Exceptional $\mathfrak{g}_2$ deformations and gauge symmetries

TL;DR

This work develops a Clifford-algebra-parametrized framework to deform the octonion-based exceptional Lie algebra $\mathfrak{g}_2$ by introducing the $u$-product and its variants, coupling octonions to full multivector elements of $\mathrm{Cl}_{0,7}$. The authors show how these deformations induce new derivations $(\mathfrak{g}_2)_u$ and twisted automorphism groups $G_{2,u}$, with explicit constructions $M_{h,u}$ and $N_{h,u}$ that satisfy the Leibniz rule for the deformed products. They demonstrate that $\mathfrak{su}(3)$-like subalgebras arise as residual symmetries within these deformed structures, illustrating how inequivalent embeddings of $\mathfrak{su}(3)$ inside $\mathfrak{g}_2$ can be interpolated by Clifford-parametrized deformations. The framework provides a versatile geometric-algebraic setting linking exceptional algebra, octonions, and Clifford theory, with potential implications for higher-dimensional unification, non-associative gauge-like dynamics, and QCD-like symmetries emerging from exceptional structures.

Abstract

Deformed $\mathfrak{g}_2$ exceptional applications are introduced via the Clifford algebra-parametrized formalism. Using the products between multivectors of $\cl_{0,7}$, the Clifford algebra over the metric vector space $\RR^{0,7}$, and octonions, resulting in an octonion, we generalize the exceptional Lie algebra $\mathfrak{g}_2$ applications, also associated with the transformation rules for bosonic and fermionic fields on the 7-sphere $S^7$. The emergence of $SU(3)$-like subalgebras within the exceptional Lie algebra $\mathfrak{g}_2$ provides an algebraic framework reminiscent of the $SU(3)$ gauge symmetry of QCD.