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

Abstract

We develop invariant theory for the quantum group ${\mathrm{U}}_q$ 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 ${\mathrm{U}}_q$-module. A set of ${\mathrm{U}}_q$-invariants in ${\mathcal{A}}_m$ attached to certain acyclic trivalent graphs is obtained, which spans the subalgebra ${\mathcal{A}}_m^{{\mathrm{U}}_q}$ of invariants as vector space. A finite set of homogeneous elements is constructed explicitly, which generates ${\mathcal{A}}_m^{{\mathrm{U}}_q}$ 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 ${\mathrm{U}}_q$. The algebra ${\mathcal{A}}_m$ is a non-flat quantisation of the coordinate ring of ${\mathbb{C}}^7\otimes {\mathbb{C}}^m$. As ${\mathrm{U}}_q$-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.