Special Identities for Comtrans Algebras

Autor: Bremner, Murray R., Elgendy, Hader A.
Rok vydání: 2018
Předmět:
Druh dokumentu: Working Paper
Popis: Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Gr\"obner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator $[x,y,z] = xyz-yxz$ and special translator $\langle x, y, z \rangle = xyz-yzx$ in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Gr\"obner bases to construct the universal associative envelope for the special comtrans algebra of $2 \times 2$ matrices.
Comment: 18 pages, 4 figures. Sections 3.2 and 3.3 have been rewritten to emphasize the importance of the forgetful functor from symmetric operads to shuffle operads
Databáze: arXiv