Double-Bosonisation and the Construction of {$U_q(g)$}

S. Majid

paper

Double-Bosonisation and the Construction of { $U_{q} (g)$ }

arXiv: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 $U_{q} (s l_{3})$ from $U_{q} (s l_{2})$ 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.

LATEX 57 pages and epsf figures

References in corpus (1)

Algebraic { $q$ }-Integration and Fourier Theory on Quantum and Braided Spaces

Double-Bosonisation and the Construction of {Uq​(g)}

Abstract

References in corpus (1)

Cited by in corpus (5)

Double-Bosonisation and the Construction of { $U_{q} (g)$ }