Lie supermodules and representations

Let $\mathfrak{g}$ be a complex Lie superalgebra. A left $\mathfrak{g}$-supermodule is a complex vector space with grading $V=V_{\bar{0}}\oplus V_{\bar{1}}$ and a complex-bilinear action $\mathfrak{g}\times V\to V$, denoted $(x,v)\mapsto x\cdot v$, such that ${\mathfrak{g}}_{\bar{i}}\cdot V_{\bar{j}}\subseteq V_{\overline{i+j}}$ and, for homogeneous $x,y\in \mathfrak{g}$, $[x,y]\cdot v=x\cdot (y\cdot v)-(-1{)}^{\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }y\mathrm{\mid }}y\cdot (x\cdot v)(v\in V)\mathrm{.}$ A parity-preserving homomorphism of supermodules is a complex-linear map $T:V\to W$ preserving the two graded summands and satisfying $T(x\cdot v)=x\cdot T(v)$. A representation is the same action data written as an even linear map $\rho :\mathfrak{g}\to \mathrm{End}(V)$, where an endomorphism has parity $i$ when it maps $V_{\bar{j}}$ to $V_{\overline{i+j}}$; composition gives the supercommutator from Associative superalgebra over the complex numbers.

No finite-dimensionality, complete reducibility, topology, or unitarity is part of this definition.