A representation of the group element (also known as ``universal ${ \cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$ g = \left(\prod_{ s=1}^{ d_B}\phantom.^>\ { \cal E}_{ 1/q_{ i(s)}}(\chi^{ (s)}T_{ -i(s)})\right) q^{ 2\vec\phi\vec H} \left(\prod_{ s=1}^{ d_B}\phantom.^<\ { \cal E}_{ q_{ i(s)}}(\psi^{ (s)} T_{ +i(s)})\right)$$ 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'' $\chi,\ \vec\phi,\ \psi$ form a Heisenberg-like algebra: $\psi^{ (s)}\psi^{ (s')} = q^{ -\vec\alpha_{ i(s)} \vec\alpha_{ i(s')}} \psi^{ (s')}\psi^{ (s)}, & \chi^{ (s)}\chi^{ (s')} = q^{ -\vec\alpha_{ i(s)}\vec\alpha_{ i(s')}} \chi^{ (s')}\chi^{ (s)}& { \rm for} \ s<s', \\ q^{ \vec h\vec\phi}\psi^{ (s)} = q^{ \vec h\vec\alpha_{ i(s)}} \psi^{ (s)}q^{ \vec h\vec\phi}, & q^{ \vec h\vec\phi}\chi^{ (s)} = q^{ \vec h \vec\alpha_{ i(s)}}\chi^{ (s)}q^{ \vec h\vec\phi}, & \\ &\psi^{ (s)} \chi^{ (s')} = \chi^{ (s')}\psi^{ (s)} & { \rm for\ any}\ s,s'.$ 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 $g \rightarrow g'\cdot g''$. 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 $${ \cal R} = q^{ -\sum_{ ij}^{ r_G}||\vec\alpha_i||^2|| \vec\alpha_j||^2 (\vec\alpha\vec\alpha)^{ -1}_{ ij}H_i \otimes H_j}\prod_{ \vec\alpha > 0}^{ d_B}{ \cal E}_{ q_{ \vec\alpha}}\left(-(q_{ \vec\alpha}- q_{ \vec\alpha}^{ -1})T_{ \vec\alpha}\otimes T_{ -\vec\alpha}\right).$$
Categories: hep-th