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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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A Geometric Model for Syzygies Over 2-Calabi–Yau Tilted Algebras II
Abstract In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra $$B$$, we construct a polygon $$\mathcal {S}$$ with a checkerboard pattern in its interior, which defines a category $$\text {Diag}(\mathcal {S})$$. The indecomposable objects of $$\text {Diag}(\mathcal {S})$$ are the 2-diagonals in $$\mathcal {S}$$, and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category $$\text {Diag}(\mathcal {S})$$ is equivalent to the stable syzygy category of the algebra $$B$$. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type $$\mathbb {A}$$. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon $$\mathcal {S}$$ is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver.
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- PAR ID:
- 10509953
- Publisher / Repository:
- IMRN
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnad078
