[Paper Review] A family of new simple modules over the Schrödinger-Virasoro algebra

Haibo Chen, Yanyong Hong+1 more|arXiv (Cornell University)|Aug 29, 2016

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper constructs a large family of new simple modules over the Schrödinger-Virasoro algebra and the $W(2,2)$-algebra by inducing from simple modules over finite-dimensional quotient algebras of certain subalgebras. The construction includes highest weight and Whittaker modules as special cases and identifies all simple modules with locally finite actions of elements in a positive part as belonging to this class, yielding new non-weight, non-Whittaker simple modules.

ABSTRACT

In this article, a large class of simple modules over the Schrödinger-Virasoro algebra $\mathcal{G}$ are constructed, which include highest weight modules and Whittaker modules. These modules are determined by the simple modules over the finite-dimensional quotient algebras of some subalgebras. Moreover, we show that all simple modules of $\mathcal{G}$ with locally finite actions of elements in a certain positive part belong to this class of simple modules. Similarly, a large class of simple modules over the $W$-algebra $W(2,2)$ are constructed.

Motivation & Objective

• To construct a large class of new simple modules over the Schrödinger-Virasoro algebra that generalize highest weight and Whittaker modules.
• To characterize all simple modules with locally finite actions of elements in a positive part of the algebra as belonging to this constructed class.
• To extend the construction method to the $W(2,2)$-algebra and produce new simple modules there as well.
• To unify the representation theory of these algebras by identifying a common framework for modules with locally finite or locally nilpotent actions.

Proposed method

• Inducing modules from simple modules over finite-dimensional quotient algebras of subalgebras of the Schrödinger-Virasoro algebra.
• Using a principal total order on $\mathbb{M} \times \mathbb{M}$ to define a filtration and analyze the structure of induced modules.
• Defining conditions on the action of generators $W_i, L_j$ in the $W(2,2)$-algebra to ensure simplicity of induced modules.
• Applying the notion of locally finite and locally nilpotent actions to classify simple modules over both algebras.
• Verifying that the induced modules are simple under specific conditions on the actions of $W_t$ and $L_j$ for large indices.
• Establishing equivalence between locally finite actions and induced structures via the $\mathcal{W}_d$-subalgebra framework.

Experimental results

Research questions

• RQ1 Can a unified construction method generate both highest weight and Whittaker modules over the Schrödinger-Virasoro algebra as special cases?
• RQ2 What conditions on the action of generators ensure that an induced module over the Schrödinger-Virasoro algebra is simple?
• RQ3 Are all simple modules with locally finite actions of elements in a positive part of the Schrödinger-Virasoro algebra captured by this construction?
• RQ4 Can the same method be extended to construct new simple modules over the $W(2,2)$-algebra?
• RQ5 What is the precise relationship between locally finite actions and induced modules in the context of these algebras?

Key findings

• The construction recovers highest weight modules as a special case when the induced module arises from a one-dimensional module over a subalgebra.
• The construction recovers Whittaker modules as a special case when the induced module is defined using a character on a subalgebra with specific action conditions.
• All simple modules of the Schrödinger-Virasoro algebra with locally finite actions of $M_i, (1-\delta_{i,0})Y_{i-1/2}, L_i$ for sufficiently large $i$ are isomorphic to modules constructed in this paper.
• For the $W(2,2)$-algebra, the induced modules $\mathrm{Ind}(V)$ are simple if $V$ is a simple $\mathcal{W}_d$-module satisfying injectivity and vanishing conditions on $W_t$ and $L_j$ for large indices.
• The paper establishes that locally finite and locally nilpotent actions of $W_i, L_j$ for large $i,j$ are equivalent for simple $\mathcal{W}$-modules.
• Any simple $\mathcal{W}$-module with such locally finite actions is isomorphic to an induced module $\mathrm{Ind}(V)$ from a simple $\mathcal{W}_d$-module satisfying the conditions in Theorem 5.1.