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Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.
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Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings
Abstract The elliptic algebras in the title are connected graded $$\mathbb {C}$$ -algebras, denoted $$Q_{n,k}(E,\tau )$$ , depending on a pair of relatively prime integers $$n>k\ge 1$$ , an elliptic curve E and a point $$\tau \in E$$ . This paper examines a canonical homomorphism from $$Q_{n,k}(E,\tau )$$ to the twisted homogeneous coordinate ring $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ on the characteristic variety $$X_{n/k}$$ for $$Q_{n,k}(E,\tau )$$ . When $$X_{n/k}$$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $$Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ is surjective, the relations for $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ are generated in degrees $$\le 3$$ and the noncommutative scheme $$\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $$X_{n/k}=E^g$$ and $$\tau =0$$ , the results about $$B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$$ show that the morphism $$\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.
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- Award ID(s):
- 2001128
- PAR ID:
- 10225264
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 9
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2020.60
