[Paper Review] The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

Yang Zhang|arXiv (Cornell University)|Mar 6, 2017

Algebraic structures and combinatorial models25 references3 citations

TL;DR

This paper establishes the first and second fundamental theorems of invariant theory for the quantum general linear supergroup $\mathrm{U}_q(\mathfrak{gl}_{m|n})$ by constructing a non-commutative polynomial superalgebra $\mathcal{P}^{k|l}_{r|s}$ as a quantum analogue of the classical supersymmetric algebra. Using quantum Howe duality, it proves that the subalgebra of $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-invariants is finitely generated, determines its generators, and identifies the relations among them, with an isomorphism to a braided supersymmetric algebra if and only if $m \geq \min\{k,r\}$ and $n \geq \min\{l,s\}$.

ABSTRACT

We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup ${ m{ U}}_q(\mathfrak{gl}_{m|n})$. A non-commutative ${ m{ U}}_q(\mathfrak{gl}_{m|n})$-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 ${ m{ U}}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{\,r|s}$ by using the quantum super analogue of Howe duality. The subalgebra of ${ m{ U}}_q(\mathfrak{gl}_{m|n})$-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 ${ m{ U}}_q(\mathfrak{gl}_{m|n})$. 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 ${ m{ U}}_q(\mathfrak{gl}_{m|n})$. We consider two applications of our results. A complete treatment of the non-commutative polynomial version of invariant theory for ${ m{ U}}_q(\mathfrak{gl}_{m})$ 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., $q o 1$) limit of our results.

Motivation & Objective

• To develop a non-commutative polynomial version of invariant theory for the quantum general linear supergroup $\mathrm{U}_q(\mathfrak{gl}_{m|n})$.
• To construct a quantum analogue $\mathcal{P}^{k|l}_{r|s}$ of the classical supersymmetric algebra over $\mathbb{C}^{k|l} \otimes \mathbb{C}^{m|n} \oplus \mathbb{C}^{r|s} \otimes (\mathbb{C}^{m|n})^*$.
• To determine a finite set of generators for the subalgebra of $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{r|s}$, establishing the first fundamental theorem.
• To describe the algebraic relations among the generators of the invariant subalgebra, yielding the second fundamental theorem.
• To recover the classical FFT and SFT of invariant theory for $\mathfrak{gl}_{m|n}$ in the $q \to 1$ limit, and to provide a complete treatment for the case $n=0$ (i.e., $\mathrm{U}_q(\mathfrak{gl}_m)$).

Proposed method

• Construct the non-commutative $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-module superalgebra $\mathcal{P}^{k|l}_{r|s}$ as a quantum deformation of the classical supersymmetric algebra using braided supersymmetric algebras.
• Apply the quantum super analogue of Howe duality to analyze the structure of the subalgebra of $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{r|s}$.
• Establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto the invariant subalgebra, proving the first fundamental theorem.
• Prove that this homomorphism is an isomorphism if and only if $m \geq \min\{k,r\}$ and $n \geq \min\{l,s\}$, and construct a PBW basis in this case.
• When the homomorphism is not injective, describe the kernel representation-theoretically to determine the relations among generators, yielding the second fundamental theorem.
• Recover the classical FFT and SFT of invariant theory for $\mathfrak{gl}_{m|n}$ by taking the $q \to 1$ limit of the quantum results.

Experimental results

Research questions

• RQ1 What is an appropriate quantum polynomial superalgebra $\mathcal{P}^{k|l}_{r|s}$ that deforms the classical supersymmetric algebra $\mathcal{S}^{k|l}_{r|s}$ in the non-commutative setting?
• RQ2 What is the structure of the subalgebra of $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{r|s}$, and how can it be finitely generated despite non-commutativity?
• RQ3 When is the natural homomorphism from the braided supersymmetric algebra to the invariant subalgebra an isomorphism, and what is the PBW basis in that case?
• RQ4 What are the algebraic relations among the generators of the invariant subalgebra when the homomorphism is not injective?
• RQ5 How do the quantum FFT and SFT for $\mathrm{U}_q(\mathfrak{gl}_{m|n})$ reduce to the classical results in the $q \to 1$ limit?

Key findings

• The subalgebra of $\mathrm{U}_q(\mathfrak{gl}_{m|n})$-invariants in $\mathcal{P}^{k|l}_{r|s}$ is finitely generated, with generators explicitly determined via quantum Howe duality.
• A surjective superalgebra homomorphism exists from a braided supersymmetric algebra onto the invariant subalgebra, establishing the first fundamental theorem.
• This homomorphism is an isomorphism if and only if $m \geq \min\{k,r\}$ and $n \geq \min\{l,s\}$, in which case a PBW basis for the invariant subalgebra is constructed.
• When the homomorphism is not injective, the kernel is described representation-theoretically, yielding the relations among generators and proving the second fundamental theorem.
• The classical FFT and SFT for $\mathfrak{gl}_{m|n}$ are recovered as the $q \to 1$ limit of the quantum results.
• The case $n=0$ yields a complete treatment of the non-commutative invariant theory for $\mathrm{U}_q(\mathfrak{gl}_m)$, with an explicit SFT that is believed to be new.