dorsal/arxiv
View SchemaDual Affine Quantum Groups
| Authors | Fabio Gavarini |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9712013 |
| URL | https://arxiv.org/abs/q-alg/9712013 |
| DOI | 10.1007/s002090050502 |
| Journal | Mathematische Zeitschrift 234 (2000), no. 1, 9-52 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.
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"abstract": "Let $\\hat{\\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its\nSklyanin-Drinfel\u0027d structure of Lie bialgebra, and let $\\hat{\\mathfrak{h}}$ be\nthe dual Lie bialgebra. By dualizing the quantum double construction - via\nformal Hopf algebras - we construct a new quantum group\n$U_q(\\hat{\\mathfrak{h}})$, dual of $U_q(\\hat{\\mathfrak{g}})$. Studying its\nrestricted and unrestricted integer forms and their specializations at roots of\n1 (in particular, their classical limits), we prove that\n$U_q(\\hat{\\mathfrak{h}})$ yields quantizations of $\\hat{\\mathfrak{h}}$ and\n$\\hat{G}^\\infty$ (the formal group attached to $\\hat{\\mathfrak{g}}$), and we\nconstruct new quantum Frobenius morphisms. The whole picture extends to the\nuntwisted affine case the results known for quantum groups of finite type.",
"arxiv_id": "q-alg/9712013",
"authors": [
"Fabio Gavarini"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1007/s002090050502",
"journal_ref": "Mathematische Zeitschrift 234 (2000), no. 1, 9-52",
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Dual Affine Quantum Groups",
"url": "https://arxiv.org/abs/q-alg/9712013"
},
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