Archives

Odd Singular Vector Formula for General Linear Lie Superalgebras
by Jie Liu   Li Luo   Weiqiang Wang

Vol. 14 No. 4 (2019) P.389~P.402
 DOI: https://doi.org/10.21915/BIMAS.2019401 10.21915/BIMAS.2019401

ABSTRACT

We establish a closed formula for a singular vector of weight $\lambda-\beta$ in the Verma module of highest weight $\lambda$ for Lie superalgebra ${\mathfrak {gl}}(m|n)$ when $\lambda$ is atypical with respect to an odd positive root $\beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $\lambda$ and $\lambda-\beta$ are dominant integral.

KEYWORDS
Lie superalgebra, singular vector

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 17B10

MILESTONES