$w_{\infty}$ 3-algebra

TL;DR

This paper constructs a $w_{ abla}$ 3-algebra from the $W_{ abla}$ generators using a double scaling limit, verifying that it satisfies the fundamental identity condition of 3-algebras.

Contribution

It introduces a new $w_{ abla}$ 3-algebra derived from $W_{ abla}$ generators and confirms its mathematical consistency through the fundamental identity.

Findings

Successfully derived $w_{ abla}$ 3-algebra from $W_{ abla}$ generators.

Verified the fundamental identity condition for the constructed 3-algebra.

Provides a new algebraic structure relevant to mathematical physics.

Abstract

We use `lone-star' product of the $W_{\infty}$ generators as well as their commutation relations to obtain a $w_{\infty}$ 3-algebra by applying appropriate double scaling limits on the generators. We show explicitly that "Fundamental Identity Condition" (FI) of 3-algebra is satisfied.