Basic Twist Quantization of the Exceptional Lie Algebra G_2

TL;DR

This paper develops a detailed twist quantization method for the exceptional Lie algebra G_2, involving a sequence of four twist factors, and introduces new generators to simplify the coproduct calculations.

Contribution

It provides explicit formulas for twist quantization of G_2, including new methods for similarity transformations and generators that simplify coproducts.

Findings

Explicit coproduct formulas for each twist step

New formulas for similarity transformations on tensor products

Introduction of new generators simplifying algebraic structures

Abstract

We present the formulae for twist quantization of $g_2$, corresponding to the solution of classical YB equation with support in the 8-dimensional Borel subalgebra of $g_2$. The considered chain of twists consists of the four factors describing the four steps of quantization: Jordanian twist, the two twist factors extending Jordanian twist and the deformed Jordanian or in second variant additional Abelian twist. The first two steps describe as well the $sl(3)$ quantization. The coproducts are calculated for each step in explicite form, and for that purpose we present new formulas for the calculation of similarity transformations on tensor product. We introduce new basic generators in universal enveloping algebra $U(g_2)$ which provide nonlinearities in algebraic sector maximally simplifying the deformed coproducts.