dorsal/arxiv
View SchemaDouble-Bosonisation and the Construction of {$U_q(g)$}
| Authors | S. Majid |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9511001 |
| URL | https://arxiv.org/abs/q-alg/9511001 |
Abstract
We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the {\em double-bosonisation}, associated to every braided group $B$ in the category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$ appears as the `positive root space', $H$ as the `Cartan subalgebra' and the dual braided group $B^*$ as the `negative root space' of $U(B)$. The choice $B=f$ recovers Lusztig's construction of $U_q(g)$, where $f$ is Lusztig's algebra associated to a Cartan datum; other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the quantum-braided plane $A_q^2$. We provide a fundamental representation of $U(B)$ in $B$. A projection from the quantum double, a theory of double biproducts and a Tannaka-Krein reconstruction point of view are also provided.
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"abstract": "We introduce a quasitriangular Hopf algebra or `quantum group\u0027 $U(B)$, the\n{\\em double-bosonisation}, associated to every braided group $B$ in the\ncategory of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$\nappears as the `positive root space\u0027, $H$ as the `Cartan subalgebra\u0027 and the\ndual braided group $B^*$ as the `negative root space\u0027 of $U(B)$. The choice\n$B=f$ recovers Lusztig\u0027s construction of $U_q(g)$, where $f$ is Lusztig\u0027s\nalgebra associated to a Cartan datum; other choices give more novel quantum\ngroups. As an application, our construction provides a canonical way of\nbuilding up quantum groups from smaller ones by repeatedly extending their\npositive and negative root spaces by linear braided groups; we explicitly\nconstruct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the\nquantum-braided plane $A_q^2$. We provide a fundamental representation of\n$U(B)$ in $B$. A projection from the quantum double, a theory of double\nbiproducts and a Tannaka-Krein reconstruction point of view are also provided.",
"arxiv_id": "q-alg/9511001",
"authors": [
"S. Majid"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Double-Bosonisation and the Construction of {$U_q(g)$}",
"url": "https://arxiv.org/abs/q-alg/9511001"
},
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