<p>The algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\frak{q}_{\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a subalgebra of both the Ramond and Neveu-Schwarz sectors of the <i>N</i> = 2 superconformal algebra, with its even part being the Heisenberg-Virasoro algebra. Based on their previous work on simple Whittaker modules over the algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frak{q}_{\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, the authors construct and analyze two categories of non-weight modules for the algebra <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frak{q}_{\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. The first category consists of modules whose restriction to the universal enveloping algebra of the Cartan subalgebra (modulo the center) is free of rank 1. The second category involves modules whose restriction to the universal enveloping algebra of a subalgebra larger than Cartan subalgebra is free of rank 1. In particular, the authors determine the isomorphism classes and irreducibility of these modules and classify all simple cuspidal modules and then classify all simple Harish-Chandra modules for the algebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frak{q}_{\varepsilon}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.</p>

Non-weight Modules and Harish-Chandra Modules for a Subalgebra of N = 2 Superconformal Algebra

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发表日期：

2026年6月24日

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Pengfa Xu,
Naihuan Jing,
Honglian Zhang

摘要

The algebra $\frak{q}_{\varepsilon}$ $q_{ε}$ is a subalgebra of both the Ramond and Neveu-Schwarz sectors of the N = 2 superconformal algebra, with its even part being the Heisenberg-Virasoro algebra. Based on their previous work on simple Whittaker modules over the algebra $\frak{q}_{\varepsilon}$ $q_{ε}$ , the authors construct and analyze two categories of non-weight modules for the algebra $\frak{q}_{\varepsilon}$ $q_{ε}$ . The first category consists of modules whose restriction to the universal enveloping algebra of the Cartan subalgebra (modulo the center) is free of rank 1. The second category involves modules whose restriction to the universal enveloping algebra of a subalgebra larger than Cartan subalgebra is free of rank 1. In particular, the authors determine the isomorphism classes and irreducibility of these modules and classify all simple cuspidal modules and then classify all simple Harish-Chandra modules for the algebra $\frak{q}_{\varepsilon}$ $q_{ε}$ .