<p>In this paper, we give a unified construction of vertex algebras from infinite-dimensional Lie algebras, including the affine Kac–Moody algebras, Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds and deformations. More specifically, we introduce a notion of what we call quasi vertex Lie algebra to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\frak{g}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> </math></EquationSource> </InlineEquation>, we construct a corresponding vertex Lie algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\frak{g}{0}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mrow> <mn mathvariant="fraktur">0</mn> </mrow> </math></EquationSource> </InlineEquation>, and establish a canonical isomorphism between the category of restricted <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frak{g}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> </math></EquationSource> </InlineEquation>-modules and that of equivariant <i>φ</i>-coordinated quasi <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V_{\frak{g}{0}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>V</mi> <mrow> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mrow> <mn mathvariant="fraktur">0</mn> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation>-modules, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$V_{\frak{g}{0}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>V</mi> <mrow> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mrow> <mn mathvariant="fraktur">0</mn> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> is the universal enveloping vertex algebra of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\frak{g}{0}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">g</mi> </mrow> <mrow> <mn mathvariant="fraktur">0</mn> </mrow> </math></EquationSource> </InlineEquation>. This unifies all the previous constructions of vertex algebras from infinite-dimensional Lie algebras.</p>

A unified construction of vertex algebras from infinite-dimensional Lie algebras

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发表日期：

2025年6月29日

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Fulin Chen,
Xiaoling Liao,
Shaobin Tan,
Qing Wang

摘要

In this paper, we give a unified construction of vertex algebras from infinite-dimensional Lie algebras, including the affine Kac–Moody algebras, Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds and deformations. More specifically, we introduce a notion of what we call quasi vertex Lie algebra to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie algebra $\frak{g}$ $g$ , we construct a corresponding vertex Lie algebra $\frak{g}{0}$ $g 0$ , and establish a canonical isomorphism between the category of restricted $\frak{g}$ $g$ -modules and that of equivariant φ-coordinated quasi $V_{\frak{g}{0}}$ $V_{g 0}$ -modules, where $V_{\frak{g}{0}}$ $V_{g 0}$ is the universal enveloping vertex algebra of $\frak{g}{0}$ $g 0$ . This unifies all the previous constructions of vertex algebras from infinite-dimensional Lie algebras.