Braided symmetric algebras and a first fundamental theorem of invariant theory for ${\rm U}_q(G_2)$

TL;DR

This paper develops invariant theory for the quantum group U_q(G_2), constructing generators and relations for invariants in braided symmetric algebras, providing a non-commutative first fundamental theorem of invariant theory.

Contribution

It introduces a new invariant theory framework for U_q(G_2), explicitly constructs generators, and describes their relations in braided symmetric algebras, a novel non-flat quantisation.

Findings

Constructed a spanning set of invariants using acyclic trivalent graphs.

Explicitly generated the invariant subalgebra with a finite set of homogeneous elements.

Determined the commutation relations among the algebraic generators.

Abstract

We develop invariant theory for the quantum group ${\rm U}_q$ of $G_2$ at generic $q$ in the setting of braided symmetric algebras. Let ${\mathcal A}_m$ be the braided symmetric algebra over $m$-copies of the $7$-dimensional simple ${\rm U}_q$-module. A set of ${\rm U}_q$-invariants in ${\mathcal A}_m$ attached to certain acyclic trivalent graphs is obtained, which spans the subalgebra ${\mathcal A}_m^{{\rm U}_q}$ of invariants as vector space. A finite set of homogeneous elements is constructed explicitly, which generates ${\mathcal A}_m^{{\rm U}_q}$ as algebra. Commutation relations among the algebraic generators are determined. These results may be regarded as a non-commutative first fundamental theorem of invariant theory for ${\rm U}_q$. The algebra ${\mathcal A}_m$ is a non-flat quantisation of the coordinate ring of ${\mathbb C}^7\otimes{\mathbb C}^m$. As ${\rm U}_q$-module, ${\mathcal A}_m={\mathcal A}_1^{\otimes m}$ and we decompose ${\mathcal A}_1$ into simple submodules. The affine scheme associated to the classical limit of ${\mathcal A}_m$ is described. This is a rare case where the structure of a non-flat quantisation is understood.