 
Summary: THE FIRST FUNDAMENTAL THEOREM OF COINVARIANT
THEORY FOR THE QUANTUM GENERAL LINEAR GROUP
K. R. Goodearl, T. H. Lenagan, and L. Rigal
Abstract. We prove First Fundamental Theorems of Coinvariant Theory for the standard
coactions of the quantum groups Oq (GL t (K)) and Oq (SL t (K)) on the quantized algebra
Oq (Mm;t
(K))\Omega Oq (M t;n (K)). (Here K is an arbitrary field and q an arbitrary nonzero
scalar.) In both cases, the set of coinvariants is a subalgebra of Oq (Mm;t
(K))\Omega Oq (M t;n (K)),
which we identify.
Introduction
One of the highlights of classical invariant theory is the determination of the algebra
of invariant functions for the standard action of the general linear group on the variety of
pairs of matrices over a field K. More precisely, the standard action of GL t = GL t (K) on
the variety V := M m;t (K) \Theta M t;n (K) induces an action of GL t on O(V ), and the classical
theorem determines the algebra of invariants, O(V ) GL t . We recall the details below, since,
if we assume that K is algebraically closed, the method of proof we follow has an easy
geometric translation.
The main theorem of this paper, Theorem 4.5, gives a quantum analog of the above
theorem. Since the quantum group O q (GL t ) is not a group but a Hopf algebra, we first
