left ideals in Lie super algebras

Question

Let $\mathfrak g$ be a Lie superalgebra.

If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?

For me as sets they are linear span of $[a,x]$ and $[x,a]$ and hence they are same. But in book it is given they are different and the author has defined left and right ideal separately.

I am reading the book "Lie superalgebras and enveloping algebras by Ian M.Musson" Proposition 1.2.2.

Kindly help me with this.

Thank you.

Answer 1

The linear spans of $[a,x]$ and $[x,a]$, in a Lie superalgebra (i.e. a $\mathbb{Z}_2$-graded Lie algebra) are generally not the same (unlike the Lie algebras case):

Since $\mathfrak a$ is not a graded subspace of $\mathfrak g$, then in general its elements are not homogeneous. So for $a\in \mathfrak a$ we generally have: $a=a_0+a_1$, i.e. its decomposition into even and odd parts.
Thus, if we take for example an odd element $x\in \mathfrak g_1$: $$ [a,x]=[a_0,x]+[a_1,x]=-[x,a_0]+[x,a_1] $$ while $$ [x,a]=[x,a_0]+[x,a_1] $$ Consequently, the elements $[a,x]$ and $[x,a]$, are -in general- not collinear, as has already been indicated in the comments to the OP.