Twisted affine Lie superalgebras and integrability

Abstract

A twisted affine Lie superalgebra is either a twisted affine Lie algebra or of one of the types $X=A{(2m-1,2n-1)}^{(2)}$ ($m,n\ne 0,$ $(m,n)\ne (1,1)$), $A{(2m,2n)}^{(4)},A{(2m,2n-1)}^{(2)}$ or $D{(m+1,n)}^{(2)}$ ($m\ge 0,n>0$). It is known that irreducible integrable highest weight modules over a twisted affine Lie superalgebra of type X do not exist if $m\ne 0.$ In this paper, we show that nonzero level irreducible integrable finite weight modules over a twisted affine Lie superalgebra of type X do not exist if $m\ne 0.$

Keywords:

• Finite weight modules
• integrable modules
• twisted affine Lie superalgebras

2020 Mathematics Subject Classification:

• 17B10
• 17B65

Disclosure statement

There are no relevant financial or non-financial competing interests to report.

Notes

1 It is the subalgebra of all diagonal matrices.

2 It means that $S=-S$ and $(S+S)\cap R\subseteq S.$ We mention that in this case, $0\in S$ and $\sum _{\alpha \in S}{\mathfrak{L}}^{\alpha }$ is a subalgebra of $\mathfrak{L}$ containing $\mathfrak{h}.$

3 By an irreducible module, we mean a nonzero module whose trivial submodules are the only submodules.

Additional information

Funding

This research was in part supported by a grant from IPM (No. 1401170215) and is partially carried out in IPM-Isfahan Branch.