Cubical Species and Nonassociative Algebras

✍ Scribed by Gábor Hetyei; Gilbert Labelle; Pierre Leroux

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 500 KB
• Volume: 21
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1998.0614

✦ Synopsis

We lay down the foundations of a theory of cubical species, as a variant of Joyal's Ž Ž . . classical theory of species A. Joyal, Ad¨. Math. 42 1981 , 1᎐82 . Informally, a cubical species associates in a functorial way a set of structures to each hypercube. In this context, the hyperoctahedral groups replace the symmetric groups. We analyze cubical species, molecular cubical species, and basic operations among them, along with explicit examples. We show, in particular, that the cubical product gives rise, in a natural way, to a commutative nonassociative ring of formal power series. We conclude with a detailed analysis of this nonassociative ring. ᮊ 1998