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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.
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