Special Identities for Comtrans Algebras

TL;DR

This paper investigates the algebraic structure of comtrans algebras, determining their identities and Gr"obner bases, and constructs their universal associative envelope, advancing understanding in web geometry and algebraic identities.

Contribution

It provides a comprehensive analysis of identities in comtrans algebras, including new identities at degree 5 and 7, and constructs the universal associative envelope for specific cases.

Findings

Degree 3 identities generate all identities for comtrans algebras.

New identities at degree 5 relate the operations.

Degree 7 analysis shows no new identities relating the operations.

Abstract

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.