Highest-weight vectors and modules for a Lie superalgebra

Fix a triangular decomposition datum $({\mathfrak{n}}^{-},\mathfrak{h},{\mathfrak{n}}^{+})$ as in Triangular decomposition of a Lie superalgebra and a left $\mathfrak{g}$-supermodule $V$. Here ${\mathfrak{h}}^{\ast }=\mathcal{L}(\mathfrak{h},\mathbb{C})$ is the algebraic dual from Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$. A highest-weight vector of weight $\lambda \in {\mathfrak{h}}^{\ast }$ is a nonzero homogeneous $v\in V$ such that $h\cdot v=\lambda (h)v(h\in \mathfrak{h}),n\cdot v=0(n\in {\mathfrak{n}}^{+})\mathrm{.}$ The highest-weight module generated by $v$ is the smallest graded $\mathfrak{g}$-invariant subspace containing $v$, equivalently the span of $v$ and all vectors $x_1\cdots x_r\cdot v$ with $r\ge 1$ and homogeneous $x_i\in \mathfrak{g}$. If this span is all of $V$, then $V$ is a highest-weight module of weight $\lambda$.

This definition makes no finite-dimensionality, integrability, semisimplicity, root-system, or classification assertion. In particular, the ordinary highest-weight classification for finite-dimensional complex semisimple Lie algebras does not apply to a general Lie superalgebra.