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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $ 1$ by an automorphism of order $ 2$ and Whittaker vectors

Author: Kenichiro Tanabe
Journal: Proc. Amer. Math. Soc. 145 (2017), 4127-4140
MSC (2010): Primary 17B69; Secondary 17B68
Published electronically: July 7, 2017
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Abstract: Let $ M(1)$ be the vertex operator algebra with the Virasoro element $ \omega $ associated to the Heisenberg algebra of rank $ 1$ and let $ M(1)^{+}$ be the subalgebra of $ M(1)$ consisting of the fixed points of an automorphism of $ M(1)$ of order $ 2$. We classify the simple weak $ M(1)^{+}$-modules with a non-zero element $ w$ such that for some integer $ s\geq 2$, $ \omega _iw\in \mathbb{C}w$ ( $ i=\lfloor s/2\rfloor +1,\lfloor s/2\rfloor +2,\ldots ,s-1$), $ \omega _{s}w\in \mathbb{C}^{\times }w$, and $ \omega _iw=0$ for all $ i>s$. The result says that any such simple weak $ M(1)^{+}$-module is isomorphic to some simple weak $ M(1)$-module or to some $ \theta $-twisted simple weak $ M(1)$-module.

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Kenichiro Tanabe
Affiliation: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

Keywords: Vertex operator algebras, weak modules, Whittaker vectors
Received by editor(s): September 15, 2016
Published electronically: July 7, 2017
Additional Notes: The author’s research was partially supported by Grant-in-Aid (No. 15K04770) for Scientific Research, JSPS
Communicated by: Kailash Misra
Article copyright: © Copyright 2017 American Mathematical Society