  Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are
embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo
in the same way as ordinary Lie algebras are embedded into their enveloping
algebras. The quantum Lie product on $\qlie{g}$ is induced by the quantum
adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to
$U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the
quantum root systems, which are now functions of the quantum parameter $q$.
They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.
