In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$V^k(sl_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>V</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq7.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$k=-n+\frac{n-1}{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mo>-</mo> <mi>n</mi> <mo>+</mo> <mfrac> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>q</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is generated by two singular vectors of conformal weight 3q if <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and by one singular vector of conformal weight 2q if <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\geqslant 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. We next determine the associated varieties of the simple vertex operator algebras <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$L_k(sl_3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all the non-admissible levels <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq11.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$k=-3+\frac{2}{2m+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mo>-</mo> <mn>3</mn> <mo>+</mo> <mfrac> <mn>2</mn> <mrow> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq12.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The varieties of the associated simple affine W-algebras <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$W_k(sl_3,f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for nilpotent elements f of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5291_Article_IEq14.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$sl_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <msub> <mi>l</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, are also determined.

Associated Varieties of Simple Affine VOAs $L_k(sl_3)$ and W-algebras $W_k(sl_3,f)$

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发表日期：

2025年4月4日

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Cuipo Jiang,
Jingtian Song

摘要

In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra $V^k(sl_n)$ $V^{k} (s l_{n})$ for $k=-n+\frac{n-1}{q}$ $k = - n + \frac{n - 1}{q}$ is generated by two singular vectors of conformal weight 3q if $n=3$ $n = 3$ , and by one singular vector of conformal weight 2q if $n\geqslant 4$ $n ⩾ 4$ . We next determine the associated varieties of the simple vertex operator algebras $L_k(sl_3)$ $L_{k} (s l_{3})$ for all the non-admissible levels $k=-3+\frac{2}{2m+1}$ $k = - 3 + \frac{2}{2 m + 1}$ , $m\geqslant 0$ $m ⩾ 0$ . The varieties of the associated simple affine W-algebras $W_k(sl_3,f)$ $W_{k} (s l_{3}, f)$ , for nilpotent elements f of $sl_3$ $s l_{3}$ , are also determined.

Associated Varieties of Simple Affine VOAs \(L_k(sl_3)\) and W-algebras \(W_k(sl_3,f)\)

摘要