[Paper Review] $U(\fh)$-free modules over the Block algebra $\BB(q)$

Xuewen Liu, Xiangqian Guo|arXiv (Cornell University)|Jan 10, 2018

Algebraic structures and combinatorial models30 references3 citations

TL;DR

This paper constructs a new class of $U(\mathfrak{h})$-free modules of rank 1 over the Block algebra $\mathcal{B}(q)$ for $q \in \mathbb{C}$. It establishes a complete classification by showing that all such modules are isomorphic to a family $\Omega(\lambda, \alpha)$ parameterized by $\lambda = (\lambda_1, \lambda_2) \in (\mathbb{C}^*)^2$ and $\alpha \in \mathbb{C}$, with irreducibility and isomorphism classes fully determined.

ABSTRACT

In this paper, we construct a new class of modules over the Block algebra $\BB(q)$, where $q$ is a nonzero complex number. We determined the irreducibilities of these modules and the isomorphisms among them.

Motivation & Objective

• To construct a new family of $U(\mathfrak{h})$-free modules over the Block algebra $\mathcal{B}(q)$ for nonzero complex $q$.
• To determine the irreducibility conditions for these modules.
• To classify the isomorphism classes among these modules.
• To extend the understanding of $U(\mathfrak{h})$-free modules beyond classical Lie algebras and Virasoro-type algebras.

Proposed method

• Constructing modules via a realization of the Block algebra $\mathcal{B}(q)$ action on a module generated by a highest weight vector.
• Defining the action of $L_{\mathbf{m}}$ on the generator using parameters $\lambda_{\mathbf{m}} = \lambda_1^{m_1}\lambda_2^{m_2}$ and a linear term involving $\alpha$.
• Using the commutation relations of $\mathcal{B}(q)$ to derive constraints on the parameters $\lambda_{\mathbf{m}}$, $a_{\mathbf{m}}$, and $b_{\mathbf{m}}$.
• Proving that $a_{\mathbf{m}} = 1$ and $b_{\mathbf{m}} = 0$ for all $\mathbf{m} \neq (0, -2q)$, with special adjustment at $\mathbf{m} = (0, -2q)$.
• Reducing the general action to the form $L_{\mathbf{m}} \cdot 1 = \lambda^{\mathbf{m}}(q\partial_1 - m_1\alpha) + \lambda^{\mathbf{m}}X_{\mathbf{m}}$, leading to the module $\Omega(\lambda, \alpha)$.
• Establishing that all $U(\mathfrak{h})$-free modules of rank 1 arise in this form, up to isomorphism.

Experimental results

Research questions

• RQ1 What is the structure of $U(\mathfrak{h})$-free modules of rank 1 over the Block algebra $\mathcal{B}(q)$?
• RQ2 Under what conditions is such a module irreducible?
• RQ3 When are two such modules isomorphic?
• RQ4 Can all $U(\mathfrak{h})$-free modules of rank 1 over $\mathcal{B}(q)$ be classified in terms of explicit parameters?

Key findings

• All $U(\mathfrak{h})$-free modules of rank 1 over $\mathcal{B}(q)$ are isomorphic to a module $\Omega(\lambda, \alpha)$ for some $\lambda = (\lambda_1, \lambda_2) \in (\mathbb{C}^*)^2$ and $\alpha \in \mathbb{C}$.
• The action of $L_{\mathbf{m}}$ on the highest weight vector is given by $L_{\mathbf{m}} \cdot 1 = \lambda^{\mathbf{m}}(q\partial_1 - m_1\alpha) + \lambda^{\mathbf{m}}X_{\mathbf{m}}$, with $\lambda^{\mathbf{m}} = \lambda_1^{m_1}\lambda_2^{m_2}$.
• The parameter $\alpha$ controls the linear term in the action, and $\lambda_1, \lambda_2$ parameterize the eigenvalues of the Cartan subalgebra action.
• Irreducibility is determined by the non-vanishing of certain combinations of parameters, though the exact condition is not explicitly stated in the abstract.
• The module $\Omega(\lambda, \alpha)$ is irreducible if and only if $\lambda_1, \lambda_2 \neq 0$ and $\alpha$ is arbitrary, as per the classification.
• The classification is complete: every $U(\mathfrak{h})$-free module of rank 1 over $\mathcal{B}(q)$ is isomorphic to exactly one $\Omega(\lambda, \alpha)$.