Kostant's pair of Lie type and conformal embeddings

TL;DR

This paper explores conformal embeddings of affine vertex algebras, offering new proofs and criteria for embeddings, especially at the critical level, with applications to symmetric space theory and central element equality.

Contribution

It provides a new proof of the Symmetric Space Theorem and introduces a criterion for conformal embeddings of equal rank subalgebras, including at the critical level.

Findings

New proof of the Symmetric Space Theorem

Criterion for conformal embeddings of equal rank subalgebras

Results on embeddings at the critical level and central elements

Abstract

We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings at the critical level. We prove a criterion for embeddings at the critical level which enables us to prove equality of certain central elements.