The matrix-extended $W_{1+\infty}$ algebra
Lorenz Eberhardt, Tom\'a\v{s} Proch\'azka

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 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.
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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic structures and combinatorial models · Advanced Topics in Algebra
