In the present paper, we give a complete classification of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1954_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$U(\mathfrak {h})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">h</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-free modules of rank 1 over the planar Galilean conformal algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1954_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1954_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {C}L_0\oplus \mathbb {C}I_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msub> <mi>L</mi> <mn>0</mn> </msub> <mo>⊕</mo> <mi mathvariant="double-struck">C</mi> <msub> <mi>I</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is a subalgebra of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1954_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation> of degree-0 part. It is dependent on the classification result of W-algebra W(2, 2) by Chen and Guo.

Non-Weight Representations Over the Planar Galilean Conformal Algebra

期刊论文
发表日期：

2025年8月21日

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Yongliang Wang,
Haibo Chen

摘要

In the present paper, we give a complete classification of $U(\mathfrak {h})$ $U (h)$ -free modules of rank 1 over the planar Galilean conformal algebra $\mathcal {G}$ $G$ , where $\mathbb {C}L_0\oplus \mathbb {C}I_0$ $C L_{0} \oplus C I_{0}$ is a subalgebra of $\mathcal {G}$ $G$ of degree-0 part. It is dependent on the classification result of W-algebra W(2, 2) by Chen and Guo.