The First Fundamental Theorem of Coinvariant Theory for the Quantum General Linear Group

Abstract

We prove First Fundamental Theorems of Coinvariant Theory for the standard coactions of the quantum groups ${\mathcal{O}}_q({GL}_t(K))$ and ${\mathcal{O}}_q({SL}_t(K))$ on the quantized algebra ${\mathcal{O}}_q(M_{m,t}(K))\otimes {\mathcal{O}}_q(M_{t,n}(K))$. (Here $K$ is an arbitrary field and $q$ an arbitrary nonzero scalar.) In both cases, the set of co-invariants is a subalgebra of ${\mathcal{O}}_q(M_{m,t}(K))\otimes {\mathcal{O}}_q(M_{t,n}(K))$, which we identify.