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.

中文翻译：

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量子广义线性超群不变论的第一和第二基本定理

我们为量子通用线性超群 ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$ 开发了不变理论的非交换多项式版本。非交换 ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-模超代数 $\mathcal{P}^{k|l}_{\,r|s} $ 被构造，它是 $\mathbb{C}^{k|l}\otimes \mathbb{C}^{m|n}\oplus \mathbb{C}^{r| 上超对称代数的量子模拟s}\otimes (\mathbb{C}^{m|n})^{\ast}$。我们分析了$\mathcal{P}^{k|l}_{\中${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-invariants的子代数的结构， r|s}$ 通过使用 Howe 对偶的量子超级模拟。$\mathcal{P}^{k|l}_{\,r|s}$中${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-不变量的子代数被证明是有限生成的。我们确定它的生成元，并从一个编织的超对称代数到它上面建立一个满射超代数同态。这建立了 ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$ 不变论的第一个基本定理。我们证明上述超代数同态是同构当且仅当 $m\geq \min\{k,r\}$ 和 $n\geq \min\{l,s\}$，并获得单项式基对于这种情况下的不变量的子代数。当同态不是单射时，我们给出了与分区 $((m+1)^{n+1})$ 相关联的内核的生成元素的表示理论描述，产生了 $ 不变论的第二个基本定理{\rm{ U}}_q(\mathfrak{gl}_{m|n})$。我们考虑我们结果的两种应用。对于 ${\rm{ U}}_q(\mathfrak{gl}_{m})$ 的不变论的非交换多项式版本的完整处理是作为 $n=0$ 的特殊情况获得的，其中证明了显式 SFT，我们认为这是新的。一般线性超代数不变理论的 FFT 和 SFT 是从我们结果的经典（即 $q\to 1$）极限中恢复的。