Yamaguti algebras and noncrossing partitions
Frédéric Chapoton1; Vladimir Dotsenko1
1Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France
Mathematics Research Reports, Volume 7 (2026), pp. 59-69

Recently, Das defined a new type of algebras, the Yamaguti algebras, which are supposed to serve as envelopes of Lie–Yamaguti algebras appearing naturally in differential geometry. We show that the nonsymmetric operad of Yamaguti algebras admits a combinatorial description via noncrossing partitions without singleton blocks, and a representation-theoretic description as the equivariant endomorphism operad of the adjoint module of the Lie algebra $\mathfrak{sl}_2$.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.27
Classification: 18M65, 18M70, 18M80, 05A18, 17B10
Keywords: Yamaguti algebras, noncrossing partitions, cyclic operad, equivariant, endomorphism operad

Frédéric Chapoton  1 ; Vladimir Dotsenko  1

1 Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000 Strasbourg, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Frédéric Chapoton; Vladimir Dotsenko. Yamaguti algebras and noncrossing partitions. Mathematics Research Reports, Volume 7 (2026), pp. 59-69. doi: 10.5802/mrr.27
@article{MRR_2026__7__59_0,
     author = {Fr\'ed\'eric Chapoton and Vladimir Dotsenko},
     title = {Yamaguti algebras and noncrossing partitions},
     journal = {Mathematics Research Reports},
     pages = {59--69},
     year = {2026},
     publisher = {MathOA foundation},
     volume = {7},
     doi = {10.5802/mrr.27},
     language = {en},
     url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.27/}
}
TY  - JOUR
AU  - Frédéric Chapoton
AU  - Vladimir Dotsenko
TI  - Yamaguti algebras and noncrossing partitions
JO  - Mathematics Research Reports
PY  - 2026
SP  - 59
EP  - 69
VL  - 7
PB  - MathOA foundation
UR  - https://mrr.centre-mersenne.org/articles/10.5802/mrr.27/
DO  - 10.5802/mrr.27
LA  - en
ID  - MRR_2026__7__59_0
ER  - 
%0 Journal Article
%A Frédéric Chapoton
%A Vladimir Dotsenko
%T Yamaguti algebras and noncrossing partitions
%J Mathematics Research Reports
%D 2026
%P 59-69
%V 7
%I MathOA foundation
%U https://mrr.centre-mersenne.org/articles/10.5802/mrr.27/
%R 10.5802/mrr.27
%G en
%F MRR_2026__7__59_0

[1] Christos A. Athanasiadis; Christina Savvidou The local h-vector of the cluster subdivision of a simplex, Sém. Lothar. Combin., Volume 66 (2011/12), p. Art. B66c, 21 | MR | Zbl

[2] Pilar Benito; Alberto Elduque; Fabián Martín-Herce Irreducible Lie-Yamaguti algebras, J. Pure Appl. Algebra, Volume 213 (2009) no. 5, pp. 795-808 | DOI | MR | Zbl

[3] Pilar Benito; Alberto Elduque; Fabián Martín-Herce Irreducible Lie-Yamaguti algebras of generic type, J. Pure Appl. Algebra, Volume 215 (2011) no. 2, pp. 108-130 | DOI | MR | Zbl

[4] Murray R. Bremner A polynomial identity for the bilinear operation in Lie-Yamaguti algebras, Linear Multilinear Algebra, Volume 62 (2014) no. 12, pp. 1671-1682 | DOI | MR | Zbl

[5] Murray R. Bremner; Vladimir Dotsenko Algebraic operads: An algorithmic companion, CRC Press, Boca Raton, FL, 2016, xvii+365 pages | MR | DOI | Zbl

[6] S. Chmutov; S. Duzhin; J. Mostovoy Introduction to Vassiliev knot invariants, Cambridge University Press, Cambridge, 2012, xvi+504 pages | DOI | MR | Zbl

[7] S. V. Chmutov; A. N. Varchenko Remarks on the Vassiliev knot invariants coming from sl 2 , Topology, Volume 36 (1997) no. 1, pp. 153-178 | DOI | MR | Zbl

[8] Apurba Das Associative-Yamaguti algebras (2025) | arXiv | Zbl

[9] Vladimir Dotsenko; Willem Heijltjes Operadic Gröbner bases calculator, https://irma.math.unistra.fr/~dotsenko/Operads.html, 2014

[10] Vladimir Dotsenko; Pedro Tamaroff Endofunctors and Poincaré–Birkhoff–Witt theorems, Int. Math. Res. Not. IMRN (2021) no. 16, pp. 12670-12690 | DOI | MR | Zbl

[11] Kurusch Ebrahimi-Fard; Loïc Foissy; Joachim Kock; Frédéric Patras Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, Adv. Math., Volume 369 (2020), p. 107170, 55 | DOI | MR | Zbl

[12] V. T. Filippov On a variety of Mal’tsev algebras, Algebra i Logika, Volume 20 (1981) no. 3, p. 300-314, 361 | MR | Zbl

[13] Imma Gálvez-Carrillo; Andrew Tonks; Bruno Vallette Homotopy Batalin-Vilkovisky algebras, J. Noncommut. Geom., Volume 6 (2012) no. 3, pp. 539-602 | DOI | MR | Zbl

[14] Anton Khoroshkin; Dmitri Piontkovski On generating series of finitely presented operads, J. Algebra, Volume 426 (2015), pp. 377-429 | DOI | MR | Zbl

[15] Michihiko Kikkawa Geometry of homogeneous Lie loops, Hiroshima Math. J., Volume 5 (1975) no. 2, pp. 141-179 http://projecteuclid.org/euclid.hmj/1206136626 | MR | Zbl

[16] Michael K. Kinyon; Alan Weinstein Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., Volume 123 (2001) no. 3, pp. 525-550 | MR | DOI | Zbl

[17] G. Kreweras Sur les partitions non croisées d’un cycle, Discrete Math., Volume 1 (1972) no. 4, pp. 333-350 | DOI | MR | Zbl

[18] Jean-Louis Loday; Bruno Vallette Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, Heidelberg, 2012, xxiv+634 pages | DOI | MR | Zbl

[19] Jean-Baptiste Meilhan; Sakie Suzuki Riordan trees and the homotopy sl 2 weight system, J. Pure Appl. Algebra, Volume 221 (2017) no. 3, pp. 691-706 | DOI | MR | Zbl

[20] Alexander A. Mikhalev; Ivan P. Shestakov PBW-pairs of varieties of linear algebras, Comm. Algebra, Volume 42 (2014) no. 2, pp. 667-687 | DOI | MR | Zbl

[21] S. P. Mishchenko Varieties of Lie algebras that do not contain a three-dimensional simple algebra, Mat. Sb., Volume 183 (1992) no. 6, pp. 87-96 | DOI | MR | Zbl

[22] Katsumi Nomizu Invariant affine connections on homogeneous spaces, Amer. J. Math., Volume 76 (1954), pp. 33-65 | DOI | MR | Zbl

[23] OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences, 2025 (Published electronically at https://oeis.org)

[24] L. E. Positselskiuı Nonhomogeneous quadratic duality and curvature, Funktsional. Anal. i Prilozhen., Volume 27 (1993) no. 3, p. 57-66, 96 | DOI | MR | Zbl

[25] Ju. P. Razmyslov The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero, Algebra i Logika, Volume 12 (1973), p. 83-113, 121 | MR | Zbl

[26] Yunhe Sheng; Jia Zhao Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras, Comm. Algebra, Volume 50 (2022) no. 9, pp. 4056-4073 | DOI | MR | Zbl

[27] Jonatan Stava On the free Lie-Yamaguti algebra (2024) | arXiv | Zbl

[28] Kiyosi Yamaguti On the Lie triple system and its generalization, J. Sci. Hiroshima Univ. Ser. A, Volume 21 (1957/58), pp. 155-160 | MR | Zbl

Cited by Sources: