We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum groups \mathcal{O}
q(
GLt(
K)) and \mathcal{O}
q(
SLt(
K)) on the quantized algebra \mathcal{O}
q(
Mm, t(
K))⊗\mathcal{O}
q(
Mt, 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 \mathcal{O}
q(
Mm, t(
K))⊗\mathcal{O}
q(
Mt, n(
K)), which we identify.
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