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We establish a noncommutative analogue of the first fundamental theorem ofclassical invariant theory. For each quantum group associated with a classicalLie algebra, we construct a noncommutative associative algebra whose underlyingvector space forms a module for the quantum group and whose algebraic structureis preserved by the quantum group action. The subspace of invariants is shownto form a subalgebra, which is finitely generated. We determine generators ofthis subalgebra of invariants and determine their commutation relations. Ineach case considered, the noncommutative modules we construct are flatdeformations of their classical commutative analogues. Thus by taking the limitas $q\to 1$, our results imply the first fundamental theorem of classicalinvariant theory, and therefore generalise them to the noncommutative case.

A quantum analogue of the first fundamental theorem of invariant theory