A Lie superalgebra $L$ is a $Z_2$-graded vector space such that $L=L_{\bar{0}}\oplus L_{\bar{1}}$ equipped with a bilinear superbracket structure $[~,~]: L \otimes L \to L$ satisfying the requirements of graded anti-symmetry and super Jacobi identity. As we have elements of parities zero and one, how a one-dimensional Lie superalgebra is descrobed?
if we assume that $\{x\} $ is a basis, then is one dimensional Lie superalgebra is a non trivial one? Because, for two vectors $u=\alpha x$ and $v=\beta x$ we get
$ [u,v]=2 \alpha \beta x_{\bar{1}}$ Does it mean $L$ as a one-dimensional Lie superalgebra is $K x_{\bar{1}}$?