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

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 6, page 1737-1768
- ISSN: 1435-9855

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topLinshaw, 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 -

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