Title: The first and second fundamental theorems of invariant theory for the quantum general linear supergroup,
Abstract: We develop the non-commutative polynomial version of the invariant theory for
the quantum general linear supergroup ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. A
non-commutative ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-module superalgebra
$\mathcal{P}^{k|l}_{\,r|s}$ is constructed, which is the quantum analogue of
the supersymmetric algebra over $\mathbb{C}^{k|l}\otimes \mathbb{C}^{m|n}\oplus
\mathbb{C}^{r|s}\otimes (\mathbb{C}^{m|n})^{\ast}$. We analyse the structure of
the subalgebra of ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in
$\mathcal{P}^{k|l}_{\,r|s}$ by using the quantum super analogue of Howe
duality.
The subalgebra of ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in
$\mathcal{P}^{k|l}_{\,r|s}$ is shown to be finitely generated. We determine its
generators and establish a surjective superalgebra homomorphism from a braided
supersymmetric algebra onto it. This establishes the first fundamental theorem
of invariant theory for ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$.
We show that the above mentioned superalgebra homomorphism is an isomorphism
if and only if $m\geq \min\{k,r\}$ and $n\geq \min\{l,s\}$, and obtain a
monomial basis for the subalgebra of invariants in this case. When the
homomorphism is not injective, we give a representation theoretical description
of the generating elements of the kernel associated to the partition
$((m+1)^{n+1})$, producing the second fundamental theorem of invariant theory
for ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$.
We consider two applications of our results. A complete treatment of the
non-commutative polynomial version of invariant theory for ${\rm{
U}}_q(\mathfrak{gl}_{m})$ is obtained as the special case with $n=0$, where an
explicit SFT is proved, which we believe to be new. The FFT and SFT of the
invariant theory for the general linear superalgebra are recovered from the
classical (i.e., $q\to 1$) limit of our results.