Yamaguti algebras and noncrossing partitions

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔰𝔩</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> .