Free-Field Representation of Group Element for Simple Quantum Group

Abstract

A representation of the group element (also known as ``universal $\mathcal{T}$-matrix'') which satisfies $\Delta (g)=g\otimes g$, is given in the form $$g=\left(\prod _{s=1}^{d_B}{}^{>}{\mathcal{E}}_{1/q_{i(s)}}({\chi }^{(s)}{T}_{-i(s)})\right)q^{2\vec{\varphi }\vec{H}}\left(\prod _{s=1}^{d_B}{}^{<}{\mathcal{E}}_{q_{i(s)}}({\psi }^{(s)}{T}_{+i(s)})\right)$$ where $d_B=\frac{1}{2}(d_G-r_G)$, $q_i=q^{||{\vec{\alpha }}_i|{|}^2/2}$ and $H_i=2\vec{H}{\vec{\alpha }}_i/||{\vec{\alpha }}_i|{|}^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi ,\vec{\varphi },\psi$ form a Heisenberg-like algebra: $ψ^{(s)}ψ^{(s')} = q^{-\vecα_{i(s)} \vecα_{i(s')}} ψ^{(s')}ψ^{(s)}, & χ^{(s)}χ^{(s')} = q^{-\vecα_{i(s)}\vecα_{i(s')}} χ^{(s')}χ^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vecϕ}ψ^{(s)} = q^{\vec h\vecα_{i(s)}} ψ^{(s)}q^{\vec h\vecϕ}, & q^{\vec h\vecϕ}χ^{(s)} = q^{\vec h \vecα_{i(s)}}χ^{(s)}q^{\vec h\vecϕ}, & \\ &ψ^{(s)} χ^{(s')} = χ^{(s')}ψ^{(s)} & {\rm for\ any}\ s,s'.$ We argue that the $d_G$-parametric ``manifold'' which $g$ spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication $g\to g^{\prime }\cdot g^{\prime \prime }$. The universal $\mathcal{R}$-matrix with the property that $\mathcal{R}(g\otimes I)(I\otimes g)=(I\otimes g)(g\otimes I)\mathcal{R}$ is given by the usual formula $$\mathcal{R}=q^{-{\sum }_{ij}^{r_G}||{\vec{\alpha }}_i|{|}^2||{\vec{\alpha }}_j|{|}^2(\vec{\alpha }\vec{\alpha }{)}_{ij}^{-1}{H}_i\otimes H_j}\prod _{\vec{\alpha }>0}^{d_B}{\mathcal{E}}_{q_{\vec{\alpha }}}\left(-(q_{\vec{\alpha }}-q_{\vec{\alpha }}^{-1})T_{\vec{\alpha }}\otimes T_{-\vec{\alpha }}\right).$$

68 pages

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