| We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For a quantum group associated with each classical Lie algebra, we construct a noncommutative associative algebra which admits an action of the quantum group. We prove the finite generation of the subalgebra of invariants, construct its generators and determine their commutation relations. In the limit q going to 1, the results recover the first fundamental theorem of classical invariant theory. This is joint work with Gus Lehrer and H. Zhang. |