Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 066, 15 pages      arXiv:2003.11127      https://doi.org/10.3842/SIGMA.2020.066

Dendriform Algebras Relative to a Semigroup

Marcelo Aguiar
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Received April 09, 2020, in final form June 29, 2020; Published online July 11, 2020

Abstract
Loday's dendriform algebras and its siblings pre-Lie and zinbiel have received attention over the past two decades. In recent literature, there has been interest in a generalization of these types of algebra in which each individual operation is replaced by a family of operations indexed by a fixed semigroup $S$. The purpose of this note is twofold. First, we add to the existing work by showing that a similar extension is possible already for the most familiar types of algebra: commutative, associative, and Lie. Second, we show that these concepts arise naturally and in a unified manner from a categorical perspective. For this, one simply has to consider the standard types of algebra but in reference to the monoidal category of $S$-graded vector spaces.

Key words: dendriform algebra; monoidal category; dimonoidal category.

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