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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.07211 |
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| _version_ | 1866911176340799488 |
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| author | Hu, Hongmei Zhang, Ruibin |
| author_facet | Hu, Hongmei Zhang, Ruibin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07211 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Braided symmetric algebras and a first fundamental theorem of invariant theory for ${\rm U}_q(G_2)$ Hu, Hongmei Zhang, Ruibin Quantum Algebra Representation Theory 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. |
| title | Braided symmetric algebras and a first fundamental theorem of invariant theory for ${\rm U}_q(G_2)$ |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2505.07211 |