Invariant theory and the ${𝒲}_{1+\infty }$ algebra with negative integral central charge

Linshaw, Andrew. "Invariant theory and the $\mathcal {W}_{1+\infty }$ algebra with negative integral central charge." Journal of the European Mathematical Society 013.6 (2011): 1737-1768. <http://eudml.org/doc/277667>.

@article{Linshaw2011,
abstract = {The vertex algebra $\mathcal \{W\}_\{1+\infty ,c\}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that $\mathcal \{W\}_\{1+\infty ,-n\}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of $\mathcal \{W\}_\{1+\infty ,-n\}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of $\text\{GL\}_n$, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of $\mathcal \{W\}_\{1+\infty ,-n\}$ are parametrized by a closed subvariety of $\mathbb \{C\}^\{n^2+2n\}$.},
author = {Linshaw, Andrew},
journal = {Journal of the European Mathematical Society},
keywords = {invariant theory; vertex algebra; $\mathcal \{W\}_\{1+\infty \}$ algebra; orbifold construction; strong finite generation; invariant theory; vertex algebra; algebra; orbifold construction; strong finite generation},
language = {eng},
number = {6},
pages = {1737-1768},
publisher = {European Mathematical Society Publishing House},
title = {Invariant theory and the $\mathcal \{W\}_\{1+\infty \}$ algebra with negative integral central charge},
url = {http://eudml.org/doc/277667},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Linshaw, Andrew
TI - Invariant theory and the $\mathcal {W}_{1+\infty }$ algebra with negative integral central charge
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 6
SP - 1737
EP - 1768
AB - The vertex algebra $\mathcal {W}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that $\mathcal {W}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of $\mathcal {W}_{1+\infty ,-n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation of $\text{GL}_n$, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of $\mathcal {W}_{1+\infty ,-n}$ are parametrized by a closed subvariety of $\mathbb {C}^{n^2+2n}$.
LA - eng
KW - invariant theory; vertex algebra; $\mathcal {W}_{1+\infty }$ algebra; orbifold construction; strong finite generation; invariant theory; vertex algebra; algebra; orbifold construction; strong finite generation
UR - http://eudml.org/doc/277667
ER -