The matrix-extended $W_{1+\infty}$ algebra

TL;DR

This paper constructs a quadratic basis for the matrix-extended $ ext{W}_{1+ ext{infinity}}$ algebra, proposes a formula for its operator product expansions, and explores its truncations and gluing structure.

Contribution

It introduces a quadratic basis via a generalized Miura transformation and conjectures a closed-form for the algebra's operator product expansions.

Findings

Explicit low-level calculations confirm the truncation structure.

Truncations follow simple gluing rules despite algebra complexity.

The approach generalizes previous constructions of $ ext{W}_{1+ ext{infinity}}$ algebra.

Abstract

We construct a quadratic basis of generators of matrix-extended $\mathcal{W}_{1+\infty}$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.