A quantum analogue of the first fundamental theorem of invariant theory

TL;DR

This paper develops a noncommutative version of the first fundamental theorem of invariant theory for quantum groups, constructing invariant subalgebras that deform classical invariants and generalize classical results.

Contribution

It introduces noncommutative associative algebras for quantum groups, identifies their invariant subalgebras, and shows these are finitely generated deformations of classical invariants.

Findings

Invariant subalgebras are finitely generated.

Constructed noncommutative modules are flat deformations.

Results recover classical invariant theory as q approaches 1.

Abstract

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Thus by taking the limit as $q\to 1$, our results imply the first fundamental theorem of classical invariant theory, and therefore generalise them to the noncommutative case.