A commutant realization of Odake's algebra
Quantum Algebra
2021-05-21 v3 Representation Theory
Abstract
The bc\beta\gamma-system W of rank 3 has an action of the affine vertex algebra V_0(sl_2), and the commutant vertex algebra C =Com(V_0(sl_2), W) contains copies of V_{-3/2}(sl_2) and Odake's algebra O. Odake's algebra is an extension of the N=2 superconformal algebra with c=9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V_{-3/2}(sl_2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of sl_2.
Keywords
Cite
@article{arxiv.1209.6132,
title = {A commutant realization of Odake's algebra},
author = {Thomas Creutzig and Andrew R. Linshaw},
journal= {arXiv preprint arXiv:1209.6132},
year = {2021}
}
Comments
Minor corrections, discussion of Odake's algebra in Section 2 expanded, final version