Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915883207622656 |
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| author | Dey, Ayan |
| author_facet | Dey, Ayan |
| contents | Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected Lie group with $\mathrm{Lie}(G)=\g$ with compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG = \RAzm \cdot \RAzp$. This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all complex simple Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21868 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited Dey, Ayan Quantum Algebra Operator Algebras Representation Theory Primary: 17B37, 20G42, Secondary: 46L67, 81R50 Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected Lie group with $\mathrm{Lie}(G)=\g$ with compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG = \RAzm \cdot \RAzp$. This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all complex simple Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup for all connected, simply connected, compact simple Lie groups $K$. |
| title | Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited |
| topic | Quantum Algebra Operator Algebras Representation Theory Primary: 17B37, 20G42, Secondary: 46L67, 81R50 |
| url | https://arxiv.org/abs/2603.21868 |