Howe Pairs in the Theory of Vertex Algebras

Abstract

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

A few typos corrected, final version

References in corpus (2)

• Freely generated vertex algebras and non-linear Lie conformal algebras
• Chiral Equivariant Cohomology I

Cited by in corpus (19)

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• The structure of the Kac-Wang-Yan algebra
• Invariant subalgebras of affine vertex algebras
• Invariant chiral differential operators and the W_3 algebra
• A Hilbert theorem for vertex algebras
• Arc spaces and the vertex algebra commutant problem
• A commutant realization of W^(2)_n at critical level
• The ${\mathbb{Z}}_2$-orbifold of the ${\mathcal{W}}_3$-algebra
• The global sections of the chiral de Rham complex on a Kummer surface
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• A commutant realization of Odake's algebra
• The varieties of semi-conformal vectors of affine vertex operator algebras
• Twisted Heisenberg-Virasoro vertex operator algebra
• Chiral equivariant cohomology of a point: a first look
• Jet schemes, invariant chiral differential operators, and Howe duality
• The Z_2 Orbifold of the Universal Affine Vertex Algebra