whose product x⋅yx\cdot y satisfies (x⋅y)⋅z−x⋅(y⋅z)=(y⋅x)⋅z−y⋅(x⋅z)(x\cdot y)\cdot z-x\cdot(y\cdot z)=(y\cdot x)\cdot z-y\cdot(x\cdot z)).

In associative language, Aguiar polarized the associative Yang–Baxter equation r13r12−r12r23+r23r13=0r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=0 to the pair form {r,s}:=r13s12−r12s23+r23s13\{r,s\}:=r_{13}s_{12}-r_{12}s_{23}+r_{23}s_{13}; the equation {r,s}=0\{r,s\}=0 is called the polarized associative Yang–Baxter equation (PAYBE). Gao–Guo–Zhang, motivated by the multiple pre-Lie algebras that Bruned, Hairer and Zambotti isolated in the algebraic renormalisation of regularity structures for stochastic PDEs, introduced matching structures with a parameter set Ω\Omega: a matching Rota–Baxter algebra carries a family {Pω}ω∈Ω\{P_\omega\}_{\omega\in\Omega} such that every pair obeys a joint identity (so the family is closed under linear combinations, unlike single Rota–Baxter operators), a matching dendriform algebra splits the product into a family of operations ≺ω,≻ω\prec_\omega,\succ_\omega, a matching PostLie algebra carries a family of Lie brackets [⋅,⋅]ω[\cdot,\cdot]_\omega together with a family of products ∘ω\circ_\omega satisfying joint compatibility identities (a PostLie algebra, the one-family case, pairs a single Lie bracket with a single compatible product and underlies Lie–Butcher series in geometric numerical integration), and a matching Rota–Baxter Lie algebra of weights λΩ\lambda_\Omega is a Lie algebra with operators PωP_\omega satisfying [Pα(x),Pβ(y)]=Pα([x,Pβ(y)])+Pβ([Pα(x),y])+λβPα([x,y])[P_\alpha(x),P_\beta(y)]=P_\alpha([x,P_\beta(y)])+P_\beta([P_\alpha(x),y])+\lambda_\beta P_\alpha([x,y]) for all x,y∈gx,y\in\mathfrak{g} and α,β∈Ω\alpha,\beta\in\Omega. In the associative case they proved (their Proposition 2.11) that a PAYBE solution family {rω}\{r_\omega\} in R⊗RR\otimes R that additionally satisfies the commutation condition r12s23=s12r23r_{12}s_{23}=s_{12}r_{23} for every pair (r,s)=(rα,rβ)(r,s)=(r_\alpha,r_\beta) induces matching Rota–Baxter operators Pω(x)=∑iuω,i x vω,iP_\omega(x)=\sum_i u_{\omega,i}\,x\,v_{\omega,i}; this condition is sufficient but not known to be necessary. The tensor-to-operator dictionary at the Lie level — which tensor equation on a family of tensors corresponds to the weight-zero matching Rota–Baxter Lie equation — was explicitly left open by the authors, who announced it for a separate paper that has not yet appeared.

Problem Statement

Formulate the polarized classical Yang–Baxter equation at the Lie level and prove its correspondence with matching Rota–Baxter Lie algebras of weight zero. Concretely, let g\mathfrak{g} be a Lie algebra over a base field (or commutative ring) and Ω\Omega a nonempty parameter set, and consider the weight-zero matching Rota–Baxter Lie equation [Pα(x),Pβ(y)]=Pα([x,Pβ(y)])+Pβ([Pα(x),y])for all x,y∈g, α,β∈Ω.[P_\alpha(x),P_\beta(y)]=P_\alpha([x,P_\beta(y)])+P_\beta([P_\alpha(x),y])\quad\text{for all }x,y\in\mathfrak{g},\ \alpha,\beta\in\Omega. The task is to define explicit quadratic tensor equations — the Lie analogue of the polarized associative Yang–Baxter equation r13s12−r12s23+r23s13=0r_{13}s_{12}-r_{12}s_{23}+r_{23}s_{13}=0 — satisfied by a parameterized family of tensors {rω}\{r_\omega\} (in an appropriate setting). These equations must be intrinsic conditions on the Lie bracket and the tensor coefficients alone — quadratic combinations of the tensor legs built from the bracket, as in the classical Yang–Baxter equation [[r,r]]=0[[r,r]]=0 — and must neither refer to nor reencode the induced operators or the matching Rota–Baxter identity itself, so that whether a family of tensors solves them is checkable from the bracket and the tensor coefficients alone. The equations are required to be such that: (i) the operators induced from the tensors form a matching Rota–Baxter Lie family of weight zero whenever the tensors solve the equations, including whatever tensor commutation hypotheses play the role of the associative condition r12s23=s12r23r_{12}s_{23}=s_{12}r_{23} (which has no direct meaning for tensors built on a Lie bracket); (ii) conversely, delimit the reverse direction by a definite, checkable result: either prove a converse — explicitly stating the hypotheses under which a matching Rota–Baxter Lie operator family of weight zero arises from a solution of the tensor equations — or exhibit an explicit family of matching Rota–Baxter Lie operators of weight zero that does not arise from any such tensor solution; note that in the associative parallel only the sufficiency of PAYBE together with r12s23=s12r23r_{12}s_{23}=s_{12}r_{23} is established and necessity is open there as well.

References

  1. Xing Gao, Li Guo, Yi Zhang, Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras, Journal of Algebra 552 (2020) 134-170. DOI: 10.1016/j.jalgebra.2020.02.011; arXiv:1909.10577. https://arxiv.org/abs/1909.10577
  2. Marcelo Aguiar, On the associative analog of Lie bialgebras, Journal of Algebra 244 (2001) 492-532. DOI: 10.1006/jabr.2001.8877. https://doi.org/10.1006/jabr.2001.8877
  3. I. Z. Golubchik, V. V. Sokolov, Generalized operator Yang-Baxter equations, integrable ODEs and nonassociative algebras, Journal of Nonlinear Mathematical Physics 7 (2000) 184-197. DOI: 10.2991/jnmp.2000.7.2.8. https://doi.org/10.2991/jnmp.2000.7.2.8
  4. Tomasz Brzeziński, Rota-Baxter systems, dendriform algebras and covariant bialgebras, Journal of Algebra 460 (2016) 1-25. DOI: 10.1016/j.jalgebra.2016.04.018. https://doi.org/10.1016/j.jalgebra.2016.04.018
  5. Yvain Bruned, Martin Hairer, Lorenzo Zambotti, Algebraic renormalisation of regularity structures, Inventiones mathematicae 215 (2019) 1039-1156. DOI: 10.1007/s00222-018-0841-x. https://doi.org/10.1007/s00222-018-0841-x
  6. Yi Zhang, Shuangjian Guo, Matching Rota-Baxter Systems and Gröbner-Shirshov Bases, Algebra Colloquium 31 (2024) 389-406. DOI: 10.1142/S1005386724000294. https://doi.org/10.1142/S1005386724000294
  7. Ramkrishna Mandal, Apurba Das, Matching relative Rota-Baxter algebras, matching dendriform algebras and their cohomologies, Communications in Algebra 53 (2025) 96-123. DOI: 10.1080/00927872.2024.2372389. https://doi.org/10.1080/00927872.2024.2372389
  8. Dan Chen, Xiao-Song Peng, Chia Zargeh, Yi Zhang, Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras, Advances in Mathematical Physics 2020 (2020), Article 9792726. DOI: 10.1155/2020/9792726. https://doi.org/10.1155/2020/9792726
  9. Jun He, Xiaosong Peng, Yi Zhang, Multiple Rota-Baxter algebra and multiple Rota-Baxter modules (2025). arXiv:2504.16643. https://arxiv.org/abs/2504.16643
  10. Apurba Das, Shuangjian Guo, Yufei Qin, L∞-structures and cohomology theory of compatible O-operators and compatible dendriform algebras, Journal of Mathematical Physics 65 (2024) 031701. DOI: 10.1063/5.0161898. https://doi.org/10.1063/5.0161898
  11. Loïc Foissy, Dominique Manchon, Yuanyuan Zhang, A general construction of family algebraic structures, Letters in Mathematical Physics 114 (2024), Article 110. DOI: 10.1007/s11005-024-01851-7. https://doi.org/10.1007/s11005-024-01851-7
  12. Chengming Bai, Li Guo, Guilai Liu, Tianshui Ma, Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras, Algebras and Representation Theory 27 (2024) 1347-1372. DOI: 10.1007/s10468-024-10261-1. https://doi.org/10.1007/s10468-024-10261-1