The partially alternating ternary sum in an associative dialgebra

TL;DR

This paper introduces a new partially alternating ternary operation in associative dialgebras, characterizes its polynomial identities up to degree 9 using computer algebra, and identifies a new variety of such algebras.

Contribution

It defines the partially alternating ternary sum in associative dialgebras and determines its polynomial identities up to degree 9, revealing a new algebraic variety.

Findings

Identities in degrees 3 and 5 define a new variety of partially alternating ternary algebras.

A 49-dimensional space of identities exists in degree 7.

No new identities are found in degree 9.

Abstract

The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the center of each term: $a \dashv b \dashv c - a \dashv c \dashv b - b \vdash a \dashv c + c \vdash a \dashv b + b \vdash c \vdash a - c \vdash b \vdash a$. We use computer algebra to determine the polynomial identities in degree $\le 9$ satisfied by this new trilinear operation. In degrees 3 and 5 we obtain $[a,b,c] + [a,c,b] \equiv 0$ and $[a,[b,c,d],e] + [a,[c,b,d],e] \equiv 0$; these identities define a new variety of partially alternating ternary algebras. We show that there is a 49-dimensional space of multilinear identities in degree 7, and we find equivalent nonlinear identities. We use the representation theory of the symmetric group to show that there are no new identities in degree 9.