Free-Field Representation of Group Element for Simple Quantum Group

Abstract

A representation of the group element (also known as ``universal ${\cal T}$-matrix&#39;&#39;) which satisfies $Δ(g) = g\otimes g$, is given in the form $$ g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal E}_{1/q_{i(s)}}(χ^{(s)}T_{-i(s)})\right) q^{2\vecϕ\vec H} \left(\prod_{s=1}^{d_B}\phantom.^<\ {\cal E}_{q_{i(s)}}(ψ^{(s)} T_{+i(s)})\right)$$ where $d_B = \frac{1}{2}(d_G - r_G)$, $q_i = q^{|| \vecα_i||^2/2}$ and $H_i = 2\vec H\vecα_i/||\vecα_i||^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields&#39;&#39; $χ,\ \vecϕ,\ ψ$ form a Heisenberg-like algebra: $ψ^{(s)}ψ^{(s&#39;)} = q^{-\vecα_{i(s)} \vecα_{i(s&#39;)}} ψ^{(s&#39;)}ψ^{(s)}, & χ^{(s)}χ^{(s&#39;)} = q^{-\vecα_{i(s)}\vecα_{i(s&#39;)}} χ^{(s&#39;)}χ^{(s)}& {\rm for} \ s<s&#39;, \\ q^{\vec h\vecϕ}ψ^{(s)} = q^{\vec h\vecα_{i(s)}} ψ^{(s)}q^{\vec h\vecϕ}, & q^{\vec h\vecϕ}χ^{(s)} = q^{\vec h \vecα_{i(s)}}χ^{(s)}q^{\vec h\vecϕ}, & \\ &ψ^{(s)} χ^{(s&#39;)} = χ^{(s&#39;)}ψ^{(s)} & {\rm for\ any}\ s,s&#39;.$ We argue that the $d_G$-parametric ``manifold&#39;&#39; which $g$ spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication $g \rightarrow g&#39;\cdot g&#39;&#39;$. The universal ${\cal R}$-matrix with the property that ${\cal R} (g\otimes I)(I\otimes g) = (I\otimes g)(g\otimes I){\cal R}$ is given by the usual formula $${\cal R} = q^{-\sum_{ij}^{r_G}||\vecα_i||^2|| \vecα_j||^2 (\vecα\vecα)^{-1}_{ij}H_i \otimes H_j}\prod_{ \vecα> 0}^{d_B}{\cal E}_{q_{\vecα}}\left(-(q_{\vecα}- q_{\vecα}^{-1})T_{\vecα}\otimes T_{-\vecα}\right).$$